# Extreme values of the argument of the Riemann zeta function

Alexander Dobner  Department of Mathematics, UCLA
520 Portola Plaza
Los Angeles CA 90095<EMAIL_ADDRESS>
###### Abstract.

Let
$S(t)=\frac{1}{\pi}\operatorname{Im}\log\zeta\mathopen{}\mathclose{{}\left(\frac{1}{2}+it}\right)$.
Using Soundararajan’s resonance method we prove an unconditional lower bound
on the size of the tails of the distribution of $S(t)$. In particular we
reproduce the best unconditional $\Omega$ result for $S(t)$ which is due to
Tsang,

$S(t)=\Omega_{\pm}\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\frac{\log
t}{\log\log t}}\right)^{1/3}}\right),$

and get a bound on how often large values of $S(t)$ occur. We also give a
probabilistic argument for why this $\Omega$ result may be the best possible
given our current knowledge of the zeros of the zeta function.

##  1\. Introduction

Let

$S(t)\coloneqq\frac{1}{\pi}\operatorname{Im}\log\zeta\mathopen{}\mathclose{{}\left(\frac{1}{2}+it}\right)$

where $\zeta(s)$ denotes the Riemann zeta function. The logarithm of
$\zeta(s)$ is defined with branch cuts extending horizontally to the left of
each zero and pole and with the branch chosen such that $\log\zeta(2)$ is
real. When $\frac{1}{2}+it$ lies on a branch cut, $S(t)$ has a jump
discontinuity, and by convention we set $S(t)\coloneqq\lim_{\varepsilon\to
0}\mathopen{}\mathclose{{}\left(S(t+\varepsilon)+S(t-\varepsilon)}\right)/2$
in this case.

The behavior of $S(t)$ is closely connected to the locations of the zeros of
the zeta function as one may see from the Riemann-von Mangoldt formula,

(1)
$N(t)=\frac{t}{2\pi}\log\frac{t}{2\pi}-\frac{t}{2\pi}+\frac{7}{8}+S(t)+O\mathopen{}\mathclose{{}\left(t^{-1}}\right),\quad
t\geq 1.$

Here $N(t)$ denotes the number of zeta zeros $\rho=\beta+i\gamma$ for which
$0<\gamma<t$ (and any zeros with imaginary part equal to $t$ are counted with
weight $1/2$). The classical bound $S(t)=O(\log t)$ which is due to Backlund
shows that the macroscopic behavior of $N(t)$ is dictated by the first two
terms in the Riemann-von Mangoldt formula, but at smaller scales the
irregularities in the vertical distribution of zeta zeros are governed by the
oscillations of the $S(t)$ term.

In this paper we shall study the distribution of $S(t)$ as $t$ varies between
$T$ and $2T$. In particular our focus will be on understanding the tails of
this distribution. An important fact to keep in mind about $S(t)$ is the
Selberg central limit theorem which states that for any fixed $\Delta>0$,

(2)
$\frac{1}{T}\operatorname{meas}\mathopen{}\mathclose{{}\left\\{t\in[T,2T]\colon\frac{\pi
S(t)}{\sqrt{\frac{1}{2}\log\log
T}}\geq\Delta}\right\\}=\frac{1}{\sqrt{2\pi}}\int_{\Delta}^{\infty}e^{-u^{2}/2}\,du+o(1)$

as $T$ tends to infinity. This theorem does not directly say anything about
very large values of $S(t)$, but it does provide us with the heuristic that
$S(t)$ behaves like a Gaussian with mean zero and variance
$\frac{1}{2\pi^{2}}\log\log T$. By extending this heuristic to a large
deviations regime, one might make the prediction that the measure of the set
$\\{t\in[T,2T]\colon S(t)\geq V\\}$ is uniformly bounded below by
$T\exp(-cV^{2}/\log\log T)$ for some positive absolute constant $c$ even if
$V$ chosen to be much larger than $\sqrt{\log\log T}$. A lower bound of this
shape was proved by Soundararajan [7] for the analogous problem of
$\operatorname{Re}\log\zeta(\frac{1}{2}+it)$ rather than $S(t)$ for the range
$\log\log T\ll V\ll(\log T)^{\frac{1}{2}-\varepsilon}$. One might expect that
a straightforward adaptation of Soundararajan’s method would yield the same
result for $S(t)$, but unfortunately this turns out not to be the case.
Indeed, the possibility of zeros off the critical line leads to complications
which do not appear for $\operatorname{Re}\log\zeta(\frac{1}{2}+it)$. We are
able to prove the following result instead.

######  Theorem 1.

Let $0<a<1/3$. There exist constants $\kappa,c>0$ depending on $a$ such that
for all $T$ sufficiently large and all $V$ in the range $(\log T)^{a}\leq
V\leq\kappa\mathopen{}\mathclose{{}\left(\frac{\log T}{\log\log
T}}\right)^{\frac{1}{3}}$ we have

$\operatorname{meas}\,\mathopen{}\mathclose{{}\left\\{t\in[T,2T]\colon
S(t)\geq V}\right\\}\geq T\exp\mathopen{}\mathclose{{}\left(-cV^{2}}\right).$

The same is true if one replaces $S(t)$ with $-S(t)$.

The proof of this theorem combines three components each of which have been
used for studying large values of $S(t)$ in the past. The first is a
convolution formula of Selberg [6] which expresses a smoothed version of
$S(t)$ in terms of the primes and the zeros of the zeta function. The second
is an upper bound on the contribution of zeta zeros off the critical line
which is also due Selberg. The third is an application of Soundararajan’s
“resonance” method, which is a flexible method for proving $\Omega$
results.111An $\Omega$ result is any statement of the form $f(x)=\Omega(g(x))$
meaning that $\limsup_{x\to\infty}\frac{|f(x)|}{g(x)}>0$. The notation
$f(x)=\Omega_{\pm}(g(x))$ indicates that
$\limsup_{x\to\infty}\frac{f(x)}{g(x)}>0$ and
$\liminf_{x\to\infty}\frac{f(x)}{g(x)}<0$.

To compare Theorem 1 with what is already known, we note that the best
$\Omega$ results for $S(t)$ are

$S(t)=\begin{cases}\Omega_{\pm}\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\frac{\log
t}{\log\log t}}\right)^{\frac{1}{3}}}\right)\\\
\Omega_{\pm}\mathopen{}\mathclose{{}\left(\sqrt{\frac{\log t\log\log\log
t}{\log\log t}}}\right)\text{ on RH.}\end{cases}$

The unconditional result is due to Tsang [9] which is itself an improvement in
the power of the $\log\log t$ factor over a result of Selberg [6]. In the
conditional result, the order of magnitude comes from a recent breakthrough of
Bondarenko and Seip [2] which improves a longstanding theorem of Montgomery
[4] by a $\sqrt{\log\log\log t}$ factor. The fact that the Bondarenko and Seip
method yields large values of either sign was noted by Chirre and Mahatab [3].
Bondarenko and Seip’s proof uses a variant of the resonance method which was
initially developed by Aistleitner [1] to study $\log\zeta(\sigma+it)$ for
$\sigma>1/2$.

We now see that Theorem 1 implies a quantitative strengthening of Tsang’s
$\Omega$ result by giving a lower bound on how often large values of $S(t)$
occur. By comparison, Tsang’s proof (which does not use the resonance method)
does not give such a bound. To our knowledge, the only other measure result of
this kind in the literature is a theorem of Radziwiłł [5, Rmk. (1)] which
gives an asymptotic for the measure of $\\{t\in[T,2T]\colon S(t)\geq V\\}$ but
for the range $V\ll(\log\log T)^{\frac{1}{2}+\frac{1}{10}-\varepsilon}$
instead.

In the next section we will give a probabilistic argument for why the upper
limit $\kappa\mathopen{}\mathclose{{}\left(\frac{\log T}{\log\log
T}}\right)^{\frac{1}{3}}$ in our theorem appears to be optimal barring some
new idea for bounding the contribution of zeta zeros off the critical line.
The measure bound at this upper limit should also be optimal, but with more
work it may be possible to show that in the reduced range $(\log T)^{a}\leq
V\ll\mathopen{}\mathclose{{}\left(\log T}\right)^{\frac{1}{3}-\varepsilon}$
our lower bound can be improved to $T\exp(-cV^{2}/\log\log T)$ for some
positive constant $c$ depending on $\varepsilon$. The lower limit $(\log
T)^{a}$ on the range of $V$ in the theorem is a technical restriction coming
from the convolution formula that we use. It is possible to remove this
restriction by using a different kernel in the convolution formula such as the
kernel used by Bondarenko and Seip in [2, Section 5], but this comes at the
expense of only being able to produce large values of $|S(t)|$ rather than
large values of both $S(t)$ and $-S(t)$.

As a final remark we note that if one assumes the Riemann hypothesis, then it
is possible to prove an $S(t)$ analogue of Soundararajan’s measure result for
$\operatorname{Re}\log(\frac{1}{2}+it)$ but we will only consider the
unconditional case as it perhaps more interesting since it has not been
studied previously using the resonance method.

##  2\. Proof Outline

We now give a heuristic proof of Theorem 1. This will be helpful for
explaining the choice of some parameters in the full proof and for
illustrating where the upper limit on the range of $V$ in the theorem comes
from.

Like many of the proofs of $\Omega$ results for $S(t)$, we make use of a
convolution formula of Selberg. This formula states, roughly speaking, that if
we convolve $S(t)$ with an approximation to the identity of width
$\lambda^{-1}$ (where $\lambda$ is some parameter) then the resulting smoothed
version of $S(t)$ is equal to

(3) $\operatorname{Re}\sum_{\log
p\ll\lambda}\frac{1}{\sqrt{p}}p^{-it}+\text{zeros contribution}.$

The zeros contribution comes from zeta zeros lying strictly to the right of
the critical line and is dominated by the zeros $\rho=\beta+i\gamma$
satisfying $|t-\gamma|\ll\lambda^{-1}$. Such a formula is a manifestation of
the Fourier duality between the primes and the zeta zeros.

In order to find large values of $S(t)$, we will look for large values of the
smoothed version of $S(t)$ instead. If we assume the Riemann hypothesis then
there is no zeros contribution, in which case the problem is reduced to
finding large values of the real part of a certain Dirichlet polynomial. This
can be done by computing moments of the Dirichlet polynomial (see Tsang [9]),
using Dirchlet’s approximation theorem (see Montgomery [4]), or using the
resonance method (see Bondarenko and Seip [2]). If we do not assume the
Riemann hypothesis then we must take a different approach. Tsang’s method in
this case relies on a clever lemma which allows one to deduce a lower bound on
the maximum value of $S(t)$ by comparing a high moment of the Dirichlet
polynomial with a high moment of the zeros contribution. Our approach will be
more direct. The idea will be to show that the magnitude of the zeros
contribution is less than $V$ for all $t\in[T,2T]$ outside of some exceptional
set of small measure. We then show that the Dirichlet polynomial is greater
than $2V$ on a set whose measure is much greater than that of the exceptional
set. Combining these bounds we may then deduce a lower bound on how often the
sum of Dirichlet polynomial and the zeros contribution is greater than $V$.

We consider the Dirichlet polynomial first. We observe that the terms coming
from primes smaller than $V$ (say) are inconsequential because they can
contribute at most $\sum_{p\leq V}\frac{1}{\sqrt{p}}\ll\frac{\sqrt{V}}{\log
V}$ to the total. One implication of this is that the parameter $\lambda$ must
satisfy $\lambda\gg\log V$ otherwise the sum can never be greater than $2V$
for trivial reasons. In particular if $V$ is a power of $\log T$ then we must
take $\lambda\gg\log\log T$. We may now model the remaining terms in the sum
by a random sum since the phases $p^{-it}$ should behave roughly like i.i.d
random variables with mean zero. The variance of this random sum model is
approximately $\log\lambda-\log\log V$ by Mertens’ theorem, so if we assume it
has a roughly Gaussian distribution then we expect it to be greater than $2V$
with probability at least $\exp(-c_{0}V^{2}/(\log\lambda-\log\log V))$ for
some positive constant $c_{0}$. Hence we expect that our deterministic sum is
greater than $2V$ on a subset of $[T,2T]$ of measure

(4)
$T\exp\mathopen{}\mathclose{{}\left(-\frac{c_{0}V^{2}}{\log\lambda-\log\log
V}}\right).$

Getting a rigorous proof of a bound like is where we will use Soundararajan’s
resonance method. This will be explained in a later section.

We now consider the zeros contribution. As mentioned above, at any point
$t\in[T,2T]$ this contribution is dominated by the zeros $\rho=\beta+i\gamma$
for which $\beta>\frac{1}{2}$ and $|t-\gamma|\ll\lambda^{-1}$. Roughly
speaking, it will be the case that each individual zero contributes at most
$\lambda^{2}(\beta-\frac{1}{2})^{2}$, and we expect that there are
$\ll\lambda^{-1}\log T$ such zeros. Hence, letting
$\theta_{t}\coloneqq\max|\beta-\frac{1}{2}|$ where the maximum is taken over
some sufficiently large window of zeros at height $t$, we expect that total
contribution at $t$ will be

(5)
$\ll\mathopen{}\mathclose{{}\left(\lambda^{2}\theta_{t}^{2}}\right)\mathopen{}\mathclose{{}\left(\lambda^{-1}\log
T}\right)=\lambda\theta_{t}^{2}\log T.$

One difficulty in giving a rigorous proof of this bound is that the upper
bound $O(\lambda^{-1}\log T)$ on the number of zeta zeros in an interval of
length $\lambda^{-1}$ is not known to hold unconditionally if $\lambda\neq
O(1)$. This is problematic because we saw in the previous paragraph that for
our application we will need to choose $\lambda\gg\log\log T$. Fortunately, it
turns out that if $\lambda\asymp\log\log T$, then it is still possible to
prove a bound like (5) unconditionally. This was shown to be the case by
Selberg via an ingenious method, and he used this bound to prove his $\Omega$
theorem for $S(t)$ (which was later improved upon by Tsang). We will see
shortly that this choice of $\lambda$ turns out to be optimal if one wants $V$
to be as large as possible.

Continuing with the proof sketch, we note that by (5), the zeros contribution
is less than $V$ at any point $t\in[T,2T]$ where

$\theta_{t}\ll\mathopen{}\mathclose{{}\left(\frac{V}{\lambda\log
T}}\right)^{\frac{1}{2}}.$

By applying a zero density estimate, it is then possible to deduce that the
measure of the set of points where this is not the case is at most

(6)
$T\exp\mathopen{}\mathclose{{}\left(-c_{1}\mathopen{}\mathclose{{}\left(\frac{V\log
T}{\lambda}}\right)^{\frac{1}{2}}}\right)$

for some positive constant $c_{1}$.

A routine calculation now shows that if we want the lower bound (4) to be
larger than the upper bound (6) for as large a $V$ as possible then it is
optimal to pick the parameter $\lambda$ so that the variance in the random
model sum is $\log\lambda-\log\log V\asymp 1$, in which case we can take $V$
up to

$\ll\mathopen{}\mathclose{{}\left(\frac{\log
T}{\lambda}}\right)^{\frac{1}{3}}\asymp\mathopen{}\mathclose{{}\left(\frac{\log
T}{\log\log T}}\right)^{\frac{1}{3}}.$

As long as the implicit constant in the inequality above is small enough, the
quantity (4) will always be (say) twice as large as the quantity (6), and we
will be able to deduce the measure bound in the theorem.

##  3\. Convolution formula

We now begin the full proof Theorem 1. For the rest of the paper we let
$\lambda\coloneqq\log\log T$ and we will assume $t\in[T,2T]$ where $T$ is
large. We also fix the parameter $0<a<1/3$ from the statement of the theorem
and we assume that $(\log T)^{a}\leq
V\leq\kappa\mathopen{}\mathclose{{}\left(\frac{\log T}{\log\log
T}}\right)^{1/3}$ where $\kappa$ is to be chosen later.

We start by quoting a lemma given in [9] which is a general form of the
convolution formula for $\log\zeta$ discovered by Selberg.

######  Lemma 1.

Suppose $1/2\leq\sigma<1$ and let $K(x+iy)$ be an analytic function in the
horizontal strip $\sigma-2\leq y\leq 0$ satisfying the growth condition

$V(t)\coloneqq\sup_{\sigma-2\leq y\leq
0}|K(x+iy)|=O\mathopen{}\mathclose{{}\left((|x|\log^{2}|x|)^{-1}}\right)$

as $|x|\to\infty$. For any $t\neq 0$, we have

(7)
$\int_{-\infty}^{\infty}\log\zeta\mathopen{}\mathclose{{}\left(\sigma+i(t+u)}\right)K(u)\,du=\\\
\sum_{n=2}^{\infty}\frac{\Lambda(n)}{\log n}\hat{K}(\log n)n^{-\sigma-
it}+2\pi\sum_{\begin{subarray}{c}\rho=\beta+i\gamma\\\
\beta>\sigma\end{subarray}}\int_{0}^{\beta-\sigma}K(\gamma-
t-i\alpha)\,d\alpha+O(V(t)).$

The second summation above runs over all zeta zeros lying strictly to the
right of the line $\operatorname{Re}(s)=\sigma$. The convention used here for
the Fourier transform is $\hat{K}(v)\coloneqq\int K(u)e^{-ivu}\,du$.

To prove Theorem 1, we will need to choose two kernels $K_{+}$ and $K_{-}$
depending on whether we want to produce large positive or large negative
values of $S(t)$ respectively. To define these kernels, we first recall the
following Fourier pair,

$\omega(u)\coloneqq\frac{1}{2\pi}\mathopen{}\mathclose{{}\left(\frac{\sin(u/2)}{u/2}}\right)^{2}\iff\hat{\omega}(v)\coloneqq\max(0,1-|v|).$

We then let

$K_{\pm}(z)\coloneqq a\lambda\omega\mathopen{}\mathclose{{}\left(a\lambda
z}\right)\mathopen{}\mathclose{{}\left(\pm
1-\frac{1}{2i}e^{i\frac{2}{3}\lambda z}+\frac{1}{2i}e^{-i\frac{2}{3}\lambda
z}}\right)=a\lambda\omega\mathopen{}\mathclose{{}\left(a\lambda
z}\right)\mathopen{}\mathclose{{}\left(\pm
1-\sin\mathopen{}\mathclose{{}\left(\frac{2}{3}\lambda z}\right)}\right).$

One may verify that $K_{\pm}$ has the following properties:

  1. (i)

$K_{+}(z)\geq 0$ and $K_{-}(z)\leq 0$,

  2. (ii)

$\int|K_{\pm}(z)|\,dz\leq 2\int\omega(u)\,du=2$,

  3. (iii)

$\operatorname{Im}K_{\pm}(x+iy)\ll\frac{\lambda^{2}|y|e^{\lambda|y|}}{1+(a\lambda
x)^{2}+(a\lambda y)^{2}}$,

  4. (iv)

$\hat{K}_{\pm}(v)=\pm\hat{\omega}\mathopen{}\mathclose{{}\left(\frac{v}{a\lambda}}\right)+\frac{i}{2}\hat{\omega}\mathopen{}\mathclose{{}\left(\frac{1}{a\lambda}\mathopen{}\mathclose{{}\left(v-\frac{2}{3}\lambda}\right)}\right)-\frac{i}{2}\hat{\omega}\mathopen{}\mathclose{{}\left(\frac{1}{a\lambda}\mathopen{}\mathclose{{}\left(v+\frac{2}{3}\lambda}\right)}\right)$.

Property (iii) follows from Selberg’s bound on the imaginary part of
$\omega(u+iv)$ in [6, pg. 52].

The reason for the presence of the constant $2/3$ in the definition of
$K_{\pm}$ is that the kernel must be chosen such that the Dirichlet series we
get after applying Lemma 1 includes primes at least as big as $(\log
T)^{\frac{2}{3}}$. As discussed in the heuristic proof in the previous
section, this is necessary for trivial reasons because otherwise the Dirichlet
polynomial would be $\ll\frac{(\log T)^{\frac{1}{3}}}{\log\log T}$ for every
$t$. It would also be admissible to replace $2/3$ with some larger constant,
but this would not lead to any improvement in our theorem.

Applying the convolution formula with $K_{\pm}$, we can now prove that a short
average of $S(t)$ can be expressed in terms of the real part of a Dirichlet
polynomial and some contribution from zeta zeros.

######  Lemma 2.

For any $t\in[T,2T]$,

$\int_{-\log T}^{\log T}\pm
S(t+u)K_{\pm}(u)\,du=D(t)+Z_{\pm}(t)+O\mathopen{}\mathclose{{}\left((\log
T)^{a/2}}\right)$

where

$\displaystyle D(t)$
$\displaystyle=\frac{1}{2\pi}\operatorname{Re}\sum_{n=2}^{\infty}\frac{\Lambda(n)}{\log
n}\hat{\omega}\mathopen{}\mathclose{{}\left(\frac{1}{a\lambda}\mathopen{}\mathclose{{}\left(\log
n-\frac{2}{3}\lambda}\right)}\right)n^{-\frac{1}{2}-it},$ $\displaystyle
Z_{\pm}(t)$
$\displaystyle=2\operatorname{Im}\sum_{\beta>\frac{1}{2}}\int_{0}^{\beta-\frac{1}{2}}K_{\pm}(\gamma-
t-i\alpha)\,d\alpha.$

###### Proof.

Applying Lemma 1 with the kernel $K_{\pm}$ and $\sigma=1/2$, the resulting
Dirichlet series is

$\sum_{n=2}^{\infty}\frac{\Lambda(n)}{\log
n}\mathopen{}\mathclose{{}\left(\pm\hat{\omega}\mathopen{}\mathclose{{}\left(\frac{1}{a\lambda}\log
n}\right)+\frac{i}{2}\hat{\omega}\mathopen{}\mathclose{{}\left(\frac{1}{a\lambda}\mathopen{}\mathclose{{}\left(\log
n-\frac{2}{3}\lambda}\right)}\right)-\frac{i}{2}\hat{\omega}\mathopen{}\mathclose{{}\left(\frac{1}{a\lambda}\mathopen{}\mathclose{{}\left(\log
n+\frac{2}{3}\lambda}\right)}\right)}\right)n^{\frac{1}{2}-it}$

by property (iv). Since $\operatorname{supp}\hat{\omega}=[-1,1]$ and $a<1/3$,
we note
$\hat{\omega}\mathopen{}\mathclose{{}\left(\frac{1}{a\lambda}\mathopen{}\mathclose{{}\left(\log
n+\frac{2}{3}\lambda}\right)}\right)=0$ for every $n$ in the sum. Also,

$\pm\sum_{n=2}^{\infty}\frac{\Lambda(n)}{\log
n}\hat{\omega}\mathopen{}\mathclose{{}\left(\frac{1}{a\lambda}\log
n}\right)n^{-\frac{1}{2}-it}\ll\sum_{n\leq(\log
T)^{a}}\frac{1}{\sqrt{n}}\ll(\log T)^{a/2}.$

Hence, we see that

$\int_{-\infty}^{\infty}\log\zeta\mathopen{}\mathclose{{}\left(\frac{1}{2}+i(t+u)}\right)K_{\pm}(u)\,du=\frac{i}{2}\sum_{n=2}^{\infty}\frac{\Lambda(n)}{\log
n}\hat{\omega}\mathopen{}\mathclose{{}\left(\frac{1}{a\lambda}\mathopen{}\mathclose{{}\left(\log
n-\frac{2}{3}\lambda}\right)}\right)n^{-\frac{1}{2}-it}\\\
+2\pi\sum_{\begin{subarray}{c}\rho=\beta+i\gamma\\\
\beta>\frac{1}{2}\end{subarray}}\int_{0}^{\beta-\frac{1}{2}}K_{\pm}(\gamma-
t-i\alpha)\,d\alpha+O\mathopen{}\mathclose{{}\left((\log T)^{a/2}}\right).$

Taking imaginary parts and dividing through by $\pi$ we get

(8)
$\int_{-\infty}^{\infty}S(t+u)K_{\pm}(u)\,du=D(t)+Z_{\pm}(t)+O\mathopen{}\mathclose{{}\left((\log
T)^{a/2}}\right).$

Using the classical bound $S(t)=O(\log(|t|+2))$ and a trivial bound on the
decay of the kernel, we can bound the tails of the integral by

$\int_{|u|>\log T}S(t+u)K_{\pm}(u)\,du\ll\int_{\log T}^{\infty}(\log T+\log
u)\lambda^{-1}u^{-2}\,du\ll 1.$

Hence, the tails may be absorbed into the error term. ∎

The error term in Lemma 2 will be the reason for the lower limit on the range
of $V$ in Theorem 1.

##  4\. Contribution of the zeros

We follow Selberg’s method in [6] to bound $Z_{\pm}(t)$. Keeping our notation
in line with the heuristic proof in section 2, we shall define

$\theta_{t}\coloneqq\max_{\begin{subarray}{c}\beta>\frac{1}{2}\\\
|\gamma-t|\leq\log^{2}T\end{subarray}}\mathopen{}\mathclose{{}\left(\beta-\frac{1}{2}}\right).$

Applying property (iii) from the previous section, we see that

$\displaystyle Z_{\pm}(t)$
$\displaystyle=2\operatorname{Im}\sum_{\beta>\frac{1}{2}}\int_{0}^{\beta-\frac{1}{2}}K_{\pm}(\gamma-
t-i\alpha)\,d\alpha$
$\displaystyle\ll\sum_{\beta>\frac{1}{2}}\int_{0}^{\beta-\frac{1}{2}}\frac{\lambda^{2}\alpha
e^{\lambda\alpha}}{\mathopen{}\mathclose{{}\left(\frac{1}{a}}\right)^{2}+(\lambda(\gamma-t))^{2}+(\lambda\alpha)^{2}}\,d\alpha.$

The integrand is increasing in $\alpha$ (one may deduce this from the fact
that $\frac{e^{y}}{c+y^{2}}$ is increasing for any $c\geq 1$) so bounding the
integrand by the endpoint gives

$Z_{\pm}(t)\ll\sum_{\beta>\frac{1}{2}}\frac{\lambda^{2}\mathopen{}\mathclose{{}\left(\beta-\frac{1}{2}}\right)^{2}e^{\lambda\mathopen{}\mathclose{{}\left(\beta-\frac{1}{2}}\right)}}{\mathopen{}\mathclose{{}\left(\frac{1}{a}}\right)^{2}+(\lambda(\gamma-t))^{2}+(\lambda\mathopen{}\mathclose{{}\left(\beta-\frac{1}{2}}\right))^{2}}.$

Now we split the sum up depending on whether $|\gamma-t|\leq\log^{2}T$ or
$|\gamma-t|>\log^{2}T$. In the first instance we get the bound

$\displaystyle\sum_{\begin{subarray}{c}\beta>\frac{1}{2}\\\
|\gamma-t|\leq\log^{2}T\end{subarray}}\frac{\lambda^{2}\mathopen{}\mathclose{{}\left(\beta-\frac{1}{2}}\right)^{2}e^{\lambda\mathopen{}\mathclose{{}\left(\beta-\frac{1}{2}}\right)}}{\mathopen{}\mathclose{{}\left(\frac{1}{a}}\right)^{2}+(\lambda(\gamma-t))^{2}+(\lambda\mathopen{}\mathclose{{}\left(\beta-\frac{1}{2}}\right))^{2}}$
$\displaystyle\leq\theta_{t}^{2}e^{\lambda\theta_{t}}\sum_{\begin{subarray}{c}\beta>\frac{1}{2}\\\
|\gamma-t|\leq\log^{2}T\end{subarray}}\frac{1}{\mathopen{}\mathclose{{}\left(\frac{3}{\lambda}}\right)^{2}+(\gamma-t)^{2}+\theta_{t}^{2}}$
$\displaystyle\ll\theta_{t}^{2}e^{\lambda\theta_{t}}\lambda\log T,$

where the first inequality comes from using monotonicity again, and the second
inequality is quoted from Selberg [6, pg. 53].

The sum over the remaining zeros can be bounded more trivially. Indeed,

$\displaystyle\sum_{\begin{subarray}{c}\beta>\frac{1}{2}\\\
|\gamma-t|>\log^{2}T\end{subarray}}\frac{\lambda^{2}\mathopen{}\mathclose{{}\left(\beta-\frac{1}{2}}\right)^{2}e^{\lambda\mathopen{}\mathclose{{}\left(\beta-\frac{1}{2}}\right)}}{\mathopen{}\mathclose{{}\left(\frac{1}{a}}\right)^{2}+(\lambda(\gamma-t))^{2}+(\lambda\mathopen{}\mathclose{{}\left(\beta-\frac{1}{2}}\right))^{2}}$
$\displaystyle\ll
e^{\frac{1}{2}\lambda}\sum_{|\gamma-t|>\log^{2}T}\frac{1}{(\gamma-t)^{2}}\ll
1,$

where the second inequality follows from the classical bound
$N(T+1)-N(T)=O(\log T)$ for all $T\geq 2$.

Hence, we may conclude that

(9) $Z_{\pm}(t)\ll\theta_{t}^{2}e^{\lambda\theta_{t}}\lambda\log T+1$

which is essentially the same bound as (5) in the heuristic proof because the
exponential factor $e^{\lambda\theta_{t}}$ will be $O(1)$ for the values of
$\theta_{t}$ that we will be considering.

We would now like to prove an upper bound on the measure of the exceptional
set $E_{V}\coloneqq\\{t\in[T,2T]\colon|Z_{\pm}(t)|\geq V\\}$. There will be no
distinction between the $Z_{+}$ and $Z_{-}$ case, so we will simply allow
$E_{V}$ refer to either case from now on. Let $N(\sigma,T)$ denote the number
of zeros $\rho=\beta+i\gamma$ of $\zeta(s)$ for which $\beta>\sigma$ and
$0\leq\gamma\leq T$. Any nontrivial bound on $N(\sigma,T)$ will give a
corresponding bound on the measure of $E_{V}$. For our purposes, the following
classical result will suffice.

######  Lemma 3 (Zero density estimate).

$N(\sigma,T)\ll T^{3/2-\sigma}\log^{5}T$

###### Proof.

See Titchmarsh [8, Theorem 9.19(A)]. ∎

For $V$ is in the range $(\log T)^{a}\leq V\ll(\log T)^{1/3}$ we note that in
order to deduce $|Z_{\pm}(t)|<V$ for a given $t\in[T,2T]$, it is sufficient by
(9) to know that $\theta_{t}\leq
c^{\prime}\mathopen{}\mathclose{{}\left(\frac{K}{\lambda\log T}}\right)^{1/2}$
for some small positive constant $c^{\prime}$ which depends only on the
implicit constant in (9). Hence, by applying Lemma 3 with
$\sigma=\frac{1}{2}+c^{\prime}\mathopen{}\mathclose{{}\left(\frac{K}{\lambda\log
T}}\right)^{1/2}$, we may deduce that

$\displaystyle\operatorname{meas}(E_{V})$ $\displaystyle\ll T(\log
T)^{2}\exp\mathopen{}\mathclose{{}\left(-c^{\prime}\mathopen{}\mathclose{{}\left(\frac{V\log
T}{\lambda}}\right)^{1/2}}\right)$ $\displaystyle\ll
T\exp\mathopen{}\mathclose{{}\left(-c^{\prime\prime}\mathopen{}\mathclose{{}\left(\frac{V\log
T}{\lambda}}\right)^{1/2}}\right)$

for some positive constant $c^{\prime\prime}$.

##  5\. Contribution of the Dirichlet polynomial

The next step in the proof will be to get a lower bound on the measure of
$F_{V}\coloneqq\\{t\in[T,2T]\colon D(t)\geq 4V\\}$ using Soundararajan’s
resonance method. To start we define a Dirichlet polynomial (the “resonator”)

$R(t)\coloneqq\sum_{n\leq N}f(n)n^{-it}$

where $f\colon\mathbb{N}\to\mathbb{R}$ and $N$ are to be chosen later, and let
$\Phi$ be a smooth function supported in $[1,2]$, such that $0\leq\Phi\leq 1$
and $\Phi(t)=1$ for $t\in(5/4,7/4)$. We then define

$\displaystyle M_{1}(R,T)$
$\displaystyle\coloneqq\int_{-\infty}^{\infty}|R(t)|^{2}\Phi\mathopen{}\mathclose{{}\left(\tfrac{t}{T}}\right)\,dt,$
$\displaystyle M_{2}(R,T)$
$\displaystyle\coloneqq\int_{-\infty}^{\infty}D(t)|R(t)|^{2}\Phi\mathopen{}\mathclose{{}\left(\tfrac{t}{T}}\right)\,dt.$

The basic idea behind the resonance method is that maximum value of $D(t)$ on
$[T,2T]$ is at least $M_{2}(R,T)/M_{1}(R,T)$, so one may hope to prove a
strong lower bound on the maximum of $D(t)$ by picking a resonator which
maximizes this ratio. In our application, we require a lower bound on the
measure of $F_{V}$ instead. To get such a bound, we may note that

$\displaystyle M_{2}(R,T)$ $\displaystyle\leq
4V\,M_{1}(R,T)+\int_{F_{V}}D(t)|R(t)|^{2}\Phi\mathopen{}\mathclose{{}\left(\tfrac{t}{T}}\right)\,dt$
$\displaystyle\leq 4V\,M_{1}(R,T)+\log
T\int_{F_{V}}|R(t)|^{2}\Phi\mathopen{}\mathclose{{}\left(\tfrac{t}{T}}\right)\,dt$
$\displaystyle\leq 4V\,M_{1}(R,T)+\log
T\,\big{(}\operatorname{meas}(F_{V})\big{)}^{\frac{1}{2}}\mathopen{}\mathclose{{}\left(\int_{-\infty}^{\infty}|R(t)|^{4}\Phi\mathopen{}\mathclose{{}\left(\tfrac{t}{T}}\right)\,dt}\right)^{\frac{1}{2}}$

where we have trivially bounded $D(t)$ to get the second inequality and
applied Cauchy-Schwarz to get the third inequality. If $M_{2}(R,T)\geq
8V\,M_{1}(R,T)$ say, then rearranging the bound above gives

(10) $\operatorname{meas}(F_{V})\gg\frac{M_{2}(R,T)^{2}}{(\log
T)^{2}}\mathopen{}\mathclose{{}\left(\int_{-\infty}^{\infty}|R(t)|^{4}\Phi\mathopen{}\mathclose{{}\left(\tfrac{t}{T}}\right)\,dt}\right)^{-1}.$

Of course, to actually make use of (10), we need to be able to estimate
$M_{1}(R,T)$, $M_{2}(R,T)$, and the fourth moment
$\int|R(t)|^{4}\Phi(\tfrac{t}{T})\,dt$. This is possible using standard
techniques for estimating the mean values of Dirichlet polynomials as long as
$N$ is not too large relative to $T$. Indeed, following the exact same method
as in Soundararajan [7, pg. 4-5], one may verify that if $N\leq
T^{1-\varepsilon}$ then

(11)
$M_{1}(R,T)=T\hat{\Phi}(0)\mathopen{}\mathclose{{}\left(1+O\mathopen{}\mathclose{{}\left(T^{-1}}\right)}\right)\sum_{n\leq
N}f(n)^{2}$ (12) $M_{2}(R,T)=T\hat{\Phi}(0)\frac{1}{2\pi}\sum_{mk=n\leq
N}\frac{f(m)f(n)\Lambda(k)}{(\log
k)\sqrt{k}}\hat{\omega}\mathopen{}\mathclose{{}\left(\frac{1}{a\lambda}(\log
k-b\lambda)}\right)\\\ +O\mathopen{}\mathclose{{}\left((\log
T)^{O(1)}\,\sum_{n\leq N}f(n)^{2}}\right).$

If $N\leq T^{\frac{1}{2}-\varepsilon}$ and $f$ is chosen such that $f(n)\leq
1$, then we also have (see [7, pg. 8]),

(13)
$\int|R(t)|^{4}\Phi(\tfrac{t}{T})\,dt=T\hat{\Phi}(0)\sum_{\begin{subarray}{c}a,b,c,d\leq
N\\\
ab=cd\end{subarray}}f(a)f(b)f(c)f(d)+O\mathopen{}\mathclose{{}\left(T^{-1}}\right).$

From now on we will assume that $N$ is a fixed small power of $T$ (for
instance $N=T^{\frac{1}{3}}$) so that that the estimates above will hold. With
this choice of $N$ we also have $\log N\asymp\log T$ which will be useful to
keep in mind moving forward.

As we have just seen, in order the prove the strongest possible lower bound on
the measure of $F_{V}$, our goal will be to choose $f$ so that
$M_{2}(R,T)/M_{1}(R,T)\geq 8V$ but the fourth moment of $R(t)$ is as small as
possible. For now we will define a somewhat generic $f$ in terms of three
parameters $P,Q$ and $A$ which are allowed to depend on $N$ and $V$ and which
tend to infinity as $N$ does. In particular, we let $f$ be a multiplicative
function supported on squarefree numbers whose prime factors lie in
$\mathcal{P}\coloneqq\mathopen{}\mathclose{{}\left\\{p\text{ prime}\colon
P\leq p\leq Q}\right\\}$, and we set

$f(p)\coloneqq\frac{A}{\sqrt{p}}$

for every $p\in\mathcal{P}$. This is essentially a generalized version of the
resonator used by Soundararajan in [7, pg. 7]. We will now analyze how select
$P,Q$ and $A$ depending on $V$ and $N$.

By (11), (12), and the nonnegativity of the summands in (12), we see that

(14)
$\frac{M_{2}(R,T)}{M_{1}(R,T)}\geq(1+o(1))\mathopen{}\mathclose{{}\left(\min_{p\in\mathcal{P}}\hat{\omega}\mathopen{}\mathclose{{}\left(\frac{1}{a\lambda}(\log
p-\frac{2}{3}\lambda)}\right)}\right)\sum_{\begin{subarray}{c}mp=n\leq N\\\
p\in\mathcal{P}\end{subarray}}\frac{f(m)f(n)}{\sqrt{p}}\bigg{/}\sum_{n\leq
N}f(n)^{2}.$

By choosing the parameters $P$ and $Q$ to satisfy $(\log T)^{2/3-a/2}\leq
P\leq Q\leq(\log T)^{2/3+a/2}$ we can ensure that the $\min_{p\in\mathcal{P}}$
factor in (14) is bounded below by $1/2$ in which case we then just have

(15)
$\frac{M_{2}(R,T)}{M_{1}(R,T)}\geq\frac{1}{4}\sum_{\begin{subarray}{c}mp=n\leq
N\\\
p\in\mathcal{P}\end{subarray}}\frac{f(m)f(n)}{\sqrt{p}}\bigg{/}\sum_{n\leq
N}f(n)^{2}.$

for all sufficiently large $T$.

###  5.1. Resonator lemmas

In order to continue our analysis, we will impose some restrictions on the
relative sizes of the parameters $P,Q,A$ and $N$. These restrictions will
appear to be quite arbitrary right now, but each of them will be used in
proving the subsequent two lemmas.

The restrictions are

  1. (a)

$10A^{2}\log Q\leq\log N$,

  2. (b)

$P\geq A^{2}$,

  3. (c)

$Q\geq 2P$,

  4. (d)

$\frac{A^{2}}{\log P}\to\infty$.

The lemmas we prove now will show that under the restrictions above the mass
of the function $f$ is concentrated on integers $n\leq N$ with many divisors.

######  Lemma 4.

Assuming restriction (a) holds, we have

$\frac{1}{\sum_{i\in\mathbb{N}}f(i)^{2}}\sum_{n>N}f(n)^{2}=o_{N\to\infty}(1).$

######  Lemma 5.

Let $\mathcal{N}$ denote the set of square-free numbers whose prime factors
all lie in $\mathcal{P}$ and with at least $\frac{1}{10}A^{2}\log\frac{\log
Q}{\log P}$ factors in $\mathcal{P}$. Assuming restrictions (b), (c), and (d)
hold, we have

(16)
$\frac{1}{\sum_{i\in\mathbb{N}}f(i)^{2}}\sum_{n\not\in\mathcal{N}}f(n)^{2}=o_{N\to\infty}(1).$

Lemma 5 also holds if $\frac{1}{10}$ is replaced by any constant less than 1
but the statement above suffices for our purposes.

###### Proof of Lemma 4.

To remove the cutoff at $N$ in the sum we use Rankin’s trick, which is the
observation that $1_{n>N}\leq(n/N)^{\alpha}$ for any $\alpha>0$. Applying this
and the fact that $f$ is multiplicative we may bound the ratio in the lemma
statement by

$\displaystyle\frac{1}{\sum_{i\in\mathbb{N}}f(i)^{2}}N^{-\alpha}\sum_{n\in\mathbb{N}}n^{\alpha}f(n)^{2}$
$\displaystyle=N^{-\alpha}\,\prod_{p\in\mathcal{P}}\frac{1+p^{\alpha}f(p)^{2}}{1+f(p)^{2}}$
$\displaystyle\leq\exp\mathopen{}\mathclose{{}\left(-\alpha\log
N+\sum_{p\in\mathcal{P}}(p^{\alpha}-1)f(p)^{2}}\right).$

Choosing $\alpha=\frac{1}{2\log Q}$, we see that $p^{\alpha}-1\leq\frac{\log
p}{\log Q}$, so the bound becomes

$\displaystyle\leq\exp\mathopen{}\mathclose{{}\left(-\frac{\log N}{2\log
Q}+\frac{A^{2}}{\log Q}\sum_{p\in\mathcal{P}}\frac{\log p}{p}}\right)$
$\displaystyle\leq\exp\mathopen{}\mathclose{{}\left(-\frac{\log N}{2\log
Q}+2A^{2}}\right)$
$\displaystyle\leq\exp\mathopen{}\mathclose{{}\left(-\frac{\log N}{4\log
Q}}\right)$

by the prime number theorem and by assumption (a). This bound tends to zero
since $\log N/\log Q\geq 10A^{2}\to\infty$ by assumption (a) and by the
assumption that $A$ tends to infinity with $N$. ∎

###### Proof of Lemma 5.

Let $D\coloneqq\frac{1}{10}A^{2}\log\frac{\log Q}{\log P}$. By the definition
of $\mathcal{N}$ and by a multiplicative analogue of Rankin’s trick we see
that for any $0<b<1$,

$\sum_{n\not\in\mathcal{N}}f(n)^{2}\leq
b^{-D}\prod_{p\in\mathcal{P}}(1+b\,f(p)^{2}).$

Hence, the ratio on the left-hand side of (16) is

(17) $\leq b^{-D}\prod_{p\in\mathcal{P}}\frac{1+b\,f(p)^{2}}{1+f(p)^{2}}.$

Since $f(p)^{2}\leq 1$ (by assumption (b)) and since $0<b<1$, one may verify
the elementary inequality

$\frac{1+b\,f(p)^{2}}{1+f(p)^{2}}\leq 1+\frac{1}{2}(b-1)f(p)^{2}.$

Applying this we see that (17) is bounded above by

$\exp\mathopen{}\mathclose{{}\left(-D\log
b+\frac{1}{2}(b-1)\sum_{p\in\mathcal{P}}f(p)^{2}}\right).$

By the prime number theorem and assumption (c), we see that

$\sum_{p\in\mathcal{P}}f(p)^{2}=(1+o(1))A^{2}\log\frac{\log Q}{\log P},$

which tends to infinity by assumptions (c) and (d) and which is equal to
$10D(1+o(1))$ by the definition of $D$. Hence, we conclude that (17) is

$\leq\exp\bigg{(}D\big{(}-\log b+5(b-1)(1+o(1))\big{)}\bigg{)}.$

Taking $b=\frac{1}{2}$ (say) we see that $-\log b+5(b-1)<0$. Therefore, since
$D$ tends to infinity, the bound above goes to zero. ∎

###  5.2. Large values of $D(t)$

Applying the two lemmas above we see that if all our restriction on the
parameters $P,Q,$ and $A$ hold, then

$\displaystyle\sum_{\begin{subarray}{c}mp=n\leq N\\\
p\in\mathcal{P}\end{subarray}}\frac{f(m)f(n)}{\sqrt{p}}\bigg{/}\sum_{n\leq
N}f(n)^{2}$
$\displaystyle\geq\frac{1}{\sum_{i\in\mathbb{N}}f(i)^{2}}\sum_{\begin{subarray}{c}mp=n\leq
N\\\ p\in\mathcal{P}\\\
n\in\mathcal{N}\end{subarray}}\frac{f(m)f(n)}{\sqrt{p}}$
$\displaystyle=\frac{1}{\sum_{i\in\mathbb{N}}f(i)^{2}}\sum_{\begin{subarray}{c}n\leq
N\\\ n\in\mathcal{N}\end{subarray}}f(n)^{2}\sum_{\begin{subarray}{c}p|n\\\
p\in\mathcal{P}\end{subarray}}\frac{1}{f(p)\sqrt{p}}$
$\displaystyle\geq\frac{1}{\sum_{i\in\mathbb{N}}f(i)^{2}}\sum_{\begin{subarray}{c}n\leq
N\\\ n\in\mathcal{N}\end{subarray}}f(n)^{2}\frac{1}{20A}A^{2}\log\frac{\log
Q}{\log P}$ $\displaystyle\geq\frac{1}{25A}\log\frac{\log Q}{\log P}$

for all large enough $N$. The second to last inequality follows from the
definition of $\mathcal{N}$, and the last inequality follows from the two
lemmas in the previous section. Hence, by (15) we have

(18) $\frac{M_{2}(R,T)}{M_{1}(R,T)}\geq\frac{1}{100A}\log\frac{\log Q}{\log
P}$

for large enough $N$.

We also see that

$\displaystyle\sum_{\begin{subarray}{c}a,b,c,d\leq N\\\
ab=cd\end{subarray}}f(a)f(b)f(c)f(d)$
$\displaystyle\ll\prod_{p\in\mathcal{P}}(1+4f(p)^{2}+f(p)^{4})$
$\displaystyle\ll\exp\mathopen{}\mathclose{{}\left(5\sum_{p\in\mathcal{P}}f(p)^{2}}\right)$
$\displaystyle\ll\exp\mathopen{}\mathclose{{}\left(6A^{2}\log\frac{\log
Q}{\log P}}\right).$

Hence, by (13) we get

(19) $\int|R(t)|^{4}\Phi(\tfrac{t}{T})\,dt\ll
T\exp\mathopen{}\mathclose{{}\left(6A^{2}\log\frac{\log Q}{\log P}}\right)$

for large enough $N$.

It is now routine to check that if $P=(\log T)^{2/3}$, $Q=(\log T)^{2/3+a/2}$,
and $A=bV$ for some suitably large constant $b$ depending on $a$, then all the
constraints on these parameters that we listed previously are satisfied, and
by (18) we have $M_{2}(R,T)/M_{1}(R,T)\geq 8V$. Also, by (19) we get
$\int|R(t)|^{4}\Phi(\tfrac{t}{T})\,dt\ll T\exp(d^{\prime}V^{2})$ for some
positive constant $d^{\prime}$ depending only on $a$. Hence, we may apply (10)
and the fact that $M_{2}(R,T)\gg T$ to conclude that

$\displaystyle\operatorname{meas}(F_{V})\gg\frac{T}{(\log
T)^{2}}\exp(-d^{\prime}V^{2})$ $\displaystyle\gg
T\exp(-d^{\prime\prime}V^{2})$

for some positive constant $d^{\prime\prime}$ depending only on $a$.

##  6\. Proof of Theorem 1

We now prove the theorem by combining the measure bounds on $E_{V}$ and
$F_{V}$ proved in sections 4 and 5. We recall this bounds now:

$\displaystyle\operatorname{meas}(E_{V})$ $\displaystyle\ll
T\exp\mathopen{}\mathclose{{}\left(-c^{\prime\prime}\mathopen{}\mathclose{{}\left(\frac{V\log
T}{\lambda}}\right)^{1/2}}\right)$ $\displaystyle\operatorname{meas}(F_{V})$
$\displaystyle\gg T\exp(-d^{\prime\prime}V^{2})$

where $c^{\prime\prime},d^{\prime\prime}$ are positive constants depending
only on $a$.

As long as $V$ is restricted to the range $(\log T)^{a}\leq
V\leq\kappa\mathopen{}\mathclose{{}\left(\frac{\log T}{\log\log
T}}\right)^{1/3}$ for a sufficently small $\kappa$ depending on
$c^{\prime\prime}$ and $d^{\prime\prime}$ (and hence on $a$), we see that the
measure of $F_{V}$ is at least twice the measure of $E_{V}$ say. By the
definitions of $E_{V}$ and $F_{V}$, and by Lemma 2 we may conclude that the
set of $t\in[T,2T]$ for which

(20) $\int_{-\log T}^{\log T}\pm S(t+u)K_{\pm}(u)\,du\geq 2.5V$

has measure at least $T\exp(-2d^{\prime\prime}V^{2})$. It is then possible to
find (via a greedy algorithm) a finite collection of at least
$T\exp(-4d^{\prime\prime}V^{2})$ points in this set which are all at least
$3\log T$ apart from one another. Call this collection of points $G$.

We shall now restrict to just the case of positive values of $S(t)$ for ease
of exposition. For any $t_{0}\in G$, let
$F^{t_{0}}_{V}\coloneqq\\{t\in[t_{0}-\log T,t_{0}+\log T]\colon S(t)\geq
V\\}$. We note that if $t_{0}$ and $t_{1}$ are distinct points in $G$ then
clearly $F_{V}^{t_{0}}$ and $F_{V}^{t_{1}}$ are disjoint. Hence, to prove
Theorem 1 it is sufficient to prove a weak lower bound on the measure of the
sets $F_{V}^{t_{0}}$ such as

(21)
$\operatorname{meas}\mathopen{}\mathclose{{}\left(F_{V}^{t_{0}}}\right)\gg(\log
T)^{-O(1)}.$

Using properties (i) and (ii) of the kernel $K_{+}$ and the bound $S(t)\leq
A\log T$ for some absolute constant $A$, we have

$\displaystyle\int_{-\log T}^{\log T}S(t_{0}+u)K_{+}(u)\,du$
$\displaystyle\leq
V\int_{-\infty}^{\infty}K_{+}(u)\,du+\int_{F^{t_{0}}_{V}}S(v)K_{+}(v-t_{0})\,dv$
$\displaystyle\leq 2V+A\log T\int_{F^{t_{0}}_{V}}K_{+}(v-t_{0})\,dv$
$\displaystyle\leq 2V+A\log
T\big{(}\operatorname{meas}(F^{t_{0}}_{V})\big{)}^{\frac{1}{2}}\mathopen{}\mathclose{{}\left(\int_{-\infty}^{\infty}|K_{+}(u)|^{2}\,du}\right)^{\frac{1}{2}}.$

Since $t_{0}\in G$, we may combine (20) with the string of inequalities above
and then rearrange to conclude that (21) holds. This proves the theorem for
large positive values of $S(t)$, and the negative values case is the same but
one uses the $K_{-}$ kernel instead.

## References

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# On the dynamics of a collapsing bubble in contact with a rigid wall

Mandeep<EMAIL_ADDRESS>Erwan Tanne2  Michel
Arrigoni 2  Stephane Zaleski 1,3  Daniel Fuster1  daniel.fuster@sorbonne-
universite.fr 1Sorbonne Université and CNRS UMR 7190, Institut Jean le Rond
$\partial$’Alembert F75005 Paris, France 2ENSTA Bertagne UMR 6027 - IRDL
F-29806, Brest, France 3Institut universitaire de France, UMR 7190, Institut
Jean le Rond $\partial$’Alembert F75005 Paris, France

###### Abstract

This work reveals that the dynamic response of a spherical cap bubble in
contact with a rigid wall depends on the effective contact angle at the
instant prior to collapse. This parameter allows us to discriminate between
two regimes in which the mechanisms of interaction between the collapsing
bubble and its surrounding medium differ significantly: When the contact angle
is smaller than 90 degrees a classical jet directed towards the wall is
observed whereas if the initial contact angle is larger than 90 degrees an
annular re-entrant jet parallel to the wall appears. We show that this change
of the behaviour can be explained using the impulse potential flow theory for
small times which shows the presence of a singularity on the initial
acceleration of the contact line when the contact angle is larger then 90
degrees. Direct Numerical Simulations show that although viscosity regularises
the solution at $t>0$, the solution remains singular at $t=0$. In these
circumstances numerical and experimental results show that the collapse of
flat bubbles can eventually lead to the formation of a vortex ring that
unexpectedly induces long-range effects. The role of the bubble geometry at
the instant of maximum expansion on the overall collapse process is shown to
be well captured by the impulse potential flow theory, which can be easily
generalised to other bubble shapes. These results may find direct application
in the interpretation of geophysical flows as well as the control and design
of bio-medical, naval, manufacturing and sonochemistry applications.

###### keywords:

Cavitation $|$ Contact line$|$ Bubble dynamics $|$ Jets $|$ Singularity

##  1 Introduction

Cavitation, the process of bubble formation and collapse, is a key physics
problem because of its applications in nature, mechanics, biomedical, and many
other fields (Cole & Weller, 1948; Lohse et al., 2001; Ohl et al., 2006a, b;
Maxwell et al., 2013; Fuster, 2019). In recent studies, Hou et al. (2021) show
that cavitation bubbles can enhance the efficiency of solar absorption
refrigeration systems, Bhat & Gogate (2021) and Prado et al. (2022) reviewed
the effect of cavitation in wastewater and biomass treatment. Lyons et al.
(2019) compared the low frequency sound signals from volcanic eruptions to
that of bubble collapse and suspect the bubbles of the order of 100 m are
created by the interaction of magma with water. Bubbles can generally appear
from the nuclei sitting at walls (Crum, 1979; Atchley & Prosperetti, 1989;
Borkent et al., 2009), making the interaction with the wall an essential
aspect of bubble collapse dynamics. The wall modifies the symmetry of the
pressure field near the bubble that causes unequal interface acceleration and
thus high-speed liquid jets leading to a complex cavitation phenomena (Popinet
& Zaleski, 2002; Brenner et al., 2002; Supponen, 2017). These jets are
important to understand the destructive potential of the cavitation bubbles
for several biomedical applications such as the disruption of biological
membranes in High Intensity Focussed Ultrasound (HIFU), lithotripsy,
histotripsy and sonoporation techniques (Prentice et al., 2005; Ohl et al.,
2006b; Maxwell et al., 2011; Pishchalnikov et al., 2019). Other applications
concerning erosion, surface cleaning, noise emission and manufacturing
processes (e.g. shotpeening) are also closely linked to the phenomenon of
jetting (Krefting et al., 2004; Blake et al., 2015). The bottleneck in
development and comprehension of these techniques as well as naturally
occurring phenomenon remains the limited understanding of the interaction
amongst cavitation bubble and its surrounding medium.

It is well known that the jets formed during the collapse of bubbles in the
presence of a nearby wall are directed towards the wall and that the
characteristics of these jets can be described well using the scaling laws
given by Popinet & Zaleski (2002) and Supponen (2017). However, the
investigations on the collapse of a bubble initially attached to a wall are
reported less often. Naudé & Ellis (1961) discuss the collapse of bubble
initially in contact with wall by writing the higher order perturbation
solutions to potential flow model. They also show the collapse dynamics of
such bubbles experimentally by producing bubbles with electric spark in the
liquid close to wall. Hupfeld et al. (2020) present experimental results for
laser induced bubbles using ultra-high speed photography in the high capillary
number regimes. They study both expansion and collapse phase in several
liquids to understand the effect of viscosity on the growth, collapse and
bubble shapes including liquid microlayer and contact line evolution. Both
Naudé & Ellis (1961) and Hupfeld et al. (2020) show that during the collapse
the jet is directed towards the solid whereas Li et al. (2018) show that the
jet can be directed in opposite direction when a collapsing bubble is in
contact with a spherical solid particle. Similar problem is studied
numerically by Lechner et al. (2020), who also include the effect of bubble
expansion and the liquid microlayer that results in to different nature of
bubble collapse and jetting. This problem has also been briefly discussed in
numerical studies of Lauer et al. (2012) and Koukouvinis et al. (2016) who
observe the change in jetting behaviour as a function of bubble shape but they
do not show jets opposite to the wall. Reuter et al. (2017) speculate the
appearance of a jet parallel to the wall during the secondary collapse and
consequently the generation of vortex ring, although no direct experimental
observation is reported.

The initiation and consequences of this change of behaviour has not be
understood and the hydrodynamic description of the same is not available. In
this article we bridge this gap and describe the factors controlling the
jetting direction using comprehensive theoretical, numerical and experimental
evidences. We clarify that the change in the jetting direction is an
implication of the singularity in the pressure gradient obtained from the
potential flow model. This singularity only appears when the effective contact
angle at the instant before the collapse is larger than 90 degrees. We show
that at short instants after beginning of collapse phase and for sufficiently
large Reynolds and Weber numbers, the potential flow solution accurately
represents the results of DNS (Direct Numerical Simulation). Despite the fact
that viscosity instantaneously regularise the solution near the singularity
and predicts finite velocities, the effect of singularity is still present
outside the boundary layer and it explains the changes in the nature of the
jet generated. In addition, we show that there is a direct link between the
bubble response at short times and the formation of vortex ring that
eventually leads to surprising long range interaction effects between the
bubble and a free surface (see supplementary video) that have not been
reported before. These results provide a significant advance in the
characterisation of the interaction mechanisms between a bubble and its
surrounding medium and may be a reasoning behind unexplained delayed effects
in underwater explosions observed by Kedrinskii (1978) and tsunami generation
by underwater volcano eruptions (Paris, 2015).

##  2 Problem setup

We focus on the classical three-dimensional Rayleigh collapse problem for a
bubble that has a spherical cap shape (figure 1). This choice is motivated by
the fact that for a spherical cap the bubble geometry is defined by only two
parameters, the bubble radius and the contact angle, which simplifies the
discussion of the results compared to more complex shapes (e.g. ellipsoids)
that could be focus of future investigations. We restrict ourselves to an
axisymmetric configuration where the initial bubble pressure $p_{g,0}$ is
uniform inside the bubble and the liquid, initially at rest, is at a higher
pressure far from the bubble $p_{\infty}>p_{g,0}$. This problem represents the
solution of a bubble in equilibrium state subjected to impulsive action caused
by a sudden increase in far-field pressure, and is also a good approximation
for the dynamic response of a low pressure bubble after the expansion stage
when it reaches its maximum radius and the liquid velocity approaches to zero.

Figure 1: Problem setup and different coordinate systems used in this article
i.e. $(r_{w},\theta_{w})$ and $(r_{s},\theta_{s})$

The dynamics of this system is governed by the mass, momentum and energy
conservation laws, which for the $i$th component in a multi-component setup is
given as ∂ρi∂t + ∇⋅(ρ_i u) & = 0,
∂ρiu∂t + ∇⋅(ρ_i uu) = -∇p_i + ∇⋅τ_i,
∂(ρiEi)∂t + ∇⋅(ρ_i E_i u) = - ∇⋅(up_i) + ∇⋅(τ_i u). As usual, $\mathbf{u}$
represents the velocity vector field, $\rho_{i}$ is density, $p_{i}$ is the
pressure field and
$\tau_{i}=\mu_{i}(\nabla{\mathbf{u}}+(\nabla{\mathbf{u}})^{\rm T})$ is the
viscous stress tensor for each component. The total energy
$\rho_{i}E_{i}=\rho_{i}e_{i}+\frac{1}{2}\rho_{i}{\mathbf{u}}\cdot{\mathbf{u}}$,
is the sum of internal energy per unit volume $\rho_{i}e_{i}$ and kinetic
energy $\frac{1}{2}\rho_{i}{\mathbf{u}}\cdot{\mathbf{u}}$ per unit volume.
Finally the equation of state (EOS) is written as
$\rho_{i}e_{i}=\frac{p_{i}+\Gamma_{i}\Pi_{i}}{\Gamma_{i}-1}$ where $\Gamma$
and $\Pi$ are adopted from Johnsen & Colonius (2006) in such a way that liquid
is a slightly compressible and the bubble contents obeys an ideal gas EOS. The
system of equations is closed by adding an advection equation for the
interface position which is represented by the volume of fluid method
(Tryggvason et al., 2011). After the advection, the interface is reconstructed
using a piecewise linear interface construction. The surface-tension appears
implicitly while averaging the equation 1 in the mixed cells and it is added
as the continum surface force using the method of Brackbill et al. (1992) also
reviewed recently by Popinet (2018). All the intricate details of the method
are given in Fuster & Popinet (2018)

In the DNS, a non-slip boundary condition is imposed on the velocity field at
the wall, while a standard free flow condition is applied in all the other
boundaries. Note that although a non-slip boundary condition imposes zero
velocity at the boundary, the velocity at which the interface is advected in
the cell adjacent to the wall is different from zero. This effectively acts as
an implicit slip length for the contact line motion given as $\lambda=\Delta
x_{min}/2$, where $\Delta x_{min}$ is the smallest grid size. This influence
of this parameter will be discussed in appendix A. The static contact angle
model proposed by Afkhami & Bussmann (2008) is used to implement contact angle
boundary condition which although is not very accurate for prediction the
contact line motion but is a convenient tool that gives fair insight of the
growth of boundary layer and range of influence of viscosity. More
sophisticated models for contact line motion could be used but a significant
change in the conclusions of this work is not expected.

The initial conditions are defined by imposing the fluid to be initially at
rest ($\boldsymbol{u}=0$). In this case the divergence of the momentum
equation gives a Laplace equation for the initial liquid pressure while
assuming the liquid as an incompressible substance,

$\Delta p_{l}=0.$ (1)

This equation is readily solved if we assume that the gas pressure is
initially uniform such that the boundary conditions for pressure (as shown in
figure 1) are the following: at the interface the liquid pressure is

$p_{0,L}=p_{g,0}-\frac{2\sigma}{R_{c}},$ (2)

where $\sigma$ is the surface tension coefficient and $R_{c}$ is the radius of
curvature; at the far–away boundaries the pressure is $p_{\infty}$ and at the
wall the normal pressure gradient is null due to impermeability condition. The
initial condition for total energy is then calculated from the EOS using the
initial pressure field.

We finally introduce relevant dimensionless quantities using the radius of
curvature as characteristic length, the liquid density and the characteristic
velocity $U_{c}=\left({p_{\infty}}/{\rho_{l}}\right)^{1/2}$ obtained by
assuming that at the instant of maximum radius the liquid pressure along the
surface is negligible compared to the far–field pressure. A simple dimensional
analysis shows that the solution of the problem depends on the density and
viscosity ratios, the Reynolds number,
$\Rey=\frac{\rho_{l}U_{c}R_{c}}{\mu_{l}}$, the Weber number
$\Web=\frac{\rho_{l}U_{c}^{2}R_{c}}{\sigma}$ and the contact angle $(\alpha)$.
The characteristic velocity is constant and equal to 10 m/s for the low
pressure bubbles collapsing in water under the action of the atmospheric
pressure, therefore both, the Reynolds and Weber number, are determined only
by the radius of curvature of the bubble. In the particular experiments
considered in this study the radius of curvature remains to be of the order 1
cm obtaining characteristic values of the Weber and Reynolds number of the
order of $\Web\sim\textit{O}(10^{4})$ and $\Rey\sim\textit{O}(10^{5})$. The
influence of finite values of Reynolds is briefly discussed using Direct
Numerical Simulations.

Figure 2: $(a)$ Isocontours of the magnitude of acceleration for bubble with
$\alpha=\alpha=3\pi/4$: The isocontours in the left half are obtained from
free surface model and those in the right half are obtained from DNS. $(b)$
Non-dimensional acceleration magnitude along the wall obtained for bubble with
$\alpha=3\pi/4$ using the free surface model.

##  3 Short time dynamics

At short times, we can simplify the system of equations if we consider the
liquid as an incompressible substance and the interface as a free surface,
i.e. we neglect inertial and viscous effects inside the bubble. This
assumption is reasonable given that $\mu_{g}/\mu_{l}\ll 1$ and
$\rho_{g}/\rho_{l}\ll 1$ (in experiments the density ratio is of the order of
$10^{-5}$ when the bubble pressure equals the vapour pressure). The velocity
at short times remains a small quantity, thus we can neglect the convective
terms as compared to temporal derivatives of velocity and spatial pressure
gradients (Batchelor, 2000). This hypothesis remains true when time scales
under consideration are smaller than the advection time scales i.e.
$\frac{t_{s}U_{c}}{R_{c}}\ll 1$. Under these assumptions the solution of the
velocity is obtained from the integration of the linearised momentum equation
as

${\mathbf{u}}(t_{s})=-\frac{1}{\rho}\int_{0}^{t_{s}}\nabla
p+\frac{1}{\rho}\int_{0}^{t_{s}}\nabla\cdot{\mathsfbi\tau}.$ (3)

It is classical for this linear system to decompose the velocity field in to a
potential part (${\mathbf{u}}_{\phi}$) and a viscous correction
(${\mathbf{u}}_{\nu}$) i.e.
${\mathbf{u}}={\mathbf{u}}_{\phi}+{\mathbf{u}}_{\nu}$. The potential flow
theory gives ${\mathbf{u}}_{\phi}$ whereas the viscous correction requires
complex analysis (Saffman, 1995; Batchelor, 2000; Pope, 2001). For the short
times and large enough Reynolds numbers, the viscous contributions to velocity
are relevant only in the small region of thickness
$\delta\sim\textit{O}(\sqrt{\nu t})$ near the solid boundary. Thus in such a
situation, we expect the solution to be governed by the potential part and the
viscous part is only a correction in a thin region close to the wall. We
discuss these aspects next.

###  3.1 Potential flow solution

The potential flow solution is obtained by solving the inviscid part of the
linearised momentum equation

$\frac{\partial{\mathbf{u}}_{\phi}}{\partial t}=-\frac{1}{\rho}\nabla p.$ (4)

The magnitude of pressure gradient field (or acceleration field) at short
infinitesimal times govern the dynamics of bubble collapse at a finite time.
To obtain the acceleration we introduce the dimensionless pressure
$\tilde{p}=\frac{p-p_{0,L}}{p_{\infty}-p_{0,L}}$. Note that in the linear
problem the interface does not move, therefore surface tension effects at
short times only introduce a correction on the scaling prefactor
$(p_{\infty}-p_{0,L})$ upon which the solution depends. The divergence of
equation 4 leads to a Laplace equation ($\nabla^{2}\tilde{p}=0$) which can be
solved with appropriate boundary conditions for $\tilde{p}$. In particular we
impose $\tilde{p}=1$ far away from the bubble, $\tilde{p}=0$ at the bubble
interface and $\frac{d\tilde{p}}{dn}=0$ at the solid boundary. The solution of
this equation depends only on the geometry of the bubble which is fully
described as a function of the contact angle ($\alpha$). This simplified
representation of the fluid fields at short times is referred as ‘free surface
model’ in this article.

In figure 2a we start presenting the isocontours of the magnitude of the non-
dimensional acceleration ($|\nabla\tilde{p}|$) field obtained for a spherical
cap bubble with contact angle $\alpha=\frac{3}{4}\pi$ while figure 2b shows
the variation of same along the wall in the liquid phase with respect to the
distance from the contact line obtained from free surface model. The
isocontours in the left half are obtained numerically from the free surface
model, while on the right half are obtained from the DNS of the Navier–Stokes
equations (using setup shown in figure 4a) accounting for the presence of a
gas with non-zero density ($\rho_{l}/\rho_{g}=10^{-3}$) and a finite value of
viscosity ($\mu_{l}/\mu_{g}=10^{-2}$) at $\frac{tU_{c}}{R_{c}}=2.07\cdot
10^{-6}$. Good agreement between the two models supports the argument that
equation 1 is a good representation of the fluid dynamics at very short times.
We clearly distinguish two separate regions: the far-field where the pressure
gradient contours are hemispherical caps centred at the axis of symmetry, and
the near field where the contours for pressure gradient are intricate and
diverge on approaching the triple contact point. Next, we characterise each of
these regions:

####  3.1.1 Near field

Figure 3:  $(a)$ Non-dimensional acceleration magnitude,
${|{\mathbf{a}}_{I}|}/a_{0}$ along the interface parametrised using the angle
$\theta_{w}$ (measured in the counter-clockwise direction from the point of
contact of wall and the axis of symmetry) for
$\alpha=\frac{\upi}{3},\frac{\pi}{2},\frac{2}{3}\pi,\frac{3}{4}\pi$. Dots
represent the numerical solution from free surface model and the thick lines
are predictions using equation 6 evaluated at the interface. $(b)$ Exponent
$b_{0}$ and coefficient $C_{0}$ obtained from fitting the pressure gradient
obtained from the free surface model along the wall for different $\alpha$.
$(c)$ Non-dimensional averaged interface acceleration magnitude as a function
of the contact angle using different methods: analytical expression given by
equation 8 (blue line), the solution from the free surface model (yellow
line), the DNS solution of the Euler equations (black crosses), and the DNS
solution of the Navier–Stokes solver for $\Rey=100$ (red crosses).

The numerical solution of the interface acceleration at the initial time is
shown in figure 3a, where we use the angle with respect to the solid wall
$\theta_{w}$ to parametrise the interface position. The origin,
$\theta_{w}=0$, corresponds to the point of contact between the interface and
the solid wall and $\theta_{w}=\pi/2$ is the intersection of the interface
with the axis of symmetry. Figure 3a depicts the local non-dimensional
acceleration magnitude at bubble interface $|{\mathbf{a}}|/a_{0}$ for four
representative cases with $\alpha=\upi/3,\upi/2,2\upi/3,3\upi/4$. The value of
the reference acceleration is chosen as $a_{0}=\frac{p_{\infty}-p_{0,L}}{\rho
R_{c}}$, which is the acceleration magnitude for a spherical bubble with
radius $R_{c}$ and a known pressure difference. As expected, in the case of
$\alpha=\upi/2$, the numerical solution recovers the Rayleigh-Plesset solution
and the non-dimensional acceleration is uniform and equal to one all over the
interface. When $\alpha<\upi/2$ the interface acceleration (thus velocity)
tends to zero at the contact point, and therefore it seizes to move even in
the potential flow problem with a slip wall. For $\alpha>\upi/2$, the
appearance of the singularity is evident as the acceleration diverges as we
approach the $\theta_{w}\to 0$ limit. To interpret this result one must keep
in mind that the general solution of the Laplace equation in spherical
coordinates can be obtained by separation of variables that leads to an
infinite series. This series sometimes diverges at the edge where Dirichlet
and Neumann boundary conditions meet due to the appearance of a singularity
(Dauge, 2006). This singularity was reviewed extensively in the past (Landau &
Lifshitz, 1959; Blum & Dobrowolski, 1982; Steger, 1990; Deegan et al., 1997;
Li & Lu, 2000) but the studies where this singularity appears in the bubble
dynamics problems are rare. The asymptotic solutions close to the point where
homogeneous Dirichlet and Neumann boundary conditions meet can be
alternatively expressed by using the general solution of the Laplace equation
(Li & Lu, 2000; Dauge, 2006; Yosibash et al., 2011) and takes the form of

$\tilde{p}_{s}=\sum_{k}^{\infty}C_{k}\tilde{r}_{s}^{b_{k}}\cos\left(b_{k}\theta_{s}\right),$
(5)

where $\tilde{r}_{s}=r_{s}/R_{c}$ is the non-dimensional distance from the
contact line and $b_{k}=\frac{\upi}{\alpha}(k+1/2)$. Taking the derivative
with respect to the normal of the interface we readily find the interface
acceleration magnitude near the contact line as

$|{\mathbf{a}}_{I}|=\frac{1}{\rho}\left|\frac{\partial p}{\partial
n}\right|\approx a_{0}C_{0}b_{0}\tilde{r}_{s}^{b_{0}-1},$ (6)

This expression exhibits a singularity at the contact point ($r_{s}\to 0$)
when $b_{0}=\frac{\upi}{2\alpha}<1$ (or $\alpha>\upi/2$) implying that the
acceleration at the triple contact point is infinite. In these conditions the
first term in the expansion is the leading order term that eventually
dominates the solution in the region $r_{s}\leq R_{c}$. This can also be
clearly seen in figure 2b where the fitting curve shown with solid line is
obtained using the first term of the series solution (i.e. parameters $C_{0}$
and $b_{0}$ in equation 6). We repeat the fitting procedure for various
$\alpha$ to verify that the numerical values of $b_{0}$ match well with
theoretical predictions and find $C_{0}$ which is a constant of order one that
slightly increases with $\alpha$ (figure 3b).

Focusing on the characterisation of the regimes where the singularity appears,
figure 3a clearly shows that the numerical solution (dots) is well described
by equation 6 evaluated at the interface (solid lines), showing excellent
agreement close to the contact point, i.e. small values of $\theta_{w}$. Near
the axis of symmetry (e.g. $\theta_{w}\to\upi/2$ and $\tilde{r}_{s}\approx
1$), the errors are visible and the first term in the series does not suffice
to describe accurately the acceleration field.

####  3.1.2 Far field solution

The far field flow created by the bubble corresponds to a punctual sink
sitting at the intersection between the wall and the axis of symmetry. The
integration of the momentum equation in the radial coordinate provides the
magnitude of pressure gradient generated by a punctual sink at an arbitrary
distance from the sink as

$\left|\frac{d\tilde{p}_{\rm{far}}}{d\tilde{r}_{w}}\right|=\frac{|\overline{\mathbf{a}}_{I}|}{a_{0}}\frac{(1+\cos(\alpha))}{\tilde{r}^{2}_{w}},$
(7)

where $|\overline{\mathbf{a}}_{I}|$ is the magnitude of averaged acceleration
along the interface, which determines the strength of the punctual sink.
Figures 2b clearly shows that this equation captures well the decay of the
pressure gradient with the distance $\tilde{r}_{w}$ far from the interface.
Because the first term of the series i.e. equation 6 predicts the interface
acceleration reasonably well, we obtain an estimation of the averaged
acceleration magnitude for bubbles with $\alpha>\upi/2$ as

$|\overline{\mathbf{a}}_{I}|=\left|\frac{1}{S_{b}}\int\frac{-1}{\rho_{l}}\frac{\partial
p}{\partial n}dS_{b}\right|=a_{0}C_{0}b_{0}G(\alpha),$ (8)

where $S_{b}$ stands for the bubble interface surface and $G(\alpha)$ is a
geometrical factor that can be numerically computed assuming that the first
term in the series is indeed the leading order term along the entire bubble
interface. Figure 3c shows that the averaged nondimensional interface
acceleration magnitude obtained using this model, which compares well with the
numerical results of the free surface model and the solution of the both
viscous and inviscid Navier-Stokes equations. The results of the simplified
model are obtained using the value of $C_{0}$ numerically computed and
reported in figure 3b. For $\alpha>\upi/2$, $|\overline{\mathbf{a}}|$
increases with $\alpha$. The three proposed models agree well for the angles
tested implying that the free surface model as well as the simplified
expression given in equation 8 capture well the averaged bubble response at
short times for sufficiently large Reynolds numbers.

###  3.2 Viscous correction

The potential flow solution is only formally exact at the initial time $t=0$
when the interface is at rest. As soon as the interface is set in to motion a
thin boundary layer immediately develops regularising the flow close to the
contact point. We use the results from the DNS of the Navier–Stokes equations
at short times to investigate the influence of viscosity on the dynamic
response of the bubbles.

Figure 4:  $(a)$ The numerical setup for DNS $(b)$ Results from the DNS for
$\alpha=2\upi/3$ and $\Rey=100$. Time averaged interface acceleration in
direction parallel to the wall, $a_{r,I}=u_{r,I}/t$, as a function of the
distance from the wall at five different times (in colour where
$tU_{c}/R_{c,0}\in[7.27\cdot 10^{-5},3.6\cdot 10^{-4}]$). For reference we
include the potential flow solution given by equation 6 evaluated at the
interface (solid black line). The inset represents a zoom in to the viscous
boundary layer generated close to the wall.

We solve for the full set of equations 1 where, for the sake of simplicity, we
neglect surface tension effects. The equations are solved using the All-Mach
implementation (Popinet, 2015; Fuster & Popinet, 2018), where the one fluid
approach is used to solve the modelling equations. This solver has been
already used successfully to investigate some aspects on the interaction of an
oscillating bubble and a rigid tube by Fan et al. (2020) and interactions of
collapsing bubbles with free surface by Saade et al. (2021). The setup for
numerical simulations is shown in figure 4a, where a spherical cap-shaped
bubble is initialised in a square domain of size 100 times the initial radius
of curvature of this bubble. We use such large domains so that the pressure
waves reflected from the boundaries do not affect the solution, moreover grid
is coarsened away from the bubble to diffuse these waves. The simulations are
axisymmetric about left boundary and the bubble is initially in contact with
the bottom boundary where a non-slip boundary conditions is applied. The
faraway boundaries are given constant far-field pressure conditions and
homogeneous Neumann condition for velocity. The grid is refined progressively
near the interface to a minimum grid size of $\Delta x_{min}/R_{c}=0.00038$.
The maximum time step in the simulation is governed by acoustic CFL
restriction, which in our case is 0.4. The initial pressure field inside the
domain is given from Equation 1, velocity field is given as zero and total
energy is given from EOS.

In figure 4b we characterise the bubble motion at short non-dimensional time
$tU_{c}/R_{c}$ by showing the evolution of the time averaged interface
acceleration magnitude parallel to the wall, defined as $a_{r}(t)=u_{r}(t)/t$,
as a function of the normal distance from the wall. The results shown with
coloured dots are obtained from DNS for $\alpha=2\upi/3$ and $\Rey=100$ where
colour scale represents the non-dimensional time varying between $7.27\cdot
10^{-5}$ and $3.6\cdot 10^{-4}$. The inset figure shows the clear development
of a boundary layer very close to the wall, the thickness of this layer
defined by the height for which the velocity is maximum. Outside this region
the free surface potential flow solution (equation 6), shown with a solid
black line, predicts the interface velocity obtained from DNS relatively well,
therefore the potential flow solution is a good approximation of the collapse
process outside the viscous boundary layer. Inside the viscous boundary layer,
the interface acceleration is sensitive to the slip length imposed (see
Appendix A) and also to the movement of the contact line. In this region, the
solution of free surface model at $t=0$ cannot be extrapolated to predict the
flow field near the contact line at short times. Remarkably, although the
contact line motion is grid dependent, the average acceleration and the
maximum jet velocity is shown to be independent of the slip length imposed
(see Appendix A). This fact together with slight dependence of the average
acceleration with $\Rey$ (figure 3c) confirms that for large enough Reynolds
numbers, the bubble dynamic response is mainly governed by the potential flow
solution and not by the particular contact line motion model used. In these
conditions, the free surface potential flow model at $t=0$ is a useful tool to
predict the average dynamics of the interface at short times.

Figure 5:  $(a)$ Evolution of jet velocity for different contact angles from
(dots) DNS and (solid lines) fitting using equation 9. $(b)$ Exponent $q$ from
DNS fitting and predictions of equation 10.

In order to characterise the jet formation process we extract the maximum
interface velocity magnitude from DNS data, which we name as jet velocity
${u}_{jet}$. This velocity is fitted using a power law function of the form
(figure 5a)

$\frac{u_{jet}}{U_{c}}=A\left(\frac{tU_{c}}{R_{c}}\right)^{q},$ (9)

where $q$ is a coefficient that becomes smaller than one when the acceleration
is singular at $t=0$. A theoretical estimation of coefficient $q$ accounting
for viscous effects can be obtained from the singular solution described in
the previous section as following: The jet velocity after short time $(t_{s})$
can be approximated from the initial acceleration evaluated at the interface
at height equal to thickness of boundary layer $z=\delta(t_{s})$. The $\delta$
is given approximately by the solution of the Stokes problem for the flow near
the flat plate that is impulsively started from rest therefore, we can impose
that $\delta(t_{s})\approx C_{\delta}\sqrt{\nu t_{s}}$ where $C_{\delta}$ is a
constant of order unity. In this case the jet velocity velocity is estimated
as

$u_{jet}(t_{s})=a_{r,I}\left(t=0,z_{I}=\delta(t_{s})\right)\;t_{s}$

where we assume that the interface quickly decelerates as soon as it enters
inside the viscous boundary layer. Under these assumptions, the coefficient
$q$ is readily obtained as

$\displaystyle q=\frac{1}{2}+\frac{\upi}{4\alpha}.$ (10)

As shown in Figures 5b this model is capable of capturing the evolution of the
jet velocity at short times found from DNS simulations accounting for viscous
effects when $\alpha>\upi/2$. The exponent $q$ (figure 5b) decreases as
$\alpha$ increases because the acceleration gradient in the normal direction
increases with $\alpha$ as seen in figure 3a. Note that the DNS results
confirm that $q<1$ for $\alpha>\upi/2$ and therefore the jet acceleration is
singular at $t=0$ even in the presence of viscous effects.

Figure 6:  The vorticity is shown in colour map and velocity vectors obtained
from DNS after short time short time ($\frac{tU_{c}}{R_{c}}=0.1$) for
$\rm{Re}=100$ for $(a)$ $\alpha=2\upi/3$ and $(b)$ $\alpha=5\upi/12$.

Another interesting quantity to describe the overall interface dynamics is the
vorticity generated during the collapse. At short times, vorticity is
concentrated in a sheet vortex at the wall and at the interface. By assuming
zero stress condition at the interface for $t=0$, the only non-zero component
of the vorticity can be written as

$\omega_{\phi}=-2\frac{\partial{u}_{r}}{\partial\theta}.$

It follows from the potential flow solution (figure 3a) that the sign of the
acceleration gradient changes depending upon initial contact angle i.e. if it
is greater or less than $\upi/2$, reverting the sign of the vortex sheet
generated at the interface. This behaviour is indeed observed from the
numerical simulations as shown in figure 6 (a and b), where the strength of
the vortex sheet is plotted with saturated colour maps. The change in dominant
colour represents the opposite sign of vorticity at interface between the two
cases.

##  4 Long time dynamics

###  4.1 High Reynolds regime

After describing the short time dynamics, in this section we investigate the
long time consequences of this singular response using the experiments and
DNS. The results for two representative cases $\alpha>\pi/2$ and
$\alpha<\pi/2$ shown in figure 7 $(a)$ and $(b)$ are discussed now.

Figure 7:  The snapshots of bubble shape are shown for the two representative
cases, the numerical bubble shapes shown with red curves are overlaid at same
times and scaled to same length as in experiments. $(a)$ Case where
$\alpha>\upi/2$, each snapshots are taken every $0.1$ ms $(b)$ Case where
$\alpha<\upi/2$, each snapshots are taken every $0.125$ ms.

####  4.1.1 Dynamics of bubbles with an effective contact angle larger than
$\pi/2$

First we discuss the case where the singularity is present (e.g.
$\alpha>\pi/2$). The generation of bubbles with $\alpha>\pi/2$ can be
investigated using the setup described briefly in appendix B and references
(Bourguille et al., 2017; Tahan et al., 2020). The laser is focused at the
bottom of liquid tank where a water drop is attached. The cavitation in the
water drop induces shock wave in aluminium plate that leads to the appearance
of multiple bubbles in the water tank that interact to form a flat bubble with
shape similar to spherical cap at maximum volume (figure 7 a). The snapshots
are taken at every $0.1$ms, the interface from the second snapshot is
extracted and fitted with an approximate spherical cap which gives
$R_{c}=7.56\times 10^{-3}m$ and $\alpha=0.727\pi$.

We reproduce the experimental conditions numerically using the numerical setup
described in section 3.2. The minimum mesh size of $\Delta x=60\mu m$, the far
field pressure $p_{\infty}$ is 1 atm and the pressure inside bubble is set to
a low value of 0.1 atm which is chosen by hit and trial method to match the
numerical collapse time with the collapse time in experiments. We consider
both surface tension and viscous effects in the numerical simulations with
$\Rey\approx 75600$ and $\Web\approx 10500$. The numerically obtained bubble
shapes plotted with red contours and scaled to bubble size in snapshot 2,
these are subsequently overlaid on the experimental snapshots after same
times. A very good agreement is seen between the numerical and the
experimental bubble shapes, small differences subsist because of the
simplification of the bubble shape during the expansion, the influence of
gravity, mass transfer and other effects that are not considered in the
numerical simulations.

At the beginning of collapse phase the highest interface velocity is developed
at edge of the viscous boundary layer (see figure 4b) that leads to the
appearance of an annular jet which is generated parallel to the wall and a
mushroom-like shape of the interface contour (see figure 7a and supplementary
movie 1). These results are consistent with previous numerical works of Lauer
et al. (2012) and Koukouvinis et al. (2016). Remarkably, when the collapse is
strong enough and the jet reaches the axis of symmetry, a stagnation point
appears there and a secondary upward jet normal to the wall is generated. The
re-entrant jet observed for $\alpha>\pi/2$ is not conventional in cavitation
and generates vortex ring structures similar to those observed by Reuter et
al. (2017) which are persistent in nature and can travel large distances in
comparison to bubble size. The generation of this vortex ring is illustrated
numerically in figure 8a where the colour maps show the vorticity field. A
clear re-entrant annular jet is observed, followed by a mushroom like
structure and the bubble detaches from the wall consequently a jet in the
upwards direction which eventually leads to formation of vortex ring. In the
experiments we visualise this by adding the dye in the bottom of the tank
(figure 8b) during the collapse of a flat bubble. This vortex ring can travel
to long distance and induce unusual long range effects like free surface waves
and jetting (see supplementary movie 3).

####  4.1.2 Dynamics of bubbles with an effective contact angle smaller than
$\pi/2$

The dynamics of bubbles with contact angles smaller than 90 degrees (figure 7
b) can be obtained using a classical experiment where a laser is focused
directly in to the liquid very close to the wall. The bubble shape from
snapshot 2 in figure 7b is approximated with a spherical cap that gives
$R_{c}=7.63\times 10^{-3}m$ and $\alpha=0.389\pi$. The interface contours
obtained numerically are shown with the red curves scaled to size of the
bubble size in first snapshot. In this case, the interface acceleration is
minimum at the contact line and maximum at the tip of the spherical cap
leading to a conventional high–speed liquid jet directed towards the wall (see
figure 7b and supplementary movie 2). This jet developed towards the wall
impinges in to the solid surface causes cavitation damage. Similar dynamics
have been described in several of previous studies (Naudé & Ellis, 1961;
Gonzalez-Avila et al., 2021).

Figure 8: $(a)$ Bubble interface in black curve and the vorticity field in the
colour map as obtained from DNS is shown at (consecutively from left to right)
$\frac{tU_{c}}{R_{c}}=0,0.19,0.3,0.81,1.00,1.45,1.95,4.90$. An the annular jet
parallel to wall reaches the axis resulting in to the jet directed away from
the wall which later generates the vortex ring. The results are plotted for
$\alpha=2\upi/3$ and $\Rey=\infty$ $\Web=\infty$ $(b)$ The visualisation of
liquid flow field obtained by adding dye at the bottom of tank resulting from
collapse of bubble is shown at every 2.5ms.

###  4.2 Finite Reynolds number effects

Popinet & Zaleski (2002) showed that for the small bubbles collapsing in the
vicinity of a rigid wall, the jets can be suppressed due to the viscous
effects. In this section, we use the DNS to quantitatively predict the range
of size for which the observations reported in this manuscript apply. We take
an ideal case of an air bubble with $p_{b,0}=0.1$ atm and constant contact
angle equal to 120 degrees collapsing in water at an ambient far field
pressure. In this situation, the characteristic velocity of the process stays
constant and equal to 10 m/s and the initial radius of curvature is then left
as the unique parameter controlling the values of the Reynolds and Weber
number.

Figure 9: $(a)$ Peak of the non-dimensional velocity as a function of Reynolds
number for an air bubble at $p_{0}=0.1$ atm collapsing in water at atmospheric
pressure. $(b)$ Interface contours as a function of the Reynolds number at the
instant of minimum radius.

In figure 9a we show the maximum non-dimensional velocity reached during the
collapse as a function of the Reynolds number. For Reynolds numbers above a
given critical value $\Rey>\Rey_{c}=100$ the influence of Reynolds on the peak
velocities is only marginal. However, when the Reynolds is below this critical
value, the jet velocity drops dramatically further decaying with decreasing
$\Rey$ . This sudden change on the peak velocities is controlled by the
appearance of jetting, which is not visible for $\Rey\leq 100$ (see figure
9b). For a low pressure air bubble collapsing in water at atmospheric pressure
these results reveal that the observations reported (reversed re-entrant jet)
are applicable for bubbles larger than $R_{c}>10\mu$m. We must still emphasise
that the results reported here are limited to spherical cap bubbles. In
reality other kind of behaviour can be observed depending upon the mechanism
by which the bubbles are generated and the bubble shape reached at the instant
of maximum expansion. If the bubble shape is not a spherical cap or the
microlayer is not able to drain during the bubble collapse or the bubble size
is comparable to pits, it can lead to a variety of other collapse and jetting
dynamics discussed by Lechner et al. (2020), Reuter & Ohl (2021) and Trummler
et al. (2020) respectively.

##  5 Conclusions

In this work, we show that the impulsive potential flow theory can be used to
discuss the influence of the bubble shape on the dynamic response of
collapsing bubbles for sufficiently large Reynolds and Weber numbers. As an
example, we present the results obtained for the collapse of the spherical cap
bubbles showing that the effective contact angle at the instant of maximum
expansion controls the interface acceleration at the beginning of the collapse
phase and the jetting direction observed in Direct Numerical Simulations and
experiments. When $\alpha>90$ degrees the potential flow solution at small
times shows the appearance of a singularity which causes extremely high
accelerations close to the contact point and a change in vorticity direction
with respect to the $\alpha\leq 90$ degrees case. In the former case, an
unconventional jetting mechanism is observed and shown to be responsible for
the appearance of a vortex dipole travelling in the direction opposite to the
wall. The nature of interaction between the bubble and the surrounding medium
is then strongly influenced by the bubble shape at the instant before
collapse, appearing as a critical parameter if one wants to control or model
the physical phenomena triggered by the collapse of bubbles attached to a
wall.

The findings of this work could be used to explain and control various
processes triggered by the collapse of bubbles. Some examples include mixing
and transport processes, the development of treatment technologies based on
the bubble–tissue interactions including drug delivery and high intensity
ultrasound techniques (Prentice et al., 2005; Ohl et al., 2006b; Maxwell et
al., 2011) and the interactions between a collapsing bubble and the free
surface (Kedrinskii, 1978). Another exciting phenomenon is generation of
surface waves from the bubble collapse in deep water. The phenomenon described
in this manuscript may be eventually related to the appearance of Tsunamis
from underwater volcanic eruptions where the mechanisms linking both processes
are not fully understood yet Paris (2015). Finally the fundamental
understanding of interaction of collapsing bubble with the surrounding media
should improve the design and development of various technologies based on the
physical, mechanical and chemical effects induced by the collapse of bubbles.
The relation of the singularity of potential flow to the well known force
singularity during the motion of contact line described by Huh & Scriven
(1971) remains another open research problem where the solutions reported in
this work may inspire new ideas.

[Acknowledgements]Authors want to acknowledge Julien Le Clanche, research
engineer at ENSTA Bretagne for his kind support.

[Funding]This research is supported by European Union (EU) under the MSCA-ITN
grant agreement number 813766, under the project named ultrasound cavitation
in soft matter (UCOM). Part of this work was part of the PROBALCAV program
supported by The French National Research Agency (ANR) and cofunded by DGA
(French Minisitry of Defense Procurment Agency) under reference Projet
ANR-21-ASM1-0005 PROBALCAV.

[Declaration of interests]The authors report no conflict of interest.

[Author contributions]D.F., M.A. and S.Z. designed the research program. M.S.
conducted the simulations and analyzed data. M.S. and D.F. did the theoretical
calculus. E.T. and M.A. performed the experiments. M.S, D.F, M.A., E.T. and
S.Z discussed the results. M.S. and D.F. wrote the paper.

##  Appendix A Convergence study

Figure 10: The grid convergence of viscous solution $(a)$ The evolution of
velocity of contact line is plotted for different grid-size. $(b)$ The
evolution of jet velocity is plotted for various grid size.

The convergence properties of the contact line velocity and jet velocity are
discussed in this section. The velocity of the contact line (figure 10a where
$|{\mathbf{u}}_{TP}|$ is contact line velocity) changes as the mesh is refined
since slip length changes implicitly ($\lambda=\Delta x_{min}/2$). It is also
evident from the same plot that in very short times the well known scaling for
the thickness of the boundary layer ($t^{1/2}$) is not recovered as the
boundary layer is not well resolved. At slightly longer times the boundary
layer growth recovers the expected scaling law. Despite of the sensitivity of
the contact line motion to the slip length imposed, the peak interface
velocity (jet velocity $u_{jet}$) plotted in Figure 10b reveal that the
numerical results are converged and therefore independent of the slip length.

##  Appendix B Experimental method

Figure 11: Experimental setup used to create flat bubbles.

The experiments are performed with pulsed Nd:YAG laser (model quanta ray pro
350-10) that provides maximum of 3.5J of energy with Gaussian distribution in
time having half maximum at 9.2 ms. This laser beam passes through a quarter
wave plate and polariser for modulating the energy and then focused using a
planoconvex lens of focal length 250mm that creates a focal spot of 4mm in
diameter on the aluminium plate. For bubbles shown with $\alpha>\upi/2$ the
laser is focused on the water droplet attached to bottom of an aluminium plate
(1 mm thick) kept at bottom of a water tank and for the bubbles with
$\alpha<\upi/2$ the laser if focused on the top of this aluminium plate kept
at the bottom of same tank (see figure 11). The laser energy is set to 50% of
maximum in case where $\alpha<\upi/2$ and we observe the cavitation phenomenon
in the water tank using high speed camera.

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