# Density estimates for the zeros of the
Beurling $\zeta$ function in the critical strip††thanks: Supported in part by
Hungarian National Foundation for Scientific Research, Grant # T-72731,
T-049301, K-61908, K-119528 and K-132097 and by the Hungarian-French
Scientific and Technological Governmental Cooperation, Project # TÉT-F-10/04
and the Hungarian-German Scientific and Technological Governmental
Cooperation, Project # TEMPUS-DAAD # 308015.

Szilárd Gy. Révész

(Dedicated to János Pintz on the occasion of his 70th anniversary)

###### Abstract

In this paper we prove three results on the density resp. local density and
clustering of zeros of the Beurling zeta function $\zeta(s)$ close to the one-
line $\sigma:=\Re s=1$. The analysis here brings about some news, sometimes
even for the classical case of the Riemann zeta function.

As a complement to known results for the Selberg class, first we prove a
Carlson type zero density estimate. Note that density results for the Selberg
class rely on use of the functional equation of $\zeta$, not available in the
Beurling context. Our result sharpens results of Kahane, who proved an $O(T)$
estimate for zeros lying precisely just on a vertical line $\Re s=a$ in the
critical strip.

Next we deduce a variant of a well-known theorem of Turán, extending its range
of validity even for rectangles of height only $h=2$.

Finally, we extend a zero clustering result of Ramachandra from the Riemann
zeta case. A weaker result – which, on the other hand, is a strong sharpening
of the average result from the classic book of Montgomery – was worked out by
Diamond, Montgomery and Vorhauer. On our way we show that some obscure
technicalities of the Ramachandra paper can be avoided.

MSC 2000 Subject Classification. Primary 11M41; Secondary 11F66, 11M36, 30B50,
30C15.

Keywords and phrases. Beurling zeta function, analytic continuation,
arithmetical semigroups, Beurling prime number formula, zero of the Beurling
zeta function, oscillation of remainder term, density estimates for zeta
zeros.

Author information. Alfréd Rényi Institute of Mathematics
Reáltanoda utca 13-15, 1053 Budapest, Hungary
<EMAIL_ADDRESS>
##  1 Introduction

###  1.1 Beurling’s theory of generalized integers and primes.

Beurling’s theory fits well to the study of several mathematical structures. A
vast field of applications of Beurling’s theory is nowadays called
_arithmetical semigroups_ , which are described in detail e.g. by Knopfmacher,
[21].

Here ${\mathcal{G}}$ is a unitary, commutative semigroup, with a countable set
of indecomposable generators, called the _primes_ of ${\mathcal{G}}$ and
denoted usually as $p\in{\mathcal{P}}$, (with
${\mathcal{P}}\subset{\mathcal{G}}$ the set of all primes within
${\mathcal{G}}$), which freely generate the whole of ${\mathcal{G}}$: i.e.,
any element $g\in{\mathcal{G}}$ can be (essentially, i.e. up to order of
terms) uniquely written in the form $g=p_{1}^{k_{1}}\cdot\dots\cdot
p_{m}^{k_{m}}$: two (essentially) different such expressions are necessarily
different as elements of ${\mathcal{G}}$, while each element has its
(essentially) own unique prime decomposition. Moreover, there is a _norm_
$|\cdot|\leavevmode\nobreak\ :{\mathcal{G}}\to{\mathbb{R}}_{+}$ so that the
following hold. First, the image of ${\mathcal{G}}$,
$|{\mathcal{G}}|\subset{\mathbb{R}}_{+}$ is discrete, i.e. any finite interval
of ${\mathbb{R}}_{+}$ can contain the norm of only a finite number of elements
of ${\mathcal{G}}$; thus the function

${{\mathcal{N}}}(x):=\\#\\{g\in{\mathcal{G}}\leavevmode\nobreak\
:\leavevmode\nobreak\ |g|\leq x\\}$ (1)

exists as a finite, nondecreasing, right continuous, nonnegative integer
valued function on ${\mathbb{R}}_{+}$. Second, the norm is multiplicative,
i.e. $|g\cdot h|=|g|\cdot|h|$; it follows that for the unit element $e$ of
${\mathcal{G}}$ $|e|=1$, and that all other elements $g\in{\mathcal{G}}$ have
norms strictly larger than 1.

Arithmetical functions can also be defined on ${\mathcal{G}}$. We will use in
this work the identically one function $u$, the Möbius function $\mu$ and the
number of divisors function $d$: for their analogous to the classical case
definition see pages 73-79 in [21]. The generalized von Mangoldt function
$\Lambda_{{\mathcal{G}}}(g)$, will appear below in (4).

In this work we assume the so-called _"Axiom A"_ (in its normalized form to
$\delta=1$) of Knopfmacher, see pages 73-79 of his fundamental book [21].

######  Definition 1.

It is said that ${{\mathcal{N}}}$ (or, loosely speaking, $\zeta$) satisfies
_Axiom A_ – more precisely, Axiom $A(\kappa,\theta)$ with the suitable
constants $\kappa>0$ and $0<\theta<1$ – if we have111The usual formulation
uses the more natural version ${\mathcal{R}}(x):={\mathcal{N}}(x)-\kappa x$.
However, our version is more convenient with respect to the initial values at
1, as we here have ${\mathcal{R}}(1-0)=0$. All respective integrals of the
form $\int_{X}$ will be understood as integrals from $X-0$, and thus we can
avoid considering endpoint values in the partial integration formulae
involving integration starting form 1. Alternatively, we could have taken also
${\mathcal{N}}(x):=\\#\\{g\in{\mathcal{G}},|g|<x\\}$ left continuous, and
${\mathcal{R}}(x):={\mathcal{N}}(x)-\begin{cases}\kappa
x\qquad&\text{if}\leavevmode\nobreak\ x>1\\\ 0&\text{if}\leavevmode\nobreak\
x\leq 1\end{cases}$–also with this convention we would have
${\mathcal{R}}(1-0)=0$ for the remainder term, but this seemed to be less
convenient than our above choice. for the remainder term

${\mathcal{R}}(x):={\mathcal{N}}(x)-\kappa(x-1)$

the estimate

$\displaystyle\left|{\mathcal{R}}(x)\right|\leq Ax^{\theta}\quad(x\geq 1).$
(2)

It is clear that under Axiom A the Beurling zeta function

$\zeta(s):=\zeta_{{\mathcal{G}}}(s):=\int_{1}^{\infty}x^{-s}d{\mathcal{N}}(x)=\sum_{g\in{\mathcal{G}}}\frac{1}{|g|^{s}}$
(3)

admits a meromorphic, essentially analytic continuation
$\kappa\frac{1}{s-1}+\int_{1}^{\infty}x^{-s}d{\mathcal{R}}(x)$ up to $\Re
s>\theta$ with only one, simple pole at 1.

###  1.2 Analytic theory of the distribution of Beurling generalized primes.

The Beurling $\zeta$ function (3) is expressed by the generalized von Mangoldt
function

$\Lambda(g):=\Lambda_{{\mathcal{G}}}(g):=\begin{cases}\log|p|\quad\textrm{if}\quad
g=p^{k},\leavevmode\nobreak\ k\in{\mathbb{N}}\leavevmode\nobreak\
\leavevmode\nobreak\ \textrm{with some prime}\leavevmode\nobreak\
\leavevmode\nobreak\ p\in{\mathcal{G}}\\\ 0\quad\textrm{if}\quad
g\in{\mathcal{G}}\leavevmode\nobreak\ \leavevmode\nobreak\ \textrm{is not a
prime power in}\leavevmode\nobreak\ \leavevmode\nobreak\
{\mathcal{G}}\end{cases},$ (4)

as coefficients of the logarithmic derivative of the zeta function

$-\frac{\zeta^{\prime}}{\zeta}(s)=\sum_{g\in{\mathcal{G}}}\frac{\Lambda(g)}{|g|^{s}}.$
(5)

The Beurling theory of generalized primes is mainly concerned with the
analysis of the summatory function

$\psi(x):=\psi_{{\mathcal{G}}}(x):=\sum_{g\in{\mathcal{G}},\leavevmode\nobreak\
|g|\leq x}\Lambda(g).$ (6)

The generalized PNT (Prime Number Theorem) is the asymptotic equality
$\psi(x)\thicksim x$. The remainder term in this equivalence is denoted, as
usual,

$\Delta(x):=\Delta_{{\mathcal{G}}}(x):=\psi(x)-x.$ (7)

In the classical case of prime number distribution, as well as regarding some
extensions to primes in arithmetical progressions and distribution of prime
ideals in algebraic number fields, the connection between location and
distribution of zeta-zeroes and oscillatory behavior in the remainder term of
the prime number formula $\psi(x)\thicksim x$ is well understood [15, 17, 18,
20, 26, 27, 28, 29, 32, 33, 34, 35, 39, 40, 41, 42, 43, 44]. On the other hand
in the generality of Beurling primes and zeta function, investigations so far
were focused on mainly four directions. First, better and better, minimal
conditions were sought in order to have a Chebyshev type formula
$x\ll\psi(x)\ll x$, see e.g. [47, 48, 8, 9]. Understandably, as in the
classical case, this relation requires only an analysis of the $\zeta$
function of Beurling in, and on the boundary of the convergence halfplane.
Second, conditions for the prime number theorem to hold, were sought see e.g.
[3, 18, 5, 6, 10, 49, 50]. Again, this relies on the boundary behavior of
$\zeta$ on the one-line $\sigma=1$. Third, rough (as compared to our knowledge
in the prime number case) estimates and equivalences were worked out in the
analysis of the connection between $\zeta$-zero distribution and error term
behavior for $\psi(x)$ see e.g. [13], [25]. Fourth, examples were constructed
for arithmetical semigroups with very "regular" (such as satisfying the
Riemann Hypothesis RH and error estimates $\psi(x)=x+O(x^{1/2+\varepsilon})$)
and very "irregular" (such as having no better zero-free regions than (8)
below and no better asymptotic error estimates than (9)) behavior and zero- or
prime distribution, see e.g. [1], [4], [7], [13], [51]. Here we must point out
that the above citations are just examples, and are far from being a complete
description of the otherwise formidable literature222E.g. a natural, but
somewhat different direction, going back to Beurling himself, is the study of
analogous questions in case the assumption of Axiom A is weakened to e.g. an
asymptotic condition on ${\mathcal{N}}(x)$ with a product of $x$ and a sum of
powers of $\log x$, or sum of powers of $\log x$ perturbed by almost periodic
polynomials in $\log x$, or ${\mathcal{N}}(x)-cx$ periodic, see [3], [52],
[12], [36].. For a throughout analysis of these directions as well as for much
more information the reader may consult the monograph [11].

The main focus of our study, of which this present paper is a part only, is to
establish as precise as possible connections between distribution of the zeros
of the Beurling zeta function $\zeta$ on the one hand and order of magnitude
estimates or oscillatory properties of $\Delta(x)$ on the other hand.

Apart from generality and applicability to e.g. distribution of prime ideals
in number fields, the interest in these things were greatly boosted by a
construction of Diamond, Montgomery and Vorhauer [7]. They basically showed
that under Axiom A the Riemann hypothesis may still fail; moreover, nothing
better than the most classical zero-free region and error term [46] of

$\zeta(s)\neq 0\qquad\text{whenever}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ s=\sigma+it,\leavevmode\nobreak\ \leavevmode\nobreak\
\sigma>1-\frac{c}{\log t},$ (8)

and

$\psi(x)=x+O(x\exp(-c\sqrt{\log x})$ (9)

follows from (2) at least if $\theta>1/2$.

###  1.3 Aims and results of the paper.

The present paper is the second part of a series. In [37] we worked out in
detail a number of technical lemmas on the behavior of the Beurling zeta
function, and arrived at a Riemann-von Mangoldt type formula. Here we analyze
further the distribution of zeroes of the Beurling zeta function in the
critical strip $\theta<\Re s\leq 1$.

Our aim with this analysis is to lay the ground for the extension to the
Beurling case of a number of results of number theory nature. In Section 6 we
will briefly explain what concrete goals we have in mind, but we consider that
a lot of further number theory results–like e.g. estimates for primes in short
intervals etc.–become also accessible by use of the here presented information
on the distribution of the zeroes of the Beurling $\zeta$ function.

We start the present paper with a classical "Carlson type" density result.
Theorem 1 is somewhat surprising, because we lack a functional equation,
essential in the treatment of the Selberg class, where zero density estimates
are known to hold [16]. However, the functional equation is only used in the
Selberg class to estimate $\zeta$, and we succeed because similar estimates
can be derived directly from our conditions. A predecessor of our result was
worked out by Kahane [19], who proved that the number of Beurling zeta zeroes
lying on some vertical line $\Re s=\sigma=a$, where $a>\max(1/2,\theta)$, has
finite upper density. That is already a nontrivial fact333This particular
result enabled Kahane to draw deep number theoretical consequences regarding
the oscillation (sign changes) of the error term in the prime number formula.
Obviously, obtaining a much sharper result – estimating the total number of
zeroes in a full rectangle, not only on one individual vertical line, and with
a quantity essentially below the order of $T$ when $a$ is getting close to $1$
– provides an even stronger foothold for deriving number theoretical
consequences. because the total number of zeroes in the rectangle
$[a,1]\times[-iT,iT]$ may grow in the order $T\log T$ for any $a<1/2$.

Second, we present a Turán type local density estimate in case the Beurling
zeroes locally somewhat keep off the 1-line. In this we improve upon the local
precision allowing heights of rectangles in focus just being constants
(instead of the classical $\log\log|t|$).

Finally, the third result is a zero clustering estimate, providing an
improvement on the corresponding version of what Diamond, Montgomery and
Vorhauer proved and used in [7], which in itself was an improvement over the
weaker, averaged result in [23]. Our result is a better presented and
generalized variant of a classical, but obscurely written and thus seemingly
forgotten result of Ramachandra [31] for the Riemann zeta case.

The very fact that the proofs of these go through in this generality is
somewhat surprising. Moreover, the last two of them contain some refinements
even for the Riemann zeta function, so that they bear some novelty, however
minor, even in the most classical case.

For deriving the below density theorem we will use the following two
additional assumptions, both quite frequent and general, but still forming
some restrictions to our general treatment. One very generally used condition
is that the norm would actually map to the natural integers. In cases of
counting type problems, as well as e.g. for algebraic number fields where
certain indices are used as norms (equivalence classes modulo an ideal, e.g.),
this is all self-evident.

######  Definition 2 (Condition B).

We say that _Condition B_ is satisfied, if
$|\cdot|:{\mathcal{G}}\to{\mathbb{N}}$, that is, the norm $|g|$ of any element
$g\in{\mathcal{G}}$ is a natural number.

As is natural, we will write $\nu\in|{\mathcal{G}}|$ if there exists
$g\in{\mathcal{G}}$ with $|g|=\nu$. Under Condition B we can introduce the
arithmetical function $G(\nu):=\sum_{g\in{\mathcal{G}},\leavevmode\nobreak\
|g|=\nu}1$, which is then a super-multiplicative arithmetical function on
${\mathbb{N}}$. The next condition is a kind of "average Ramanujan condition"
for the Beurling zeta function.

######  Definition 3 (Condition G).

We say that _Condition G_ is satisfied, if with a certain $p>1$ we have for
the function

$F_{p}(X):=\frac{1}{X}\sum_{g\in{\mathcal{G}};|g|\leq
X}G(|g|)^{p}=\frac{1}{X}\sum_{\nu\in|{\mathcal{G}}|;\nu\leq
X}G(\nu)^{1+p}=\frac{1}{X}\int_{1}^{X}G^{p}(x)d{\mathcal{N}}(x)$ (10)

the property that

$\log F_{p}(X)=o(\log X)\qquad(X\to\infty),$ (11)

that is, for any fixed $\varepsilon>0$ $F_{p}(X)=O(X^{\varepsilon})$.

Note that in case $\log G(\nu)=o(\log\nu)$, i.e. when for all
${\varepsilon}>0$ we have $G(\nu)=O(\nu^{{\varepsilon}})$, then Condition G is
automatically satisfied for all $0<p<\infty$. Even this stronger
$O(\nu^{{\varepsilon}})$ order estimate is proved for many important cases,
see e.g. V.2.4. Theorem and V.2.5. Corollary of [21]. For some discussion of
these conditions see §3.

We denote the number of zeroes of the Beurling zeta function in
$[b,1]\times[-iT,iT]$ as

$N(b,T):=\\#\\{\rho=\beta+i\gamma\leavevmode\nobreak\ :\leavevmode\nobreak\
\zeta(\rho)=0,\,\beta\geq b,|\gamma|\leq T\\}.$ (12)

The main result of the paper is the following classical style density estimate
for the distribution of the zeros of the Beurling zeta function.

######  Theorem 1.

Assume that ${\mathcal{G}}$ satisfies besides Axiom A also Conditions B and G,
too. Then for any $\varepsilon>0$ there exists a constant
$C=C(\varepsilon,{\mathcal{G}})$ such that for all sufficiently large $T$ and
$\alpha>(1+\theta)/2$ we have

$N(\alpha,T)\leq CT^{\frac{6-2\theta}{1-\theta}(1-\alpha)+{\varepsilon}}.$
(13)

Note that according to the quite standard Lemma 5 below,
$N(\alpha,T)=O(T^{1+{\varepsilon}})$ for $\alpha>\theta$, always. Thus the
statement is nontrivial only if $\alpha$ is close to $1$, more precisely when
$\alpha>\frac{5-\theta}{6-2\theta}$.

For the formulation of further two results see Sections 4 and 5. These
additional results rely solely on Axiom A, and do not use the additional
assumptions of Conditions B and G. However, the density estimate above can
still be considered the main result of the paper in view of the wide range of
prospective applicability in further studies of number theory nature.

The structure of the paper is as follows. In [37], considered Part I of the
series, we described a number of auxiliary results, estimations and formulae
for the Beurling zeta function. For the convenience of the reader those also
in use here will be presented without proofs in the next section §2. In §3 we
prove Theorem 1. In §4 the reader finds a generalized (and at the same time
somewhat sharpened) version of the Turán type local density estimate mentioned
above. Finally, in §5 we prove a streamlined and generalized version of the
above mentioned zero clustering result of Ramachandra.

##  2 Lemmata on the Beurling $\zeta$ function

###  2.1 Basic properties of the Beurling $\zeta$.

The following basic lemmas are just slightly more explicit forms of 4.2.6.
Proposition, 4.2.8. Proposition and 4.2.10. Corollary of [21]. In [37] we
elaborated on their proofs for explicit handling of the arising constants in
these estimates.

######  Lemma 1.

Denote the "partial sums" (partial Laplace transforms) of
${\mathcal{N}}|_{[1,X]}$ as $\zeta_{X}$ for arbitrary $X\geq 1$:

$\zeta_{X}(s):=\int_{1}^{X}x^{-s}d{\mathcal{N}}(x).$ (14)

Then $\zeta_{X}(s)$ is an entire function and for $\sigma:=\Re s>\theta$ it
admits

$\zeta_{X}(s)=\begin{cases}\zeta(s)-\frac{\kappa
X^{1-s}}{s-1}-\int_{X}^{\infty}x^{-s}d{\mathcal{R}}(x)&\textrm{for
all}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ s\neq 1,\\\
\frac{\kappa}{s-1}-\frac{\kappa
X^{1-s}}{s-1}+\int_{1}^{X}x^{-s}d{\mathcal{R}}(x)&\textrm{for
all}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ s\neq 1,\\\
\kappa\log
X+\int_{1}^{X}\frac{d{\mathcal{R}}(x)}{x}&\textrm{for}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ s=1,\end{cases}$ (15)

together with the estimate

$\left|\zeta_{X}(s)\right|\leq\zeta_{X}(\sigma)\leq\begin{cases}\min\left(\frac{\kappa
X^{1-\sigma}}{1-\sigma}+\frac{A}{\sigma-\theta},\leavevmode\nobreak\ \kappa
X^{1-\sigma}\log
X+\frac{A}{\sigma-\theta}\right)&\textrm{if}\quad\theta<\sigma<1,\\\
\kappa\log X+\frac{A}{1-\theta}&\textrm{if}\qquad\sigma=1,\\\
\min\left(\frac{\sigma(A+\kappa)}{\sigma-1},\leavevmode\nobreak\ {\kappa}\log
X+\frac{\sigma A}{\sigma-\theta}\right)&\textrm{if}\quad\sigma>1.\end{cases}$
(16)

Moreover, the above remainder terms can be bounded as follows.

$\left|\int_{X}^{\infty}x^{-s}d{\mathcal{R}}(x)\right|\leq
A\frac{|s|+\sigma-\theta}{\sigma-\theta}X^{\theta-\sigma},$ (17)

and

$\left|\int_{1}^{X}x^{-s}d{\mathcal{R}}(x)\right|\leq
A\left(|s|\frac{1-X^{\theta-\sigma}}{\sigma-\theta}+X^{\theta-\sigma}\right)\leq
A\min\left(\frac{|s|}{\sigma-\theta},\leavevmode\nobreak\ |s|\log
X+X^{\theta-\sigma}\right).$ (18)

######  Lemma 2.

We have

$\left|\zeta(s)-\frac{\kappa}{s-1}\right|\leq\frac{A|s|}{\sigma-\theta}\qquad\qquad\qquad(\theta<\sigma,\leavevmode\nobreak\
t\in{\mathbb{R}},\leavevmode\nobreak\ \leavevmode\nobreak\ s\neq 1).$ (19)

In particular, for large enough values of $t$ it holds

$\left|\zeta(s)\right|\leq\sqrt{2}\frac{(A+\kappa)|t|}{\sigma-\theta}\qquad\qquad\qquad(\theta<\sigma\leq|t|),$
(20)

while for small values of $t$ we have

$|\zeta(s)(s-1)-\kappa|\leq\frac{A|s||s-1|}{\sigma-\theta}\leq\frac{100A}{\sigma-\theta}\qquad(\theta<\sigma\leq
4,\leavevmode\nobreak\ |t|\leq 9).$ (21)

As a consequence, we also have

$\zeta(s)\neq
0\qquad\textrm{for}\qquad|s-1|\leq\frac{\kappa(1-\theta)}{A+\kappa}.$ (22)

######  Lemma 3.

We have

$|\zeta(s)|\leq\frac{(A+\kappa)\sigma}{\sigma-1}\qquad(\sigma>1),$ (23)

and also

$|\zeta(s)|\geq\frac{1}{\zeta(\sigma)}>\frac{\sigma-1}{(A+\kappa)\sigma}\qquad(\sigma>1).$
(24)

###  2.2 Estimates for the number of zeros of $\zeta$

######  Lemma 4.

Let $\theta<b<1$ and consider the rectangle $H:=[b,1]\times[i(T-h),i(T+h)]$,
where $h:=\frac{\sqrt{7}}{3}\sqrt{(b-\theta)(1-\theta)}$ and $|T|\geq
e^{5/4}+\sqrt{3}\approx 5.222...$ is arbitrary.

Then the number $n(H)$ of zeta-zeros in the rectangle $H$ satisfy

$\displaystyle n(H)$
$\displaystyle\leq\frac{1-\theta}{b-\theta}\left(0.654\log|T|+\log\log|T|+6\log(A+\kappa)+6\log\frac{1}{1-\theta}+12.5\right)$
$\displaystyle\leq\frac{1-\theta}{b-\theta}\left(\log|T|+6\log(A+\kappa)+6\log\frac{1}{1-\theta}+12.5\right).$
(25)

Moreover, if $|T|\leq 5.23$, then we have analogously the $\log|T|$-free
estimate

$n(H)\leq\frac{1-\theta}{b-\theta}\left(6\log(A+\kappa)+6\log\frac{1}{1-\theta}+14\right).$
(26)

######  Remark 1.

In fact, this estimate includes also the total number $N$ of zeroes in the
disc $\mathcal{D}_{r}:=\\{s\leavevmode\nobreak\ :\leavevmode\nobreak\
|s-(p+iT)|\leq r:=p-q\\}$, where $p:=1+(1-\theta)$ and
$q:=\theta+\frac{2}{3}(b-\theta)$ are parameters introduced in its proof, see
[37].

######  Lemma 5.

Let $\theta<b<1$ and consider any height $T\geq 5$ together with the rectangle
$Q:=Q(b;T):=\\{z\in{\mathbb{C}}\leavevmode\nobreak\ :\leavevmode\nobreak\ \Re
z\in[b,1],\leavevmode\nobreak\ \Im z\in[-T,T]\\}$. Then the number of zeta-
zeros $N(b,T)$ in the rectangle $Q$ satisfy

$N(b,T)\leq\frac{1}{b-\theta}\left\\{\frac{1}{2}T\log
T+\left(2\log(A+\kappa)+\log\frac{1}{b-\theta}+3\right)T\right\\}.$ (27)

######  Lemma 6.

Let $\theta<b<1$ and consider any heights $T>R\geq 5$ together with the
rectangle $Q:=Q(b;R,T):=\\{z\in{\mathbb{C}}\leavevmode\nobreak\
:\leavevmode\nobreak\ \Re z\in[b,1],\leavevmode\nobreak\ \Im z\in(R,T]\\}$.

Then the number of zeta-zeros $N(b;R,T)$ in the rectangle $Q$ satisfies444Here
and below in (29) a corrected formulation is presented. In the original
calculation of [37], when collecting terms in the end of the proof of (3.15),
the term $8\pi\log T$ heading the third line of the long displayed formula
occupying lines 13-17 of page 220 of [37], was erroneously neglected in the
final count of the last line of this displayed sequence of inequalities.
Following up the consequent changes, the resulting concrete estimations are
the ones written here.

$N(b;R,T)\leq\frac{1}{b-\theta}\left\\{\frac{4}{3\pi}(T-R)\left(\log\left(\frac{11.4(A+\kappa)^{2}}{b-\theta}{T}\right)\right)+\frac{16}{3}\log\left(\frac{60(A+\kappa)^{2}}{b-\theta}{T}\right)\right\\}.$
(28)

In particular, for the zeroes between $T-1$ and $T+1$ we have for $T\geq 6$

$\displaystyle N(b;T-1,T+1)\leq\frac{1}{(b-\theta)}\left\\{6.2\log
T+6.2\log\left(\frac{(A+\kappa)^{2}}{b-\theta}\right)+24\right\\}.$ (29)

###  2.3 The logarithmic derivative of the Beurling $\zeta$

######  Lemma 7.

Let $z=a+it_{0}$ with $|t_{0}|\geq e^{5/4}+\sqrt{3}=5.222\ldots$ and
$\theta<a\leq 1$. With $\delta:=(a-\theta)/3$ denote by $S$ the (multi)set of
the $\zeta$-zeroes (listed according to multiplicity) not farther from $z$
than $\delta$. Then we have

$\displaystyle\left|\frac{\zeta^{\prime}}{\zeta}(z)-\sum_{\rho\in
S}\frac{1}{z-\rho}\right|$
$\displaystyle<\frac{9(1-\theta)}{(a-\theta)^{2}}\left(22.5+14\log(A+\kappa)+14\log\frac{1}{a-\theta}+5\log|t_{0}|\right).$
(30)

Furthermore, for $0\leq|t_{0}|\leq 5.23$ an analogous estimate (without any
term containing $\log|t_{0}|$) holds true:

$\left|\frac{\zeta^{\prime}}{\zeta}(z)+\frac{1}{z-1}-\sum_{\rho\in
S}\frac{1}{z-\rho}\right|\leq\frac{9(1-\theta)}{(a-\theta)^{2}}\left(34+14\log(A+\kappa)+18\log\frac{1}{a-\theta}\right).$
(31)

######  Lemma 8.

For any given parameter $\theta<b<1$, and for any finite and symmetric to zero
set ${\mathcal{A}}\subset[-iB,iB]$ of cardinality $\\#{\mathcal{A}}=n$, there
exists a broken line $\Gamma=\Gamma_{b}^{{\mathcal{A}}}$, symmetric to the
real axis and consisting of horizontal and vertical line segments only, so
that its upper half is

$\Gamma_{+}=\bigcup_{k=1}^{\infty}\\{[\sigma_{k-1}+it_{k-1},\sigma_{k-1}+it_{k}]\cup[\sigma_{k-1}+it_{k},\sigma_{k}+it_{k}]\\},$

with $\sigma_{j}\in[\frac{b+\theta}{2},b]$, ($j\in{\mathbb{N}}$), $t_{0}=0$,
$t_{1}\in[4,5]$ and $t_{j}\in[t_{j-1}+1,t_{j-1}+2]$ $(j\geq 2)$ and satisfying
that the distance of any ${\mathcal{A}}$-translate
$\rho+i\alpha\leavevmode\nobreak\ (i\alpha\in{\mathcal{A}})$ of a $\zeta$-zero
$\rho$ from any point $s=\sigma+it\in\Gamma$ is at least
$d:=d(t):=d(b,\theta,n,B;t)$ with555As is mentioned in the footnote there, in
Lemma 6 a slight correction of the formulation was due, entailing some
corresponding corrections also here and in the next lemma. This involves only
the values of the numerical constants, which–as long as they are some
effective constants anyway–bear no further importance for us here.

$d(t):=\frac{(b-\theta)^{2}}{4n\left(12\log(|t|+B+5)+51\log(A+\kappa)+31\log\frac{1}{b-\theta}+113\right)}.$
(32)

Moreover, the same separation from translates of $\zeta$-zeros holds also for
the whole horizontal line segments
$H_{k}:=[\frac{b+\theta}{2}+it_{k},2+it_{k}]$, $k=1,\dots,\infty$, and their
reflections $\overline{H_{k}}:=[\frac{b+\theta}{2}-it_{k},2-it_{k}]$,
$k=1,\dots,\infty$, and furthermore the same separation holds from the
translated singularity points $1+i{\alpha}$ of $\zeta$, too.

######  Lemma 9.

For any $0<\theta<b<1$ and symmetric to ${\mathbb{R}}$ translation set
${\mathcal{A}}\subset[-iB,iB]$, on the broken line
$\Gamma=\Gamma_{b}^{{\mathcal{A}}}$, constructed in the above Lemma 8, as well
as on the horizontal line segments $H_{k}:=[a+it_{k},2+it_{k}]$ and
$\overline{H_{k}}$, $k=1,\dots,\infty$ with $a:=\frac{b+\theta}{2}$, we have
uniformly for all $\alpha\in{\mathcal{A}}$

$\left|\frac{\zeta^{\prime}}{\zeta}(s+i\alpha)\right|\leq
n\frac{1-\theta}{(b-\theta)^{3}}\left(10\log(|t|+B+5)+60\log(A+\kappa)+42\log\frac{1}{b-\theta}+140\right)^{2}.$
(33)

###  2.4 Riemann-von Mangoldt type formulae of prime distribution with zeroes
of the Beurling $\zeta$

We denote the set of $\zeta$-zeroes, lying to the right of $\Gamma$, by
${\mathcal{Z}}(\Gamma)$, and denote ${\mathcal{Z}}(\Gamma,T)$ the set of those
zeroes $\rho=\beta+i\gamma\in{\mathcal{Z}}(\Gamma)$ which satisfy
$|\gamma|\leq T$. The next statement is Theorem 5.1 from [37].

######  Lemma 10 (Riemann–von Mangoldt formula).

Let $\theta<b<1$ and $\Gamma=\Gamma_{b}^{\\{0\\}}$ be the curve defined in
Lemma 8 for the one-element set ${\mathcal{A}}:=\\{0\\}$ with $t_{k}$ denoting
the corresponding set of abscissae in the construction. Then for any
$k=1,2,\ldots$, and $4\leq t_{k}<x$ we have

$\psi(x)=x-\sum_{\rho\in{\mathcal{Z}}(\Gamma,t_{k})}\frac{x^{\rho}}{\rho}+O\left(\frac{1-\theta}{(b-\theta)^{3}}\left(A+\kappa+\log\frac{x}{b-\theta}\right)^{3}\big{(}\frac{x}{t_{k}}+x^{b}\big{)}\right).$

Here let us call attention to the regrettable fact that in [37] a dumb error
occurred in the last line of the proof of Theorem 5.1 (which is in fact the
very last line of the whole paper, too). Namely, $x/t_{k}$ was erroneously
estimated by $4x^{b}$. The formulation here is thus also a correction666Note
that in formula (5.16) of [37] $\log(t_{k})$ is superfluous in the right hand
side, because it was already incorporated into the preceding log power. of the
mistake made in the formulation of Theorem 5.1 in [37] where instead of the
correct last factor $\big{(}\frac{x}{t_{k}}+x^{b}\big{)}$, only $x^{b}$ was
put down.

##  3 A density theorem for $\zeta$-zeros close to the $1$-line

###  3.1 A discussion of condition G

Recall that we have introduced Condition B and Condition G for use in this
section.

There are many natural examples where Condition G is met. E.g. if
${\mathcal{G}}$ is the ideal ring of an algebraic number field ${\mathbb{K}}$,
then a well-known result, see e.g. Lemma 4.9. on p. 143 of [24], provides the
estimate $G(m)\leq d_{n}(m)=O(m^{\varepsilon})$ for all ${\varepsilon}>0$
(where $d_{n}(m)$ is the classical $n$-term divisor function and $n$ is the
degree of the algebraic number field ${\mathbb{K}}$ in question). It is clear
that in case $G(\nu)=O(\nu^{{\varepsilon}})$ also Condition G must hold.
Actually, $d_{n}(m)\leq d^{n-1}(m)$ and by well-known number theory we also
have

$\sum_{m<X}d^{q}(m)\sim C_{q}X\log^{{2^{q}}-1}X\qquad(X\to\infty),$

so that Condition G holds for all exponent $p$.

More generally, let $\mathfrak{A}$ denote the category of all finite abelian
groups, $\mathfrak{S}$ be the category of all semisimple associative rings of
finite cardinal, and $\mathfrak{F}$ be the category of all finitely generated
torsion modules over the ring $D$ of all algebraic integers in some given
algebraic number field ${\mathbb{K}}$. One can also consider
${\mathfrak{F}}^{(\langle k\rangle)}$, for any given finite or infinite
sequence $\langle k\rangle=k_{1},\dots,k_{n},\dots$, the category of all
modules $M\in\mathfrak{F}$ such that every indecomposable direct summand of
$M$ is isomorphic to a cyclic module of the form $D/P^{r}$, where $P$ is a
prime ideal in $D$ and $r=k_{i}$ for some $i$, see [21, p. 117]. These
structures contain, as sub-semigroups, many other important arithmetical
categories like semisimple finite dimensional associative algebras over a
given field, certain Galois fields, etc.: see [21, Ch. 1] for details, in
particular page 16-21 for more detailed description of these and related
structures.

For abelian groups of finite order, the counting function
$G_{\mathfrak{A}}(m)$ has, by [21, V.1.10 Corollary], an asymptotical $k^{\rm
th}$ moment for every $k\in{\mathbb{N}}$, which is a strong form of the above
Condition G: the $o(\log X)$ function is just $C+o(1)$. In $\mathfrak{S}$ the
counting function $G_{\mathfrak{S}}(n)$ also has, by [21, V.1.13 Theorem], an
asymptotical $k^{\rm th}$ moment for every $k\in{\mathbb{N}}$, implying again
the strong form of Condition G. For $\mathfrak{F}$ this is found in [21,
V.1.9. Theorem], too. (The reason of this is the intimate relationship of the
value of $G$ on prime powers $p^{r}$ with the partition function $p(r)$, see
p. 124 of [21]).

Note that these categories (and many others) all satisfy Axiom A, see [21], p.
16-20 and 120-121, e.g.

Let us recall the following facts from the theory of arithmetical semigroups
(see [21], IV.4.1. Proposition and V.2.9. Theorem.)

######  Lemma 11 (Knopfmacher).

Let ${\mathcal{G}}$ be an arithmetical semigroup satisfying Axiom A. Then for
the divisor function $d(g)$ on ${\mathcal{G}}$ we have with any
$k\in{\mathbb{N}}$ the asymptotic equivalence formula

$\sum_{g\in{\mathcal{G}};|g|<X}d^{k}(g)\sim
A_{0}X\log^{2^{k}-1}X\qquad(X\to\infty),$ (34)

with $A_{0}=A_{0}({\mathcal{G}},k)$ a nonzero constant. We also have

$\limsup_{|g|\to\infty}\frac{\log d(g)\log\log|g|}{\log|g|}=2,$ (35)

so $\log d(g)=o(\log|g|)$, too.

Let us note that Knopfmacher uses the notation $B_{k}$ for the constant
$A_{0}({\mathcal{G}},k)$ occurring in formula (34), and gives the semi-
explicit expression
$A_{0}=\frac{\kappa^{2^{k}}}{(2^{k}-1)!}\widetilde{g_{k}}(1)$ with
$\widetilde{g_{k}}$ defined through the process of the proof of IV.3.7.
Proposition of [21]. For a further discussion on the value of the constant see
[37].

###  3.2 Proof of Theorem 1

###### Proof.

We apply the by now standard treatment of zero-detecting sums and large sieve
type estimates. The derivation here follows the relatively simple,
straightforward argument in [28], see also [30].

Let $Y$ be a large parameter, so that $T^{3}<Y<T^{\frac{3}{1-\theta}}$. We
define the arithmetical function $a(g):=a_{T}(g)$ on ${\mathcal{G}}$ as

$a(g):=\sum_{h|g;\leavevmode\nobreak\ |h|\leq T}\mu(h).$

Clearly, $a(1)=1$ and for $1<|g|\leq T$ we have $a(g)=0$: for other values of
$g$ we have $|a(g)|\leq d(g)$. For any complex number $z\in{\mathbb{C}}$ put

$H:=H(z):=H(T,Y,z):=\sum_{|g|<Y}\frac{a(g)}{|g|^{z}}=1+\sum_{T<|g|<Y}\frac{a(g)}{|g|^{z}}$
(36)

By Lemma 5, (27) we already know about the number of zeroes that
$N(\alpha,X)\ll\frac{1}{1-\theta}X\log\frac{X}{1-\theta}$, so using also
$\alpha>\frac{1+\theta}{2}$ and choosing $X:=\log^{2}T\leavevmode\nobreak\
Y^{1-\alpha}$ the number of zeroes below the height $X$ can be estimated as

$\displaystyle N(\alpha,X)\ll\left(\frac{1}{1-\theta}\log
T\right)^{3}\,Y^{1-\alpha}.$ (37)

On the other hand for the zeroes counted in $N(\alpha,T)-N(\alpha,X)$, we
select a separated "one-covering", i.e. we take (one) zero
$\rho_{1}=\beta_{1}+i\gamma_{1}$ with $\beta_{1}>\alpha$, $\gamma_{1}\geq X$
and of minimal $\gamma_{1}$, and then inductively, after $\rho_{j}$ has
already been selected, we take the next $\rho_{j+1}=\beta_{j+1}+i\gamma_{j+1}$
with minimal $\gamma_{j+1}\geq\gamma_{j}+1$ but remaining $\leq T$. Clearly
$\gamma_{j}+1\leq\gamma_{j+1}$ entails that the set
${\mathcal{Z}}:=\\{\rho_{j}\\}$ of all the zeroes selected is finite: let its
number of elements be $Z:=\\#{\mathcal{Z}}$, say.

By construction, the imaginary part of any $\zeta$-zero in the rectangle
$[\alpha,1]\times[X,T]$ is within 1 to some element $\rho_{j}$ of the set
${\mathcal{Z}}$, so by Lemma 6 (29) and taking into account
$\alpha>\frac{1+\theta}{2}$ and the symmetry of the zeroes with respect to the
real line, too, we obtain with some implied explicit constant
$A_{2}:=A_{2}(\theta,A,\kappa)$ and for sufficiently large $T$ and $X$ the
estimate

$N({\alpha},T)-N({\alpha},X)\ll Z\log T.$ (38)

Finally, we are to estimate $\\#{\mathcal{Z}}=Z$. Firstly, for any
$\rho=\rho_{j}\in{\mathcal{Z}}$, we can write

$\displaystyle H:=H(T,Y,\rho_{j})$ $\displaystyle=\sum_{|g|\leq
Y}\frac{1}{|g|^{\rho}}\sum_{h|g\leavevmode\nobreak\ |h|\leq
T}\mu(h)=\sum_{|h|\leq T}\frac{\mu(h)}{|h|^{\rho}}\sum_{f\leavevmode\nobreak\
|f|\leq Y/|h|}\frac{1}{|f|^{\rho}},$ $\displaystyle|H|$
$\displaystyle\leq\sum_{|h|\leq
T}\frac{1}{|h|^{\beta}}\left|\zeta_{Y/|h|}(\rho)\right|.$

Here using $\zeta(\rho)=0$, too, the inner expression can be estimated by
Lemma 1, (15) (first line) and (17). We gain

$\left|\zeta_{Y/|h|}(\rho)\right|\leq\frac{\kappa\left(\frac{Y}{|h|}\right)^{1-\beta}}{|\rho-1|}+A\left(\frac{Y}{|h|}\right)^{\theta-\beta}\left(\frac{|\rho|}{\beta-\theta}+1\right)\leq\frac{\kappa}{\gamma}\left(\frac{Y}{|h|}\right)^{1-\beta}+\frac{4A\gamma}{1-\theta}\left(\frac{Y}{|h|}\right)^{\theta-\beta}.$

Applying this in the above estimation of $H$ we are led to

$|H|\leq\frac{\kappa Y^{1-\beta}}{\gamma}\sum_{|h|\leq
T}\frac{1}{|h|}+\frac{4A\gamma}{1-\theta}Y^{\theta-\beta}\sum_{|h|\leq
T}\frac{1}{|h|^{\theta}}=\frac{\kappa
Y^{1-\beta}}{\gamma}\zeta_{T}(1)+\frac{4A\gamma}{1-\theta}\gamma
Y^{\theta-\beta}\zeta_{T}(\theta).$

From the second line of (15) and the second part of (18) in Lemma 1 we get

$|\zeta_{T}(\theta)|\leq\frac{\kappa
T^{1-\theta}}{1-\theta}+\left|\int_{1}^{T}x^{-\theta}d{\mathcal{R}}(x)\right|\leq\frac{\kappa
T^{1-\theta}}{1-\theta}+A+A\theta\log
T\leq\frac{A+\kappa}{1-\theta}T^{1-\theta}\log T,$

using in the end $T>\exp(\frac{1}{1-\theta})$; and from Lemma 1, (16), second
line we obtain

$\zeta_{T}(1)\leq\frac{A+\kappa}{1-\theta}\log T.$

Inserting these into the last estimation of $H$ leads to

$\displaystyle|H|$ $\displaystyle\leq\frac{\kappa(A+\kappa)\log
T\,Y^{1-\beta}}{(1-\theta)\gamma}+\left(\frac{4A(A+\kappa)}{(1-\theta)^{2}}\right)\gamma\log
T\,T^{1-\theta}Y^{\theta-\beta}$ $\displaystyle\leq(4A+\kappa)(A+\kappa)\log
T\left(\frac{Y^{1-\beta}}{(1-\theta)\gamma}+\frac{\gamma
Y^{\theta-\beta}T^{1-\theta}}{(1-\theta)^{2}}\right).$

Assuming now that

$Y\geq
Y_{0}(T,\theta):=\left(\frac{1}{1-\theta}T^{3-\theta}\right)^{\frac{1}{1-\theta}},$
(39)

the second term is always below the first in view of $|\gamma|\leq T$.
Furthermore, $X\leq\gamma$, so writing in $X:=\log^{2}T\leavevmode\nobreak\
Y^{1-\alpha}$ in place of $\gamma$ we infer

$H\leq 8(A+\kappa)^{2}\log
T\frac{Y^{1-\beta}}{(1-\theta)\gamma}\leq\frac{8(A+\kappa)^{2}}{(1-\theta)\log
T}Y^{\alpha-\beta}\leq\frac{8(A+\kappa)^{2}}{(1-\theta)\log T}<\frac{1}{2},$

if $T>\exp\left(\frac{16(A+\kappa)^{2}}{1-\theta}\right)$. Therefore, in (36)
the first constant $1$ is dominating, and via $|H-1|>1/2$ we obtain

$\displaystyle\frac{1}{4}\cdot Z$
$\displaystyle\leq\sum_{j=1}^{Z}|H(\rho_{j})-1|^{2}=\sum_{j=1}^{Z}\left|\sum_{T<|g|<Y}\frac{a(g)}{|g|^{\rho_{j}}}\right|^{2}$
(40)
$\displaystyle\leq\sum_{j=1}^{Z}\left\\{\sum_{k=0}^{[\log(Y/T)]}\left|\sum_{e^{k}T<m\leq\min\\{e^{k+1}T,Y\\}}\frac{\left(\sum_{g\leavevmode\nobreak\
:\leavevmode\nobreak\ |g|=m}a(g)\right)}{m^{\rho_{j}}}\right|\right\\}^{2}$
$\displaystyle\leq\left(\log\left(\frac{Y}{T}\right)+1\right)\sum_{k=0}^{[\log(Y/T)]}\sum_{j=1}^{Z}\left|\sum_{M_{k}<m\leq
N_{k}}\frac{\left(\sum_{g\leavevmode\nobreak\ :\leavevmode\nobreak\
|g|=m}a(g)\right)}{m^{\rho_{j}}}\right|^{2},$

with some appropriate $T\leq M_{k}<N_{k}<eM_{k}$, $N_{k}\leq Y$. First we want
to estimate the coefficients of the Dirichlet series, so let us write

$F(m):=\sum_{g\leavevmode\nobreak\ :\leavevmode\nobreak\ |g|=m}a(g)$ (41)

and use Cauchy’s inequality and the trivial upper estimate $|a(g)|\leq d(g)$
to obtain

$\displaystyle|F(m)|^{2}$ $\displaystyle\leq\sum_{g\leavevmode\nobreak\
:\leavevmode\nobreak\ |g|=m}1\cdot\sum_{g\leavevmode\nobreak\
:\leavevmode\nobreak\ |g|=m}a^{2}(g)=G(m)\sum_{g\leavevmode\nobreak\
:\leavevmode\nobreak\ |g|=m}a^{2}(g)$ (42) $\displaystyle\leq
G(m)\sum_{g\leavevmode\nobreak\ :\leavevmode\nobreak\
|g|=m}d^{2}(g)=\sum_{g\leavevmode\nobreak\ :\leavevmode\nobreak\
|g|=m}G(|g|)d^{2}(g).$

Recall that the exponents $\rho_{j}$ in the inner Dirichlet polynomials of the
last double sum in (40) are all counted in $N(\alpha,T)$, hence
$\beta\geq{\alpha}$ and $|\gamma|\leq T$. By the large sieve type inequality
of [23, Theorem 7.5] and writing $M:=M_{k},N:=N_{k}$ here, we are led to

$\displaystyle\sum_{j=1}^{Z}\left|\sum_{M<m\leq
N}\frac{F(m)}{m^{\rho_{j}}}\right|^{2}$ $\displaystyle\ll(T+N)\log
N\sum_{m=M}^{N}\frac{F^{2}(m)}{m^{2\alpha}}\bigg{(}1+\log\frac{\log 2N}{\log
2m}\bigg{)}$ $\displaystyle\ll N\log
N\sum_{m=M}^{N}\frac{F^{2}(m)}{N^{2{\alpha}}}=\log N\leavevmode\nobreak\
N^{1-2{\alpha}}\sum_{n=1}^{N}F^{2}(m).$ (43)

Here we use (42) and apply Hölder’s inequality _while summing over elements
of_ ${\mathcal{G}}$ with some exponent $p$ satisfying Condition G. This yields
with $q:=\frac{p}{p-1}$

$\sum_{n=1}^{N}F^{2}(m)\leq\sum_{g\leavevmode\nobreak\ :\leavevmode\nobreak\
|g|\leq N}G(|g|)d^{2}(g)\leq\left(\sum_{g\leavevmode\nobreak\
:\leavevmode\nobreak\ |g|\leq
N}G(|g|)^{p}\right)^{1/p}\left(\sum_{g\leavevmode\nobreak\
:\leavevmode\nobreak\ |g|\leq N}d^{2q}(g)\right)^{1/q}.$ (44)

By Lemma 11, the second sum is $O(N\log^{C(q)}N)$, while the first sum is by
Condition G $\ll N^{1+{\varepsilon}}$. Collecting (40), (3.2) and (44) leads
to

$Z\ll\sum_{k=0}^{[\log(Y/T)]}\log Y\log N_{k}\leavevmode\nobreak\
N_{k}^{1-2{\alpha}}N_{k}^{(1+\varepsilon)/p}\left(N_{k}\log^{C(q)}N_{k}\right)^{1/q}\ll
Y^{2-2{\alpha}+{\varepsilon}}.$ (45)

At last, we can choose $Y$ the smallest possible, that is $Y:=Y_{0}$, to get

$Z\ll
Y_{0}^{2-2{\alpha}+{\varepsilon}}=\left(\frac{1}{(1-\theta)^{\frac{1}{1-\theta}}}T^{\frac{3-\theta}{1-\theta}}\right)^{2-2{\alpha}+{\varepsilon}}\ll
T^{\frac{6-2\theta}{1-\theta}\left\\{(1-{\alpha})+{\varepsilon}\right\\}}.$

To finish the proof we need only to combine this estimate with (37) and (38).
∎

The result thus shows that, e.g., the functional equation, so fundamentally
present in several approaches, is not necessary for a density theorem to hold.
On the other hand positivity of the coefficients of the Dirichlet series does
play a role here. In this respect Theorem 1 is complement to the similar
density theorem of Kaczorowsky and Perelli [16], where the Selberg class of
zeta functions are shown to admit such density estimates. It would be
interesting to analyze what essentially minimal set of properties, assumed on
the class of zeta functions considered, can still imply the validity of such
density estimates.

##  4 A Turán type local density theorem for $\zeta$-zeros close to the
boundary of the zero-free region

For arbitrary $\tau>0$ and $\theta<\sigma<1$, we denote the rectangle

$Q_{\sigma,h}(\tau):=[\sigma,1]\times[i(\tau-h),i(\tau+h)].$

######  Theorem 2.

Let $(1+\theta)/2<b\leq 1$, $2\leq h$ and $\tau>\max(2h,\tau_{0})$ where
$\tau_{0}=\tau_{0}(\theta,A,\kappa)$ is a large constant depending on the
given parameters of $\zeta(s)$.

Assume that $\zeta(s)$ does not vanish in the rectangle $\sigma\geq b$,
$|t-\tau|\leq h$, denoted by $Q_{b,h}(\tau)$, i.e. that
$N(b,\tau-h,\tau+h)=0$. Then for any $\delta$ with
$15\frac{\log\log\log\tau}{\log\log\tau}<\delta<\frac{b-\theta}{10}$ we have

$M:=N(b-\delta,\tau-\delta,\tau+\delta)\ll\delta\log\tau,$

with an implied absolute constant not depending on ${\mathcal{G}}$.

The original work of Turán, aiming at "almost getting the density hypothesis",
involves the condition $h\gg\log\tau$. The present analysis reveals that this
is not necessary: we get the same result with any $h>2$ as well. Thus the
result - at least formally - gives something new even for the Riemann zeta
function.

In proving the result we follow closely the original work of Turán, cf. the
Appendix of [45].

######  Lemma 12.

Let the parameters $b,h,\tau$ be fixed as in the previous theorem, and assume
that $\tau$ is large enough. If $N(b,\tau-h,\tau+h)=0$ and
$\lambda:=5\log\log\log\tau/\log\log\tau$, then for all $s=\sigma+it$ with
$b+3\lambda\leq\sigma\leq 1.7$ and $|t-\tau|\leq h-1$ the inequality

$\left|\frac{\zeta^{\prime}}{\zeta}(s)\right|\leq\frac{\log\tau}{(\log\log\tau)^{2}}$
(46)

holds true.

###### Proof.

Let $g(s):=\log\left(\zeta(s)/\zeta(p+i\omega)\right)$, where
$\tau-h+1\leq\omega\leq\tau+h-1$ and $1<p<1+b$ is to be determined later. This
function is analytic in the disk $D:=\\{s\leavevmode\nobreak\
:\leavevmode\nobreak\ |s-(p+i\omega)|\leq R:=p-b<1\\}$, as $\zeta(s)$ does not
vanish (for $|t-\tau|\leq|\omega-\tau|+R\leq h-1+R\leq h$ and $\Re s\geq
p-R=b$). Furthermore, the real axis, (whence the pole of $\zeta(s)$ at $s=1$)
is at least as far from any point $s\in D$ as $\omega-R>\tau-h+1-R>\tau-h>h$.
Whence $g(s)\neq 0,\infty$ in $D$ and $\log g(s)$ is analytic in $D$.

Clearly, in the center of this disk $g(p+i\omega)=0$. For the upper estimation
of the real part of $g(s)$ along the circle, let us use (20) and (24), which
leads to

$\displaystyle\Re g(s)$
$\displaystyle\leq\log\left(\frac{\sqrt{2}(A+\kappa)}{b-\theta}\right)+\log(\omega+1)+\log\left(\frac{p(A+\kappa)}{p-1}\right)\log\left(\frac{5p(A+\kappa)^{2}}{(1-\theta)(p-1)}\right),$
(47)

using also that $b-\theta>\frac{1}{2}(1-\theta)$ and also that $\omega>2$,
say.

We will chose

$p:=\frac{3}{2},\qquad\Delta:=k\frac{\log\log\log\omega}{\log\log\omega}<0.1,\quad(4\leq
k\in{\mathbb{N}}).$

So let us assume that $\tau$ is large enough, say
$\tau>\tau_{1}:=30\frac{(A+\kappa)^{2}}{1-\theta}$: then by $\omega>\tau/2$ in
the last estimate of (47) the $\log\omega$ term dominates, and $\Re g(s)\leq
2\log\omega$. Let $R^{\prime}:=R-\Delta<R$: by the Borel-Carathéodory theorem,

$\max_{|s-(p+i\omega)|\leq
R^{\prime}}|g(s)|\leq\frac{2R}{\Delta}\,2\log\omega<\frac{4}{\Delta}\log\omega.$

Next we apply the three-circle theorem to the circles
$C_{1}:=\\{s\leavevmode\nobreak\ :\leavevmode\nobreak\ |s-(p+i\omega)|\leq
R^{\prime}\\}$, $C_{2}:=\\{s\leavevmode\nobreak\ :\leavevmode\nobreak\
|s-(p+i\omega)|\leq R"\\}$ with $R":=R-2\Delta$, and
$C_{3}:=\\{s\leavevmode\nobreak\ :\leavevmode\nobreak\ |s-(p+i\omega)|\leq
q\\}$, where now $q:=(p-1)/2=1/4$, so on $C_{3}$ by the estimates of Lemma 3
we have

$\max_{C_{3}}|g(s)|\leq\log\left(\frac{(A+\kappa)^{2}(p-q)^{2}}{(p-q-1)^{2}}\right)=\log\left(25(A+\kappa)^{2}\right).$

From the three-circle theorem we obtain

$\displaystyle\max_{|s-(p+i\omega)|\leq R"}$
$\displaystyle\log|g(s)|\leq\frac{\Delta}{R^{\prime}}\log\log\left(25(A+\kappa)^{2}\right)+\frac{R^{\prime}-\Delta}{R^{\prime}}\log\left(\frac{4}{\Delta}\log\omega\right)$
$\displaystyle=\frac{\Delta}{R^{\prime}}\log\left\\{\frac{\Delta}{4}\log\left(25(A+\kappa)^{2}\right)\right\\}+\log\frac{4}{\Delta}+\left(1-\frac{\Delta}{R^{\prime}}\right)\log\log\omega.$

We have $\omega>\tau-h+1>\tau/2$, whence for $\tau>\tau_{2}(A,\kappa)$ the
first expression is negative because $\Delta=\Delta(\omega)\to 0$ when
$\omega\to\infty$, entailing that eventually also
$(\Delta/4)\log(25(A+\kappa)^{2})$ becomes smaller than 1.

Moreover, the fraction $\Delta/R^{\prime}$ is at least $\Delta$, since
$R^{\prime}<R<1$. Also, for $\tau>2\exp(\exp(e))$ we have
$\omega>\exp(\exp(e))$, too; in this case we also have $4/\Delta\leq
4/k\leavevmode\nobreak\ \log\log\omega/\log\log\log\omega\leq\log\log\omega$.
Therefore, we end up with

$\displaystyle\max_{|s-(p+i\omega)|\leq R"}\log|g(s)|$
$\displaystyle\leq\log\frac{4}{\Delta}+\left(1-{\Delta}\right)\log\log\omega$
$\displaystyle\leq\log\log\log\omega+\log\log\omega-k\log\log\log\omega.$

Choosing $k=4$ we conclude

$\max_{|s-(p+i\omega)|\leq
R"}|g(s)|\leq\frac{\log\omega}{(\log\log\omega)^{3}}.$

Finally, we choose $r:=R-3\Delta=R"-\Delta$, and apply the standard Cauchy
estimate for the value of $|g^{\prime}(s)|$ for any $s$ in the disk
$|s-(p+i\omega)|\leq r$. We thus obtain

$|g^{\prime}(s)|\leq\frac{1}{\Delta}\frac{\log\omega}{(\log\log\omega)^{3}}=\frac{\log\omega}{4\log\log\log\omega(\log\log\omega)^{2}}<\frac{\log\tau}{(\log\log\tau)^{2}}.$

We have the same all over the area of these disks, and as $\tau$ is large
enough, and $k=4$ we have $\lambda>\Delta$. Hence for large enough $\tau$ the
horizontal diameter $[b+3\Delta+i\omega,2p-b-3\Delta+i\omega]$ of the disk
$|s-(p+i\omega)|\leq r$ covers $[b+3\lambda+i\omega,1.7+i\omega]$. Using this
for all choices of $\omega$ with $|\omega-\tau|\leq h-1$, we obtain that the
rectangle $[b+3\lambda,1.7]\times[\tau-h+1,\tau+h-1]i$ is contained in the
total area covered by the union of these disks.

But $g^{\prime}(s)=(\log\zeta(s))^{\prime}=\zeta^{\prime}(s)/{\zeta(s)}$, and
so at any point $s=\sigma+it$, $b+3\lambda\leq\sigma\leq 1.7$, $|t-\tau|\leq
h-1$ we thus have (46). ∎

###### Proof of Theorem 2.

Let $s=\sigma+it\in Q:=Q_{b+3\lambda,h-1}$ (with $\lambda$ as in Lemma 12) and
consider the number of zeroes $N$ in the disk $D(s,r)$ with
$r:=(\sigma-\theta)/3$. As $\theta<\sigma<1$, the disc
$K:=\\{z\leavevmode\nobreak\ :\leavevmode\nobreak\ |z-(1+(1-\theta)+it)|\leq
R:=2-2\theta-2(\sigma-\theta)/3\\}$ contains $D(s,r)$, whence by Remark 1
(applied to $\sigma$ in place of $b$) the number of $\zeta$-zeroes $N$ in
$D(s,r)$ can at most be as large as

$N\leq\frac{1-\theta}{\sigma-\theta}\left(A_{1}+\log
t\right)\leq\frac{1-\theta}{b-\theta}\left(A_{1}+\log t\right)\leq
2\left(A_{1}+\log t\right),$

with some constant $A_{1}=A_{1}(\theta,A,\kappa)$. Note that the same disk
$D(s,r)$ occurs in Lemma 7: hence we obtain now for the (multi)set $S$ of
zeroes within $D(s,r)$ that

$\Re\left\\{-\frac{\zeta^{\prime}}{\zeta}(s)+\sum_{\rho\in
S}\frac{1}{s-\rho}\right\\}\leq\left|\frac{\zeta^{\prime}}{\zeta}(s)-\sum_{\rho\in
S}\frac{1}{s-\rho}\right|\ll\left(A_{1}+\log t\right)\ll\log\tau,$ (48)

with $\tau$ (and whence $t$) large enough (e.g. for $\tau>2\exp(A_{1})$). The
real parts of the expressions with the zeroes in the sum (48) can be rewritten
for $\rho=\beta+i\gamma$ as

$\Re\frac{1}{s-\rho}=\frac{\sigma-\beta}{(\sigma-\beta)^{2}+(t-\gamma)^{2}},$

hence from (48) we obtain with a suitably large absolute constant $C$ and for
$\tau$ (and hence $t$) large enough

$\sum_{\rho\in
S}\frac{\sigma-\beta}{(\sigma-\beta)^{2}+(t-\gamma)^{2}}<\Re\frac{\zeta^{\prime}}{\zeta}(s)+C\log\tau.$
(49)

We choose now $s:=b+\delta+i\tau$, i.e. $\sigma=b+\delta$ and $t:=\tau$. By
assumption, we have $3\lambda<\delta$. That is,
$\sigma+it\in[b+3\lambda,1.7]\times[\tau-h+1,\tau+h-1]i$ and the estimations
of the previous Lemma 12 can be applied to bound the arising
$\zeta^{\prime}/\zeta(\sigma+it)$ in the right hand side. This yields

$\sum_{\rho\in
S}\frac{\sigma-\beta}{(\sigma-\beta)^{2}+(\tau-\gamma)^{2}}<\frac{\log\tau}{(\log\log\tau)^{2}}+C\log\tau\leq
2C\log\tau,$ (50)

the second term dominating.

It is clear that the terms in the sum are nonnegative, for we have assumed
that $Q_{b,h}(\tau)$ is zero-free, the radius of the disk $D(s,r)$ is less
than 1, and $|t-\tau|<h-1$. Therefore, any term can be dropped while
preserving the inequality.

Observe that $Q_{b-\delta,\delta}(\tau)\cap
S\subset[b-\delta,b]\times[\tau-\delta,\tau+\delta]i\subset D(s,r)$, so that
we can restrict the sum for summing over zeroes in
$Q_{b-\delta,\delta}(\tau)$. Therefore,

$\sum_{\rho\in
Q_{b-\delta,\delta}}\frac{\sigma-\beta}{(\sigma-\beta)^{2}+(\tau-\gamma)^{2}}<2C\log\tau.$

Now each term on the left hand side is at least $2\delta/5\delta^{2}$, hence
we arrive at

$M\frac{1}{\delta}\leq 5C\log\tau.$

The proof of the theorem concludes. ∎

##  5 Clustering of zeroes in the vicinity of the 1-line

In [7], Theorem 2 the authors prove that the zeroes $\rho=\beta+i\gamma$ of
the Beurling zeta function, close to the one-line in the sense that
$\beta>1-c/\sqrt{\log\gamma}$, show a phenomenon of clustering: they do not
occur in isolation, but instead once a zero $\rho_{0}$ occurs, there must be
further ones in the union of some small discs $D(1+i\gamma_{0},\lambda)$ and
$D(1+i2\gamma_{0},\lambda)$ around $1+i\gamma_{0}$ and $1+i2\gamma_{0}$.

This theorem is itself a sharpening of what was proved in the average by
Montgomery in his monograph [23] for the case of the Riemann zeta function.
Yet there is a less quoted, nevertheless sharper result, due to Ramachandra
[31], which provides similar clustering with closeness relaxed to
$1/\log\log\gamma$ only and still localizes to the small discs
$D(1+i\gamma_{0},\lambda)$ and $D(1+i2\gamma_{0},\lambda)$ around
$1+i\gamma_{0}$ and $1+i2\gamma_{0}$.

It is worthy to work out the result here not only for sake of generality but
also to give a somewhat more transparent deduction of the result. Indeed,
Ramachandra uses a positive trigonometric polynomial with obscure777It seems
that the reasons lie in the later, more general application of his method to
clustering around $1+i\gamma_{1}$ and $1+i\gamma_{2}$, when two close zeroes
are known at hight $\gamma_{1}$ and $\gamma_{2}$, respectively. coefficients
(like $10^{8}$). Here we analyze the method and show that the most common
$3+4\cos\theta+\cos 2\theta$ does the job as well.

Note that the further paper of Balasubramanian and Ramachandra [2], claiming
to achieve the even nicer localization of clustering right in
$D(1+i\gamma_{0},\lambda)$, contains a fatal error888Indeed, summarizing the
previous Lemmas 3 and 5 in Lemma 6 on page 11, the authors neglect a term
$-2X^{1-s}\Gamma(\frac{1-s}{2})$, which is a main term and destroys
everything., unfortunately.

######  Theorem 3.

Assume $\zeta(\rho_{0})=0$ with $\rho_{0}=\beta_{0}+i\gamma_{0}$,
$\beta_{0}>\max(\theta,0.999)$, $\gamma_{0}\geq 100$, and
$1-\beta_{0}<\frac{1-\theta}{40\log\log\gamma_{0}}$. Further, let the
parameter $0<\lambda\leq\frac{2}{3}(1-\theta)$ be arbitrary.

Then there exists an effective constant $A_{10}:=A_{10}(\theta,A,\kappa)$,
depending only on the parameters of the Beurling zeta function given in Axiom
A, so that with any value of the further parameter $Y$ satisfying
$Y>\max\left(A_{10},\frac{4}{1-\theta}\log\log\gamma_{0}\right)$ we have

$\sum_{\rho\in
D(1+i\gamma_{0},\lambda)\setminus\\{\rho_{0}\\}}e^{-Y(1-\beta)}+\sum_{\rho\in
D(1+i2\gamma_{0},\lambda)}e^{-Y(1-\beta)}\gg\frac{\lambda}{Y(1-\beta_{0})}-c_{0},$
(51)

with $c_{0}$, as well as the implied constant in $\gg$ being explicit absolute
constants.

###### Proof.

We will work with the kernel function

$K(w,x):=x^{w}\Gamma\left(\frac{w}{2}\right),$

satisfying the integral formula

$d(x,\nu):=\frac{1}{2\pi
i}\int\limits_{(2)}\frac{1}{\nu^{w}}x^{w}\Gamma\left(\frac{w}{2}\right)dw=2\exp\left(-\left(\frac{\nu}{x}\right)^{2}\right)\qquad\left(\nu\geq
1,\leavevmode\nobreak\ \nu\in{\mathbb{R}}\right),$

where, as usual, $\int_{(2)}=\int_{2-i\infty}^{2+i\infty}$. At the outset we
let $s\in{\mathbb{C}}$ be arbitrary with $\Re s>1$. We put

$f(s):=f_{\gamma_{0}}^{B}(s)f_{2\gamma_{0}}^{C}(s)\zeta^{-2D}(s),\qquad\textrm{with}\quad
f_{{\alpha}}(s):=\zeta^{2}(s)\zeta(s-i{\alpha})\zeta(s+i{\alpha}),$

so that

$\displaystyle\frac{f^{\prime}}{f}(s)$
$\displaystyle=B\frac{f^{\prime}_{\gamma_{0}}}{f_{\gamma_{0}}}(s)+C\frac{f^{\prime}_{2\gamma_{0}}}{f_{2\gamma_{0}}}(s)-2D\frac{\zeta^{\prime}}{\zeta}(s)=2(B+C-D)\frac{\zeta^{\prime}}{\zeta}(s)$
$\displaystyle\qquad\qquad+B\left(\frac{\zeta^{\prime}}{\zeta}(s-i\gamma_{0})+\frac{\zeta^{\prime}}{\zeta}(s+i\gamma_{0})\right)+C\left(\frac{\zeta^{\prime}}{\zeta}(s-2i\gamma_{0})+\frac{\zeta^{\prime}}{\zeta}(s+i2\gamma_{0})\right)$
$\displaystyle=-2\sum_{g\in{\mathcal{G}}}\frac{\Lambda(g)}{|g|^{s}}\left((B+C-D)+B\Re|g|^{i\gamma_{0}}+C\Re|g|^{i2\gamma_{0}}\right)=-2\sum_{g\in{\mathcal{G}}}\frac{\Lambda(g)P(\gamma_{0}\log|g|)}{|g|^{s}}.$

with the choice of the constant parameters $B,C,D>0$ such that

$P(u):=(B+C-D)+B\cos u+C\cos(2u)\geq 0\leavevmode\nobreak\
\leavevmode\nobreak\ (\forall u\in{\mathbb{R}}).$ (52)

For any $s$ in the right halfplane $\Re s>1$ we have

$F(s,x):=-\sum_{g\in{\mathcal{G}}}\frac{\Lambda(g)P(\gamma_{0}\log|g|)d(x,|g|)}{|g|^{s}}=\frac{1}{2\pi
i}\int_{(2)}\frac{1}{2}\frac{f^{\prime}}{f}(s+w)K(w,x)dw.$

Similarly to the von Mangoldt type formula of Lemma 10–whose proof is detailed
in [37]–we move the line of integration to the left. For that, we fix a
parameter $b$, choose
${\mathcal{A}}:=\\{-2\gamma_{0},-\gamma_{0},0,\gamma_{0},2\gamma_{0}\\}$, and
consider the curve $\Gamma:=\Gamma^{{\mathcal{A}}}_{b}$, as constructed in
Lemma 8. It is appropriate here to fix our parameters $a,b$, frequently used
below. We choose $a:=\theta+(1-\theta)/3$, $b:=\theta+2(1-\theta)/3$, so that
all the constants $a-\theta$, $1-b$, $b-\theta$ are equivalent to $1-\theta$,
that is, they are all estimated both from below and from above by a positive,
finite constant multiple of $1-\theta$. Correspondingly, in the following the
constants $A_{1},A_{2},\ldots$ will be chosen to depend on $\theta,A$ and
$\kappa$ only, but not on $a,b$. Also note that with this choice we have
$a=\frac{b+\theta}{2}$, matching with the construction of Lemma 8.

Next, we move the part $[2-it_{k},2+it_{k}]$ of the contour of integration to
the left to the corresponding part of $\Gamma-s$ along the horizontal lines
$t=\pm t_{k}$. Now it is easy to see that the "bridges" along any horizontal
segments $t=\pm t_{k}$ give $o(1)$ contribution as $t_{k}\to\infty$, hence by
letting $t_{k}\to\infty$ the whole line $\Re w=2$ of integration can be moved
to $\Gamma-s$ (so that $w+s\in\Gamma$).

Observe that for real $s$ by construction in Lemma 8 the broken line
$\Gamma-s$ does not contain any singularities of the integrand, i.e. $\Gamma$
does not contain singularities of $f^{\prime}/f$. In fact, we will only
consider parameter values $1<s<1+1/100$ in the proof. Note that for real $s$
we can also write $F(s,x)=\Re F(s,x)$ and

$\frac{1}{2}\Re\frac{f^{\prime}}{f}(s)=\frac{1}{2}\frac{f^{\prime}}{f}(s)=E\frac{\zeta^{\prime}}{\zeta}(s)+B\Re\frac{\zeta^{\prime}}{\zeta}(s-i\gamma_{0})+C\Re\frac{\zeta^{\prime}}{\zeta}(s-2i\gamma_{0})\quad(E:=B+C-D).$

We thus find, using again the notation ${\mathcal{Z}}(L)$ for the set of
$\zeta$-zeroes to the right of the curve $L$, that

$\displaystyle F(s,x)$ $\displaystyle=\frac{1}{2\pi
i}\int\limits_{\Gamma-s}\frac{1}{2}\frac{f^{\prime}}{f}(s+w)K(w,x)dw+\frac{1}{2}\sum_{r\atop\sigma_{0}-s<r<2}\textrm{Res}\big{[}K(w,x):w=r\big{]}\frac{f^{\prime}}{f}(s+r)$
$\displaystyle+E\left\\{-K(1-s,x)+\sum_{\rho\in{\mathcal{Z}}(\Gamma)}K(\rho-s,x)\right\\}$
$\displaystyle+B\Re\left\\{-K(1-s+i\gamma_{0},x)+\sum_{\rho\in{\mathcal{Z}}(\Gamma-i\gamma_{0})}K(\rho-s+i\gamma_{0},x)\right\\}$
$\displaystyle+C\Re\left\\{-K(1-s+i2\gamma_{0},x)+\sum_{\rho\in{\mathcal{Z}}(\Gamma-i2\gamma_{0})}K(\rho-s+i2\gamma_{0},x)\right\\},$

where $\sigma_{0}$ is the abscissa of the line segment of $\Gamma$ crossing
the real axis. The only singularity $r$ of $K(w,x)$ in $[\sigma_{0}-s,2]$ is
at $r=0$ (for $\Gamma(w/2)$ has singularities only at $0$ and at the negative
even integers, whilst $-1.01<\sigma_{0}-s$ excludes the occurrence of the
latter), so the first sum reduces to $2(f^{\prime}/f)(s)$ in view of the
residuum of $x^{w}\Gamma(w/2)$ at $w=0$ being exactly 2. Multiplying by the
constant factor $1/2$ in front of it and then subtracting from the formula the
resulting term $\frac{f^{\prime}}{f}(s)$, we arrive at

$\displaystyle 0$
$\displaystyle\leq\sum_{g\in{\mathcal{G}}}\frac{\Lambda(g)P(\gamma_{0}\log|g|)\left(2-2\exp(-(|g|/x)^{2})\right)}{|g|^{s}}=F(s,x)-\frac{f^{\prime}}{f}(s)$
$\displaystyle=I+ES_{0}+B\Re S_{1}+C\Re S_{2},$ (53)

where

$I:=\frac{1}{2\pi
i}\int\limits_{\Gamma-s}\frac{1}{2}\frac{f^{\prime}}{f}(s+w)K(w,x)dw,$

and

$S_{j}:=-K(1-s+ij\gamma_{0},x)+\sum_{\rho\in{\mathcal{Z}}(\Gamma-
ij\gamma_{0})}K(\rho-s+ij\gamma_{0},x)\qquad(j=0,1,2).$

Using that $s$ is chosen to be real, and that we only need $\Re S_{j}$, we can
also write $\Re S_{j}=\Re\overline{S_{j}}$, where now

$\overline{S_{j}}=-K(1-s-ij\gamma_{0},x)+\sum_{\rho\in{\mathcal{Z}}(\Gamma+ij\gamma_{0})}K(\rho-
s-ij\gamma_{0},x)\qquad(j=0,1,2).$

Recall that the curve $\Gamma$ is constructed in Lemma 8 so that the real part
of points on the curve lie between $a:=\frac{b+\theta}{2}$ and $b$. Therefore,
$-1-1/100\leq a-s\leq\Re w\leq b-s<b-1<0$ bounds occurring values of
$w=u+iv\in\Gamma-s$ within the strip $-1.01\leq u=\Re w\leq 1-b$. Stirling’s
formula for $w=u+iv$ in this strip gives
$|\Gamma(w/2)|\ll\frac{1}{1-b}(1+|v|)^{u/2-1/4}e^{-\pi v/4}$.

Next we estimate the logarithmic derivatives
$\frac{\zeta^{\prime}}{\zeta}(z+ij\gamma_{0})$ for $j=-2,-1,0,1,2$, for
arbitrary points $z\in\Gamma$. Using Lemma 9 and $\gamma_{0}\geq 100$, we are
led to

$\displaystyle\left|\frac{\zeta^{\prime}}{\zeta}(s+ij\gamma_{0})\right|$
$\displaystyle\leq 5\frac{1-\theta}{(b-\theta)^{3}}\left(6\log(|\Im
z|+2\gamma_{0}+5)+60\log(A+\kappa)+40\log\frac{1}{b-\theta}+140\right)^{2}$
$\displaystyle\leq A_{1}\log^{2}(\gamma_{0}+|\Im
z|)\qquad(A_{1}:=A_{1}(\theta,A,\kappa)\quad\textrm{a constant}).$

Applying this bound in estimating the values of
$(\zeta^{\prime}/\zeta)(s+w+ij\gamma_{0})$ for $w\in\Gamma-s$ and combining it
with the estimation coming from Stirling’s formula for $\Gamma(w/2)$ we obtain

$\displaystyle I$
$\displaystyle\ll\int_{0}^{\infty}A_{1}\log^{2}(\gamma_{0}+v)x^{b-s}\frac{(1+|v|)^{(b-s)/2-1/4}}{1-b}e^{-v\pi/4}dv\ll
A_{2}x^{b-s}\log^{2}\gamma_{0},$ (54)

with $A_{2}:=A_{2}(\theta,A,\kappa)$ a constant.

Next, we estimate the contribution of zeroes in $\overline{S_{j}}$, having
imaginary part farther from $j\gamma_{0}$ than $Q$, with a parameter $Q>1$ to
be chosen later. One summand is at most

$|K(\beta-s+i(\gamma-j\gamma_{0}),x)|\ll
x^{1-s}|\gamma-j\gamma_{0}|^{(1-s)/2-1/4}e^{-\pi|\gamma-j\gamma_{0}|/4}\leq
x^{1-s}e^{-\pi|\gamma-j\gamma_{0}|/4},$

since for $-1-1/100<\Re w<1-s$ and $|\Im w|>Q\geq 1$ we have a uniform
Stirling bound. For the corresponding sums we thus have, by an application of
Lemma 5

$\displaystyle\left|\sum_{\rho\in{\mathcal{Z}}(\Gamma+ij\gamma_{0})\atop|\gamma-j\gamma_{0}|\geq
Q}K(\rho-s-ij\gamma_{0},x)\right|$ $\displaystyle\ll
x^{1-s}\int_{|t-j\gamma_{0}|>Q}e^{-\pi|t-j\gamma_{0}|/4}dN(a,t)$
$\displaystyle=x^{1-s}\int_{Q}^{\infty}e^{-\pi
u/4}d\left(N(a,u+j\gamma_{0})+N(a,j\gamma_{0}-u)\right)$ $\displaystyle\ll
x^{1-s}\left\\{N(a,j\gamma_{0}+Q)e^{-\pi Q/4}+\int_{Q}^{\infty}e^{-\pi
u/4}N(a,u+j\gamma_{0})du\right\\}$ $\displaystyle\ll
A_{3}x^{1-s}(Q+\gamma_{0})^{2}e^{-\pi Q/4},$

where as before, $A_{3}$ is a constant depending on the parameters $\kappa,A$
and $\theta$ only. For $j=0$ we have for the small zeroes of $S_{0}$ the
estimate

$\left|\sum_{\rho\in{\mathcal{Z}}(\Gamma)\atop|\gamma|<1}K(\rho-s,x)\right|\leq\frac{A+\kappa}{\kappa(1-\theta)}N(a,1)x^{1-s}\leq
A_{4}x^{1-s}\qquad(A_{4}:=A_{4}(\kappa,A,\theta)\leavevmode\nobreak\ ).$

according to the separation of zeroes of $\zeta(s)$ from $1$, given in Lemma
2, (22) and using Lemma 4 (26), too. Similarly, for the rest of $S_{0}$ we
have, referring to the uniform Stirling bound here,

$\left|\sum_{\rho\in{\mathcal{Z}}(\Gamma)\atop
1\leq|\gamma|<Q}K(\rho-s,x)\right|\leq\sum_{q=1}^{Q}N(a,q,q+1)x^{1-s}e^{-q\pi/4}\ll
A_{5}x^{1-s}\quad(A_{5}:=A_{5}(\kappa,A,\theta)).$

Note that $1-s-i\gamma_{0}$ and $1-s-2i\gamma_{0}$ are separated by at least
$\gamma_{0}>1$ from the singularity of $\Gamma(w/2)$ at $w=0$, whence we have
$|K(1-s-ij\gamma_{0})|\ll x^{1-s}$.

Choosing e.g. $Q:=\lceil 3\log\gamma_{0}\rceil$ the above estimates applied in
(5) furnish999From now on we drop writing $A_{k}:=A_{k}(\theta,A,\kappa)$, but
all $A_{k}$ are understood as constants depending on these parameters of Axiom
A only.

$0\leq-
EK(1-s,x)+BS_{1}^{\prime}+CS_{2}^{\prime}+A_{6}\left(x^{b-s}\log^{2}\gamma_{0}+x^{1-s}\right),$
(55)

where for $j=1,2$ we denote

$Sj^{\prime}:=\Re\sum_{\rho\in{\mathcal{Z}}(\Gamma+ij\gamma_{0})\atop|\gamma-j\gamma_{0}|<Q}K(\rho-
s-ij\gamma_{0},x).$ (56)

Assume now that
$0<\lambda<\lambda_{0}:=\lambda_{0}(a):=1-a=\frac{2}{3}(1-\theta)$ (so that
$\lambda<\lambda_{0}$ is exactly as is postulated in the theorem). With such a
constant parameter value $\lambda$ we have $1-\lambda>a$, whence according to
(29) of Lemma 6 the number of zeroes in the discs $D(1+ij\gamma_{0},\lambda)$
($j=1,2$) cannot exceed $A_{7}+\log(2+j\gamma_{0})$. Similarly, the number of
summands in both sums $S^{\prime}_{j}$ ($j=1,2$) are at most

$N(a,j\gamma_{0}-Q,j\gamma_{0}+Q)\ll
Q\left(A_{8}+\log(Q+2\gamma_{0})\right)\ll A_{9}\log^{2}\gamma_{0},$ (57)

according to Lemma 6 in view of the choice of $Q:=[3\log\gamma_{0}]$.

Since $b=1-(1-\theta)/3$, we have $x^{b-s}=x^{-(1-\theta)/3+1-s}$. We will now
assume that

$x\geq X^{*}\geq\exp\left(\frac{3}{1-\theta}\log(2\log\gamma_{0})\right),$
(58)

so that $x^{b-s}\log^{2}\gamma_{0}\leq x^{1-s}$. Using this in the above (55),
we have for $x\geq X^{*}$

$0\leq-
EK(1-s,x)+BK(\rho_{0}-s-i\gamma_{0},x)+BS_{1}"+CS_{2}^{\prime}+2A_{6}x^{1-s},$

where $S_{1}"$ is essentially $S_{1}^{\prime}$, save the contribution of the
known zero at $\rho_{0}$.

Next we separate the contribution of the zeroes inside
${\mathcal{D}}:=D(1+i\gamma_{0},\lambda)\cup D(1+i2\gamma_{0},\lambda)$. Here
we use the estimate, proved in [23] and also in [31], that for any $z$ subject
to $-3/2<\Re z<0$ we have uniformly $\Re x^{z}\Gamma(z/2)\ll x^{\Re z}\log x$.
In view of this, and our choice of $1<s<1.01$, the contribution of these close
zeroes is at most a constant times

$S^{\star}:=\sum_{\rho\in{\mathcal{D}}\setminus\\{\rho_{0}\\}}x^{\beta-s}\log
x.$

Collecting the above estimates we obtain

$EK(1-s,x)-BK(\beta_{0}-s,x)-BS_{1}^{\star}-CS_{2}^{\star}-2A_{6}x^{1-s}\ll
S^{\star},$

where now the sums $S_{j}^{\star}$ denote the same as $S_{j}^{\prime}$, but
the summands chosen only from

$\\{\rho\in{\mathcal{Z}}(\Gamma-ij\gamma_{0})\leavevmode\nobreak\
:\leavevmode\nobreak\ |\gamma-j\gamma_{0}|<Q\\}\setminus
D(1+ij\gamma_{0},\lambda).$

Note that all terms in the sums $S^{\star}_{j}$ ($j=1,2$) have the form
$K(z,x)$ with $|z|>\lambda$.

Now it may be clear that any choice of the coefficients $B,C,D$, hence $E$,
which satisfy $E<B$, suffices. So we choose $B:=4$, $C:=1$ and $D:=4$ to
obtain $E=3$ (and thus $P(u)=3+4\cos u+\cos(2u)$). After multiplying by
$x^{s-1}$ we are led to

$3\Gamma\left(\frac{1-s}{2}\right)-4x^{\beta_{0}-1}\Gamma\left(\frac{\beta_{0}-s}{2}\right)-4S_{1}^{\star\star}-S_{2}^{\star\star}-2A_{6}\ll\sum_{\rho\in{\mathcal{D}}\setminus\\{\rho_{0}\\}}x^{\beta-1}\log
x,$ (59)

where

$Sj^{\star\star}:=\Re\sum_{\rho\in{\mathcal{Z}}(\Gamma+ij\gamma_{0})\atop|\gamma-j\gamma_{0}|<Q,\leavevmode\nobreak\
|\rho-ij\gamma_{0}|>\lambda}x^{\rho-ij\gamma_{0}-1}\Gamma\left(\frac{\rho-s-
ij\gamma_{0}}{2}\right).$ (60)

We want to chose our parameters $x$ and $y:=\log x$ so that we will have

$\frac{4}{1-\theta}\log\log\gamma_{0}<y:=\log x\leq\frac{0.1}{1-\beta_{0}},$
(61)

meeting the requirement in (58), too, and further that it will hold

$(1<)\leavevmode\nobreak\ 1+10(1-\beta_{0})<s<1.01.$ (62)

Note that in order to have a nonempty interval for $y$ in the first condition
it suffices to have $1-\beta_{0}<\frac{1-\theta}{40\log\log\gamma_{0}}$ which
is assumed among the conditions of the statement of Theorem 3. Further, to
have a nonempty interval for $s$ in the second condition it suffices to have
$1-\beta_{0}<0.001$, also guaranteed by the condition $\beta_{0}>0.999$ of
Theorem 3.

The above conditions allow an interval for the choice of $x$. We will now take
a subinterval, so that $\frac{4}{1-\theta}\log\log\gamma_{0}\leq Y_{0}:=\log
X_{0}<Y_{1}:=\log X_{1}\leq 0.1/(1-\beta_{0})$, and consider admissible values
$y:=\log x=Y_{0}+u_{1}+\dots+u_{m}$ with $0\leq u_{j}\leq d$, where
$d:=2e^{2}/\lambda$ is a constant, now depending also on $\lambda$, and
$m\in{\mathbb{N}}$ is an integer parameter. Moreover, we want to choose
$m:=[Y_{0}]$. Then the above parametrization of $y=\log x$ will run between
$Y_{0}$ and $Y_{0}+md$, whence
$Y_{1}=Y_{0}+md\in(Y_{0}+d(Y_{0}-1),(1+d)Y_{0}]$ must be below the bound
$0.1/(1-\beta_{0})$ for $\log X_{1}$. We thus require $(1+d)Y_{0}\leq
0.1/(1-\beta_{0})$. Whenever such a
$Y_{0}\geq\frac{4}{1-\theta}\log\log\gamma_{0}$ is chosen, the corresponding
$m$ and $Y_{1}=Y_{0}+md$ satisfy the necessary bounds, hence the interval
$[X_{0},X_{1}]$ is admissible. So from this point on we can proceed with our
argument only in case

$\frac{4}{1-\theta}\log\log\gamma_{0}\leq
Y_{0}\leq\frac{1}{1+d}\frac{0.1}{1-\beta_{0}}.$ (63)

The left hand side inequality is assumed (with $Y$ written in place of $Y_{0}$
here) among the conditions of Theorem 3. Now, if the right hand side
inequality here fails to hold, then we necessarily have
$Y_{0}>\frac{1}{1+d}\frac{0.1}{1-\beta_{0}}=\frac{\lambda}{(\lambda+e^{2})10(1-\beta_{0})}$,
hence $Y_{0}(1-\beta_{0})>\frac{\lambda}{160}$, so that for any constant
$c_{0}\geq 160$ in the statement (51) the right hand side becomes negative.
Therefore, for such parameter values $Y=Y_{0}$ there remains nothing to prove,
and it remains to derive the assertion in case (63) holds.

Observe that for any values of $\beta_{0}$ and $s$ satisfying (62) we have

$\displaystyle 3\Gamma\left(\frac{1-s}{2}\right)$
$\displaystyle-4x^{\beta_{0}-1}\Gamma\left(\frac{\beta_{0}-s}{2}\right)$
$\displaystyle=-\frac{6}{s-1}\Gamma\left(\frac{3-s}{2}\right)+\frac{8}{s-\beta_{0}}e^{-(1-\beta_{0})\log
x}\Gamma\left(\frac{2+\beta_{0}-s}{2}\right)$
$\displaystyle\geq\left\\{\frac{8}{s-1+1-\beta_{0}}e^{-0.1}-\frac{6}{s-1}\right\\}\Gamma\left(\frac{3-s}{2}\right)$
$\displaystyle\geq\left\\{\frac{8}{(s-1)1.1e^{0.1}}-\frac{6}{s-1}\right\\}\Gamma\left(\frac{3-s}{2}\right),$

because $\Gamma$ is decreasing around $1$ and thus
$\Gamma\left(\frac{2+\beta_{0}-s}{2}\right)>\Gamma\left(\frac{3-s}{2}\right)$.
So in all we get from (59)

$3\Gamma\left(\frac{1-s}{2}\right)-4x^{\beta_{0}-1}\Gamma\left(\frac{\beta_{0}-s}{2}\right)>\frac{1}{s-1}.$

Using also $Y_{0}=\log X_{0}\leq\log x\leq Y_{1}\leq(1+d)Y_{0}$ we obtain from
(59) and the above

$\frac{1}{s-1}-2A_{6}-4S_{1}^{\star\star}-S_{2}^{\star\star}\ll\sum_{\rho\in{\mathcal{D}}\setminus\\{\rho_{0}\\}}x^{\beta-1}\log
x\leq(1+d)Y_{0}\sum_{\rho\in{\mathcal{D}}\setminus\\{\rho_{0}\\}}X_{0}^{\beta-1}.$
(64)

Since we have this inequality for all $x\in[X_{0},X_{1}]$, i.e. for all values
of ${\bf u}:=(u_{1},\dots,u_{m})\in[0,d]^{m}$, we can average it on the left
hand side with respect to all the $u_{k}$. For the general term of the sums
$S_{j}^{\star\star}$ we obtain

$\displaystyle\int_{0}^{d}\int_{0}^{d}\dots\int_{0}^{d}$
$\displaystyle\left\\{\Re X_{0}^{\rho-
ij\gamma_{0}-1}e^{(u_{1}+\dots+u_{m})(\rho-
ij\gamma_{0}-1)}\Gamma\left(\frac{\rho-s-
ij\gamma_{0}}{2}\right)\right\\}du_{1}\dots du_{m}$ $\displaystyle\ll
2^{m}\frac{1}{\lambda}\frac{1}{|\rho-
ij\gamma_{0}-1|^{m}}\leq\left(\frac{2}{\lambda}\right)^{m+1},$

because $|\rho-ij\gamma_{0}-1|\geq\lambda$ as $\rho$ lies outside
$D(1+ij\gamma_{0},\lambda)$ and $|\Gamma\left(\frac{\rho-s-
ij\gamma_{0}}{2}\right)|\ll 1/\lambda$.

Taking into account (57) and the total volume $d^{m}$ of the cube $[0,d]^{m}$,
we obtain from (64) with an absolute constant $C^{\star}$ the inequality

$\frac{1}{s-1}-2A_{6}-C^{\star}A_{9}\log^{2}\gamma_{0}\leavevmode\nobreak\
d\left(\frac{2}{d\lambda}\right)^{m+1}\leq(1+d)Y_{0}\sum_{\rho\in{\mathcal{D}}\setminus\\{\rho_{0}\\}}X_{0}^{\beta-1}.$

Now in view of $d:=2e^{2}/\lambda$, we have
$\left(\frac{2}{d\lambda}\right)^{m+1}=\exp(-2(m+1))\leq\exp(-2Y_{0})\leq\log^{-2}\gamma_{0}$,
hence writing in the value of $d$ and putting $C":=2e^{2}C^{\star}$ we arrive
at

$\frac{1}{s-1}-2A_{6}-\frac{C"A_{9}}{\lambda}\ll\frac{1}{\lambda}Y_{0}\sum_{\rho\in{\mathcal{D}}\setminus\\{\rho_{0}\\}}X_{0}^{\beta-1}.$

Clearly our best choice here is to choose $s$ as small as possible. We must
meet the conditions (61), so the smallest admissible value is
$s=1+10(1-\beta_{0})$. This choice yields

$\frac{{\lambda}}{Y_{0}(1-\beta_{0})}-\frac{20A_{6}\lambda+10C"A_{9}}{Y_{0}}\ll\sum_{\rho\in{\mathcal{D}}\setminus\\{\rho_{0}\\}}e^{Y_{0}(\beta-1)}.$

Applying $\lambda<1$ and using $Y_{0}\geq A_{10}$ with a sufficiently large
value of the constant $A_{10}$ now gives the asserted inequality (51) for
$Y_{0}$ in place of $Y$.

The result is proved. ∎

##  6 A preview of further work

Let us motivate our detailed study of the distribution of zeroes of the
Beurling zeta function by recalling three rather sharp, essentially optimal
results, known to hold _for the classical case_ of natural numbers and primes
and the Riemann zeta function.

The first one is the essentially final answer to a classic problem of
Littlewood [22] as to what oscillation could be "caused" by having a
$\zeta$-zero $\rho$? For the result and its possible optimality (in particular
as regards the somewhat surprising value of the constant $\pi/2$ in it) see
[35].

######  Theorem 4 (Révész).

Let $\zeta(\rho)=0$ with $\rho=\beta+i\gamma$ be a zero of the Riemann zeta
function. Then for arbitrary ${\varepsilon}>0$ we have for some suitable,
arbitrarily large values of $x$ the lower estimate
$|\Delta(x)|\geq(\pi/2-{\varepsilon})\frac{x^{\beta}}{|\rho|}$.

Denote by $\eta(t):(0,\infty)\to(0,1/2)$ a nonincreasing function and consider
the domain

${\mathcal{D}}(\eta):=\\{s=\sigma+it\in{\mathbb{C}}\leavevmode\nobreak\
:\leavevmode\nobreak\ \sigma>1-\eta(t),\leavevmode\nobreak\ t>0\\}.$ (65)

Following Ingham [14] and Pintz [26, 27] consider also the derived function
(the _Legendre transform_ of $\eta$ in logarithmic variables)

$\omega(x):=\omega_{\eta}(x):=\inf_{y>1}\left(\eta(y)\log x+\log y\right).$
(66)

Then the following are sharp results of Pintz, see [26, 27], sharp in the
ultimate sense that iterative applications of them give back the original
estimates without any loss in the constants in the exponents.

######  Theorem 5 (Pintz).

Assume that there is no zero of the Riemann $\zeta$ function in
${\mathcal{D}}(\eta)$. Then for arbitrary ${\varepsilon}>0$ we have

$\Delta(x)=O(x\exp(-(1-{\varepsilon})\omega(x)).$

######  Theorem 6 (Pintz).

Conversely, assuming that there are infinitely many zeroes within the domain
(65), we have for any ${\varepsilon}>0$ the oscillation estimate
$\Delta(x)=\Omega(x\exp(-(1+{\varepsilon})\omega(x))$.

These results, in their original proofs relied on particular things which are
generally not available for the Beurling zeta functions. Therefore, it is
unclear how much of these precise relations can as well be stated for the
distribution of Beurling primes? Our aim with the present series is to prove
analogously sharp results for the Beurling case eventually. The above cited
theorems are much sharper than everything currently known for the Beurling
case–so that extending them would indeed mean a considerable advance. That is
our long range aim with this series.

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