# New zero-density estimates for the Beurling $\zeta$ function††thanks: This
research was supported by Hungarian National Research, Development and
Innovation Office, Project #s KKP133819, K-147153 and K-146387.

János Pintz and Szilárd Gy. Révész

###### Abstract

In two previous papers the second author proved some Carlson type density
theorems for zeroes in the critical strip for Beurling zeta functions
satisfying Axiom A of Knopfmacher. In the first of these invoking two
additonal conditions were needed, while in the second an explicit, fully
general result was obtained. Subsequently, Frederik Broucke and Gregory
Debruyne obtained, via a different method, a general Carlson type density
theorem with an even better exponent, and recently Frederik Broucke improved
this further, getting $N(\sigma,T)\leq T^{a(1-\sigma)}$ with any
$a>\dfrac{4}{1-\theta}$. Broucke employed a new mean value estimate of the
Beurling zeta function, while he did not use the method of Halász and
Montgomery.

Here we elaborate a new approach of the first author, using the classical zero
detecting sums coupled with a kernel function technique and Halász’ method,
but otherwise arguing in an elementary way avoiding e.g. mean value estimates
for Dirichlet polynomials. We will make essential use of the additional
assumptions that the Beurling system of integers consists of natural numbers,
and that the system satisfies the Ramanujan condition, too. This way we give a
new variant of the Carlson type density estimate with similar strength as
Turán’s 1954 result for the Riemann $\zeta$ function, coming close even to the
Density Hypothesis for $\sigma$ close to 1.

MSC 2020 Subject Classification. Primary 11M41; Secondary 11M36, 30B50, 30C15.

Keywords and phrases. Beurling zeta function, analytic continuation, zero of
the Beurling zeta function, zero detecting sums, method of Halász, density
estimates for zeta zeros.

##  1 Zero density results for the Riemann zeta function

In the classical case of natural numbers ${\mathbb{N}}$ and primes
${\mathbb{P}}$, it is well-known from Riemann’s times that the distribution of
the zeroes of the (analytic continuation of) the Rieman zeta function $\zeta$
is decisive in regard of questions of prime distribution. As all the
nontrivial zeroes are in the critical strip $0<\Re s<1$ and their number up to
height $T$ is asymptotically $N(T)\sim\frac{T}{2\pi}\log T$, the estimations
of the number $N(\sigma,T)$ of zeroes in a halfplane $\Re s>\sigma$ and up to
height $T$ by quantities of smaller order could point towards validity of the
Riemann Hypothesis. We are discussing here such so-called _density estimates_
first for the Riemann zeta function itself, and then, more generally, we use
some of these results to obtain new density estimates for Beurling number
systems as well.

Bohr and Landau [2] proved in 1914 that $N(\sigma,T)=o(N(T))$ if $\sigma>1/2$.
This was the first estimate of the kind what we call density estimates.

A few years later Carlson [7] proved

$N(\sigma,T)\ll_{\varepsilon}T^{A(\sigma)(1-\sigma)+\varepsilon}\ \text{ for
any }\ \varepsilon>0\ \text{ and }\ \sigma\geq 1/2,$ (1)

where he could put

$A(\sigma)\leq 4\sigma.$ (2)

Number theoretic consequences in connection with the Landau problem on the
difference of consecutive primes made it of top interest to improve these
bounds as much as possible. Landau formulated the conjecture that there is a
prime between any two squares [15]. In particular, it was shown that an almost
exact positive answer to Landau’s question would follow from the so-called
"_Density Hypothesis_ " (DH) which asserts

$N(\sigma,T)\ll T^{2(1-\sigma)}\log^{C}T\ \text{ with some }\ C>0\ \text{ for
all }\sigma\geq\frac{1}{2},$ (3)

or, in a slightly weaker form, using the notation (1),

$A(\sigma)\leq 2\ \text{ for all }\ \sigma\geq 1/2\ \
\left(\Longleftrightarrow
N(\sigma,T)\ll_{\varepsilon}T^{2(1-\sigma)+\varepsilon}\right).$ (4)

Clearly, the Riemann Hypothesis contains the Density Hypothesis. However,
Ingham [12] could demonstrate DH from another, wekaer assumption, the so-
called "_Lindelöf Hypothesis_ " (LH), which formulates

$\mu\left(\frac{1}{2}\right)=0,\ \text{ where }\ \mu(\alpha):=\text{\rm
inf}\left\\{\mu;|\zeta(\sigma+it)|\leq T^{\mu}\text{ for
}\sigma\geq\alpha,\,1<|t|\leq T\right\\}.$ (5)

In 1954 Turán [23] used his celebrated power-sum method [22], see also [24],
to give a different proof of Ingham’s result that LH implies DH, and he almost
achieved the DH in the vicinity of the boundary line $\sigma=1$. Let us put

$s=\sigma+it,\ \ \eta:=1-\sigma,\ \ B(\eta):=\frac{1}{\eta}A(1-\eta).$ (6)

Then Turán’s result was with a small positive constant $c_{1}$ [23]

$N(1-\eta,T)\ll T^{2\eta+\eta^{1.14}}\log^{6}T\ \text{ for }\ \eta<c_{1},$ (7)

or with our new notation

$B(\eta)\leq 2+\eta^{0.14}\ \text{ for }\eta<c_{1}.$ (8)

The next breakthrough improvement of estimates, yielding in particular DH for
a fixed strip near the 1-line, was obtained in [11] by Halász and Turán. In
that work they combined Turán’s celebrated power sum theory with the best
known Korobov-Vinogradov estimates on the growth of the Riemann zeta function
and Halász’ pioneering idea [10] which will play a decisive role in the main
argument of this present work, too.

A special feature of our approach to the general case of Beurling systems is
that we invoke our better developed knowledge about the behavior of the
Riemann zeta function when addressing questions of zero density for certain
Beurling systems. In particular, we will employ the best known growth bound,
due to Bourgain [3]

$|\zeta(1/2+it)|\ll_{\varepsilon}(|t|+1)^{\frac{13}{84}+{\varepsilon}}\qquad(\forall{\varepsilon}>0).$
(9)

##  2 Beurling systems and density theorems for Beurling $\zeta$ functions

Beurling systems were introduced by Beurling [1], and were subsequently put
into a more abstract – and hence more general – framework, see e.g. [14].
Beurling systems thus can be viewed in two alternate ways, one being that they
are just an arbitrary semigroup ${\mathcal{G}}$ of elements $g$, admitting an
(essentially, i.e. up to ordering) unique factorization to prime elements
$p\in{\mathcal{P}}\subset{\mathcal{G}}$. In this "arithmetical semigroup"
point of view the elements can be algebraic numbers, ideals of integer rings
of algebraic number fields, finite Abelian groups, and many more systems which
admit a suitable Krull-Schmidt type decomposition theorem leading to a
multiplicative structure. In this abstract setting a key role is played by the
multiplicative norming of elements, i.e, there is a norm
$|\cdot|:{\mathcal{G}}\to[1,\infty)$ which is multiplicative ($|gh|=|g||h|$)
and its image is locally finite. That maps, in a unique way, the abstract
structure to ${\mathbb{R}}_{+}$, so that one can view it as included in the
real number setting where the primes ${\mathcal{P}}$ are just an arbitrary
sequence of reals, nondecreasing and tending to infinity, and freely
generating the respective system of Beurling integers ${\mathcal{N}}$.
Although we will refer back to this first interpretation here or there, the
main setting of the paper is in line with the latter real number settings.

Let ${\mathcal{P}}$ be a sequence of Beurling primes and ${\mathcal{N}}$ the
generated semigroup, that is, the system of Beurling integers. Altogether,
these form a Beurling number system
${\mathcal{B}}:=({\mathcal{P}},{\mathcal{N}})$. Due to its Euler product form,
the Beurling zeta function does not vanish in its halfplane of convergence:

$\zeta_{\mathcal{B}}(s):=\sum_{n\in{\mathcal{N}}}\frac{1}{n^{s}}=\prod_{p\in{\mathcal{P}}}\frac{1}{1-1/p^{s}}\neq
0.$ (10)

For a general overview of the analytic theory of Beurling number systems we
refer to [9].

Beurling systems may or may not have an analytic continuation over the
boundary line of convergence of their Dirichlet series expansion. However, if
they do, then there are basically two possibilities: either the analytic
continuation is governed by a nice asymptotic formula with power-type error
for the number of integers ${\mathcal{N}}(x):=\\#\\{n\leq
x\leavevmode\nobreak\ :\leavevmode\nobreak\ n\in{\mathcal{N}}\\}$, or the
Beurling zeta function must show extremely irregular behavior with large
values and no reasonable (polynomial) bounds. This second possibility was
pointed out by Frederik Broucke; for an explanation see e.g. Remark 4.1 of
[6]. In turn, if $\zeta_{\mathcal{B}}$ does not admit polynomial bounds in any
larger halfplane than the halfplane of convergence, then there is no hope to
control its behavior and to derive finer results of zero- and hence prime
distributions.

Therefore, to talk about analytic continuation of $\zeta_{\mathcal{B}}$ and to
have a chance to come up with a successful analysis using it even in the
critical strip, we assume that the integers are "well-behaved". That is, 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 [14].

######  Definition 1 (Axiom A).

It is said that ${{\mathcal{N}}}$ (or, loosely speaking,
$\zeta_{\mathcal{B}}$) satisfies _Axiom A_ – more precisely, Axiom
$A(A,\kappa,\theta)$ with the suitable constants $A,\kappa>0$ and
$0\leq\theta<1$ – if for the remainder term
${\mathcal{R}}(x):={\mathcal{N}}(x)-\kappa x$ it holds the estimate

$\left|{\mathcal{R}}(x)\right|\leq Ax^{\theta}\qquad(x\geq 1).$ (11)

Historically, the first result which could be considered a weak form of a
density estimate, was worked out by Kahane in [13]. However, it was part of an
indirect proof and the assumption he made was a very strong number theoretical
condition. Under that restrictive assumption Kahane could prove, however, that
on a given line $\Re s=a$ in the critical strip, with $a>(1+\theta)/2$, the
number of $\zeta_{\mathcal{B}}$-zeros is $O(T)$.

The first zero density result for general Beurling zeta functions was obtained
in [18] under two extra assumptions on the Beurling system ${\mathcal{B}}$.
One was that the Beurling number system remains within the realm of natural
numbers – a strong, but useful assumption, emphasized also by Knopfmacher, see
pages 57–58 of [14]. We will term this as _the integrality condition_ ,
meaning that $|\cdot|:{\mathcal{G}}\to{\mathbb{N}}$, that is, the norm $|g|$
of any element $g\in{\mathcal{G}}$ is a natural number. Alternatively, we may
take it as ${\mathcal{N}}\subset{\mathbb{N}}$.

Further, in [18] an averaged form of the Ramanujan condition was assumed in
proving the density bound

$B_{\mathcal{B}}(\eta)\leq\frac{6-2\theta}{1-\theta}.$ (12)

In the present work we will also need a form of the Ramanujan condition, but
given that our main reference [16] used it in its sharper, pointwise version,
we settle with this version here, too. For its formulation, let us introduce
the arithmetical function $G(\nu):=\\#\\{n\in{\mathcal{N}}\leavevmode\nobreak\
:\leavevmode\nobreak\ |n|=\nu\\}$, the number of Beurling integers having a
given norm (value). With that the Beurling zeta function can be written as

$\zeta_{\mathcal{B}}(s)=\sum_{n\in{\mathcal{N}}}\frac{1}{n^{s}}=\sum_{\nu=1}^{\infty}\frac{G(\nu)}{\nu^{s}}.$
(13)

Then the Beurling system, or, loosely speaking, $\zeta_{\mathcal{B}}$
satisfies the Ramanujan condition, if $\log G(\nu)=o(\log\nu)$, that is, if
for any $\delta>0$ we have $G(\nu)\leq\nu^{\delta}$ for $\nu>\nu_{0}(\delta)$.

The two extra conditions (integrality and the averaged Ramanujan condition) of
the result (12) were removed in [21] at the expense of a worse exponent.
Almost simultaneously and independently, however, Frederik Broucke and Gregory
Debruyne [5] succeeded in proving an improved bound also without relying on
these additional assumptions. Moreover, they also showed that in the
generality of assuming only Axiom A, there are Beurling zeta functions
$\zeta_{\mathcal{B}}$ with $B_{\mathcal{B}}(\eta)>c_{1}$ with some
sufficiently small positive constant $c_{1}$. Finally, very recently [4]
Frederik Broucke further improved the exponent in a general density theorem
for Beurling zeta functions satisfying Axiom A, obtaining

$B_{\mathcal{B}}(\eta)\leq\frac{4}{1+2\eta-\theta}.$ (14)

At the end of this introduction let us point out that the new development of
having zero density theorems for zeta functions of well-behaved systems of
Beurling integers and primes opened up the way to achieve many strong number
theoretical advances in Beurling’s theory. These advances are likely to be
totally impossible to reach without such tools. The interested reader may
consult [19, 20, 5, 6].

##  3 The aim of the paper

Our present goal is to demonstrate that returning to the extra assumptions of
integrality and the Ramanujan Condition, a new result, significantly stronger
than (14), can be obtained. For example, if $\theta=0$, then we can almost
reach the DH for small values of $\eta$, similarly to the 1954 result of Turán
for the classical case. See below in Corollary 1

It is of interest that while Turán used his power sum method to show (8), here
we need only Halász’ idea [10] to show a general form of (16), somewhat weaker
if $\theta>0$. Namely, we will prove

######  Theorem 1.

Under Axiom A, the integrality condition, and the Ramanujan condition for a
Beurling system ${\mathcal{B}}$, we have, as long as $0<\eta<1-\theta$, the
estimates

$B_{\mathcal{B}}(\eta)=\begin{cases}\frac{2}{1-\theta-\eta}\qquad&\text{if}\quad
0<\eta<\min\left(\frac{4}{29},\frac{8+13\theta}{71}\right)\\\
\frac{26/21}{1-4\eta}&\text{if}\quad\frac{8+13\theta}{71}\leq\eta<\frac{4}{29}\qquad(\text{empty
for}\quad\theta\geq\frac{4}{29}),\\\
\frac{2}{1-\theta-\eta}&\text{if}\quad\frac{4}{29}\leq\eta<\theta\qquad(\text{empty
for}\quad\theta<\frac{4}{29}),\\\
\frac{2}{1-2\eta}&\text{if}\quad\max\left(\theta,\frac{4}{29}\right)\leq\eta\end{cases}.$
(15)

We did not directly insert into the definition of the various ranges for
$\eta$ the generally required condition that $\eta<1-\theta$. Note that taking
into account that restriction, or even already in view of the conditions
mentioned, the actual ranges for the various estimates may well be empty for
specific values of the parameter $\theta$; only the very first range, –
written out fully as
$0<\eta<\min\left(1-\theta,\frac{4}{29},\frac{8+13\theta}{71}\right)$ – needs
to be nonempty for all values of $\theta\in[0,1)$.

Specializing for $\theta=0$ and for small $\eta$ only, we obtain

######  Corollary 1.

Let ${\mathcal{B}}$ be a Beurling number system satisfying the integrality
condition and also Axiom A with $\theta=0$. Then for the Beurling zeta
function $\zeta_{\mathcal{B}}$ of this system we have a zero density bound
with

$B_{\mathcal{B}}(\eta)=\frac{2}{1-\eta}\left(=2+O(\eta)\right)\qquad\textrm{for}\quad\eta\leq\frac{8}{71}.$
(16)

##  4 The main tool

Our main tool is a slightly strengthened variant of a general zero density
type theorem proved recently by the first named author in [16]. This deals
with general (but ordinary, i.e. with integer powers in the denominator)
Dirichlet series.

Assuming $f_{1}\neq 0$, we consider the (Dirichlet inverse to each other)
arithmetical functions $f_{n}$ and $g_{n}$, both satisfying the Ramanujan
condition, and the corresponding reciprocal pair of Dirichlet series

$f(s)=\sum_{n=1}^{\infty}\frac{f_{n}}{n^{s}},\quad
M(s)=\frac{1}{f(s)}=\sum_{n=1}^{\infty}\frac{g_{n}}{n^{s}},$ (17)

which by assumption about the Ramanujan condition must be analytic for
$\sigma>1$, and satisfy $M(s)f(s)=1$ there.

######  Remark 1.

If $f_{n}$ is completely multiplicative as a function of $n$ then $f_{1}=1$
and $g(n)=\mu(n)f_{n}$.

Further we suppose that with some fixed constant $\alpha_{f}<1$, the function
$f(s)$ can be continued analytically to the halfplane $\sigma>\alpha_{f}$,
save a simple pole at $s=1$ with residue $f_{0}$. We define the respective
"generalized Lindelöf function" to any such $f$ as follows.

$\mu_{f}(\sigma_{0}):=\text{\rm inf}\left\\{\mu;|f(\sigma+it)|\leq
T^{\mu}\text{ for }\sigma\geq\sigma_{0},1\leq|t|\leq T\right\\}<\infty\text{
for }\sigma_{0}>\alpha_{f}.$ (18)

This is clearly Lindelöf’s classical $\mu$-function if $f(s)$ is chosen to be
$\zeta(s)$. In the following technical definition the function
$\lambda_{f}^{(0)}$ will depend on $\mu_{f}$; in particular,
$\lambda_{\zeta}^{(0)}$ on $\mu_{\zeta}$. Let

$\lambda_{f}^{(0)}(\eta)\\!:=\\!\\!\inf_{0<a;(a+1)\eta<1-\alpha_{f}}\frac{\mu_{f}(1\\!-\\!(a\\!+\\!1)\eta)}{a\eta},\quad\lambda_{\zeta}^{(0)}(\eta)\\!:=\\!\\!\inf_{0<b}\frac{\mu_{\zeta}(1\\!-\\!(b\\!+\\!1)\eta)}{b\eta}.$
(19)

Note that we do not assume anything about the size of the parameter $b$ in the
above definition for $\lambda_{\zeta}^{(0)}(\eta)$; this is the essential
change compared to the original version of the main auxiliary theorem what we
want to use.

In the following we do not need _the exact_ values of these Lindelöf-type
derived functions, (which is rather fortunate, given that not even the precise
value of $\mu_{\zeta}$ is known), but we will be satisfied with any function
$\lambda_{f}(\eta)\geq\lambda^{(0)}_{f}(\eta)$, in particular with
$\lambda_{\zeta}(\eta)\geq\lambda^{(0)}_{\zeta}(\eta)$. Even if precise values
are not known, strong estimates on the growth of the Riemann zeta function
$\zeta$ will furnish us reasonably strong bounds $\lambda_{\zeta}(\eta)$.

As is usual, we denote the number of zeroes (counting them with their possible
multiplicities) as

$N_{f}(1-\eta,T):=\\#\\{\rho\leavevmode\nobreak\ :\leavevmode\nobreak\
f(\rho)=0,\Re\rho\in[1-\eta,1],|\Im\rho|\leq T\\}.$ (20)

The general density estimate, what we apply in this work, reads as follows.

######  Lemma 1.

Let $f$ and $g$ be reciprocal Dirichlet series satisfying the above
assumptions and the Ramanujan condition $\log|f_{n}|=o(\log n)$ and
$\log|g_{n}|=o(\log n)$.

Let $0<\eta<1-\alpha_{f}$ be arbitrary, and let $\lambda_{f}(\eta)$ and
$\lambda_{\zeta}(\eta)$ be some estimator functions satisfying
$\lambda_{f}(\eta)\geq\lambda_{f}^{(0)}(\eta)$ and
$\lambda_{\zeta}(\eta)\geq\lambda_{\zeta}^{(0)}(\eta)$, where
$\lambda_{f}^{(0)}(\eta),\leavevmode\nobreak\ \lambda_{\zeta}^{(0)}(\eta)$ are
defined according to (18) and (19). Denote

$B_{f}(\eta):=\max\left(2\lambda_{f}(\eta),4\lambda_{\zeta}(2\eta)\right).$
(21)

Then if the condition

$\lambda_{f}(\eta)\geq\lambda_{\zeta}(2\eta)$ (22)

holds true, then we have the zero density estimate

$N_{f}(1-\eta,T)\ll_{\eta,{\varepsilon}}T^{B_{f}(\eta)\eta+{\varepsilon}}.$
(23)

###### Proof.

This is essentially a part of Theorem 2 in [16], listed under formula (2.9)
there.

However, here we formulated an alteration in that we changed also the
definition of $\lambda_{\zeta}^{(0)}(\eta)$ somewhat, allowing the
$b$-parameter, originally restricted to satisfy some upper bound in [16],
become unbounded here.

It is clear from the original proof that

  * (i)

we can allow $\lambda_{f}(\eta)=\lambda_{\zeta}(2\eta)$, too (no need for
strict inequality, as in the original formulation);

  * (ii)

we can allow in the definition of the original $\lambda_{f}(\eta)$ function
(which corresponds to $\lambda_{f}^{(0)}(\eta)$ here) any upper estimation, as
we formally described above in detail;

  * (iii)

when defining $\lambda_{\zeta}^{(0)}(\eta)$, actually there is no need for any
upper bound for $b$, as the original proof works without change even for
larger values of $b$, given the analytic continuation (and known bounds of the
growth) of the Riemann $\zeta$ function on the whole complex plane.

With these alterations we obtain a proof of the Lemma. ∎

To obtain somewhat sharper results for larger values of $\eta$ in our present
work, we needed this change for two reasons. First, the possibility of
allowing $\eta$ to extend up to the theoretical bound of $1-\theta$, we need
good estimates on $\lambda_{\zeta}(2\eta)$ in order to meet the condition (22)
for $f=\zeta_{\mathcal{B}}$. Without this amelioration, the validity of the
condition would not be ascertained for $\eta$ exceeding certain bounds.
Second, with the above amelioration, also the bound (21) improves, providing
somewhat better density results for larger values of $\eta$.

##  5 The proof of Theorem 1

###  5.1 Computing estimates for the estimator $\lambda_{\zeta}(2\eta)$ of the
Riemann $\zeta$ function

We start with finding a sufficiently good estimator function $\lambda_{\zeta}$
(which we will later use at the value of $2\eta$, not at $\eta$). Note that
here we deal with the Riemann $\zeta$ function, so that there is no upper
bound on the admissible values of $\eta$; however, as for the Beurling zeta
function we have the natural bound $0<\eta<1-\theta\leq 1$, we do not extend
over $0<\eta<1$ (or, if considering values at $2\eta$, we restrict
equivalently for $0<\eta<1/2$).

We work with two different choices of the appropriate parameter $b$, depending
on the size of $\eta$. More precisely, for $\eta\leq 4/29$ we will use
Bourgain’s estimate (9) with choosing
$b:=\frac{1}{4\eta}-1\Longleftrightarrow(b+1)2\eta=1/2$, which leads to
$\mu_{\zeta}(1-(b+1)2\eta))=\mu_{\zeta}(1/2)\leq 13/84$ and hence the
estimator function

$\lambda_{\zeta}(2\eta)=\frac{13/84}{1/2-2\eta}=\frac{13/42}{1-4\eta},$ (24)

obviously satisfying $\lambda_{\zeta}(2\eta)\geq\lambda_{\zeta}^{(0)}(2\eta)$
for this range of values of $\eta$.

If $4/29\leq\eta\leq 1/4$, then we will use the basic estimate
$\mu_{\zeta}(0)=1/2$. Choosing the value
$b:=\frac{1}{2\eta}-1\Longleftrightarrow(b+1)2\eta=1$ in this case, we obtain
from (19) that we can take

$\lambda_{\zeta}(2\eta)=\frac{1/2}{1-2\eta}.$ (25)

If $\eta>1/4$ then $\lambda_{\zeta}(2\eta)>1$, hence by (21)
$B_{\mathcal{B}}(\eta)\geq 4\lambda_{\zeta}(2\eta)>1/\eta$, which is weaker
then the trivial estimate $1$, see Section 6.

###  5.2 Estimator for the Beurling zeta function $\zeta_{\mathcal{B}}$

For the Beurling zeta function of our Beurling system ${\mathcal{B}}$, we
cannot take values with $\Re s\leq\theta$, so that we must restrict to
$\eta<1-\theta$, and in the construction of
$\lambda_{\zeta_{\mathcal{B}}}^{(0)}(\eta)$ and
$\lambda_{\zeta_{\mathcal{B}}}(\eta)$ even to $1-(a+1)\eta>\theta$, i.e.
$(a+1)\eta<1-\theta$.

We will extend $(a+1)\eta$ close to this limit. A reference to Lemma 5 of [21]
(or, with full proof, to Lemma 2.5 of [17]) furnishes

$\mu_{\zeta_{\mathcal{B}}}(\theta+{\varepsilon})\leq\frac{1-\theta+{\varepsilon}}{1-\theta},$
(26)

which in turn allows to set
$a:=\frac{1-\theta-{\varepsilon}}{\eta}-1\Longleftrightarrow(a+1)\eta=1-\theta-{\varepsilon}$
and derive with this value the estimator functions

$\lambda_{\zeta_{\mathcal{B}}}^{({\varepsilon})}(\eta)=\frac{1-\theta+{\varepsilon}}{(1-\theta)(1-\theta-\eta-{\varepsilon})}.$

As
$\lambda_{\zeta_{\mathcal{B}}}^{({\varepsilon})}(\eta)\geq\lambda_{\zeta_{\mathcal{B}}}^{(0)}(\eta)$
holds for all ${\varepsilon}>0$, we in fact can take limits with respect to
${\varepsilon}\to 0$ and obtain the new estimator function

$\lambda_{\zeta_{\mathcal{B}}}(\eta)=\frac{1}{1-\theta-\eta}.$ (27)

Unlike the estimator functions for the Riemann $\zeta$ function, where we
distinguished two different cases with two different estimates (24) and (25),
the estimate (27) is "universal" (the same formula) for all admissible values
of $0<\eta<1-\theta$.

###  5.3 Verification of the conditions of the key lemma

Next we check the conditions of the key lemma in order to apply it for
admissible values of $\eta$. We always assume that $\eta<1-\theta$ without
repeatedly telling about it.

For the small values $0<\eta\leq 4/29$, an easy calculation furnishes that
$\lambda_{\zeta_{\mathcal{B}}}(\eta)=\frac{1}{1-\theta-\eta}\geq\frac{1}{1-\eta}\geq\frac{13/42}{1-4\eta}=\lambda_{\zeta}(2\eta)$,
the condition in (22). Let now $4/29\leq\eta\leq 1/4$. Then we find
$\lambda_{\zeta_{\mathcal{B}}}(\eta)=\frac{1}{1-\theta-\eta}\geq\frac{1}{1-\eta}\geq\frac{1}{2-4\eta}=\lambda_{\zeta}(2\eta)$,
that is, we again have condition (22) of Lemma 1 satisfied.

In all, we can record that with the estimator functions worked out above for
the Riemann $\zeta$ and the Beurling zeta $\zeta_{\mathcal{B}}$, the last
condition (22) of Lemma 1, i.e. the inequality
$\lambda_{\zeta_{\mathcal{B}}}(\eta)\geq\lambda_{\zeta}(2\eta)$, holds true
for the entire domain $0<\eta<1-\theta$.

###  5.4 End of the proof of Theorem 1

Therefore, for all values of $0<\eta<1-\theta$, we can apply Theorem A, which
yields

$B_{{\mathcal{B}}}(\eta)=\max\left(2\lambda_{\zeta_{\mathcal{B}}}(\eta),4\lambda_{\zeta}(2\eta)\right)=\begin{cases}\max\left(\frac{2}{1-\theta-\eta},\frac{26/21}{1-4\eta}\right)\qquad&\text{if}\quad\eta\leq\min(4/29,1-\theta)\\\
\max\left(\frac{2}{1-\theta-\eta},\frac{2}{1-2\eta}\right)\qquad&\text{if}\quad
4/29\leq\eta\leq\min(1/4,1-\theta)\end{cases}.$ (28)

In case of $\eta\leq 4/29$, the first term is maximal precisely when
$\eta\leq(8+13\theta)/71$.

In case of $4/29\leq\eta\leq 1/4$, the first term gives the maximum if and
only if $\eta\leq\theta$.

Winding up these partial case calculations furnishes (15).

$\square$

##  6 Conclusion

Contrary to zeta functions with a functional equation, there is no asymptotic
formula for the number of zeroes of a Beurling zeta function in the critical
strip or in a halfplane strictly in the critical strip. This is not just a
weakness of methods for a proof; Beurling systems with only finitely many
zeroes (or no zeroes at all) do exist, see [19], Theorem 7.4. (though one may
recall that these phenomenon is known only for $\theta\geq 1/2$ and strips
$\Re s\geq\sigma>\theta$). Nevertheless, an upper estimate of the same order
of $T\log T$ as in the Riemann zeta case, is known, see e.g. (7) of Theorem 2
from [8] or Lemma 3.5 of [17]. Therefore, a density bound exceeding
$T^{1+{\varepsilon}}$ does not provide anything new. In light of this the part
for $\eta\geq\frac{1-\theta}{3}$ of the above results become worthless: they
can be substituted by the better $B_{\mathcal{B}}^{\star}=1/\eta$ (equivalent
to $N(\sigma,T)\ \ll T^{1+{\varepsilon}}$ for all ${\varepsilon}>0$). Also,
one may note that our above obtained bounds become inferior to the recent
results of Broucke (14) from [4] for $\eta>\frac{1-\theta}{4}$. However, for
$\eta<\frac{1-\theta}{4}$, our results provide sharper estimates, even though
only under the two assumptions on the integrality and the Ramaujan condition.

## References

  * [1] A. Beurling, Analyse de la loi asymptotique de la distribution des nombres premiers généralisés I. _Acta Math._ 68 (1937), 255–291.
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János Pintz
HUN-REN Alfréd Rényi Institute of Mathematics
Reáltanoda utca 13-15
1053 Budapest, Hungary
<EMAIL_ADDRESS>

Szilárd Gy. Révész
HUN-REN Alfréd Rényi Institute of Mathematics
Reáltanoda utca 13-15
1053 Budapest, Hungary
<EMAIL_ADDRESS>


# First Intensity Interferometry Measurements with the H.E.S.S. Telescopes

Andreas Zmija1, Naomi Vogel1, Frederik Wohlleben1,2, Gisela Anton1, Adrian
Zink1, Stefan Funk1
1Erlangen Centre for Astroparticle Physics, Friedrich-Alexander-Universität
Erlangen-Nürnberg, Nikolaus-Fiebiger-Str. 2, Erlangen 91058, Germany
2now at Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, Heidelberg,
69117, Germany
E-mail<EMAIL_ADDRESS>
(This is a pre-copyedited, author-produced PDF of an article accepted for
publication in Monthly Notices of the Royal Astronomical Society following
peer review. The version of record [Zmija et al. (2023)] is available online
at: https://doi.org/10.1093/mnras/stad3676.)

###### Abstract

Intensity interferometry for astrophysical observations has gained increasing
interest in the last decade. The method of correlating photon fluxes at
different telescopes for high resolution astronomy without access to the phase
of the incoming light is insensitive to atmospheric turbulence and doesn’t
require high-precision optical path control. The necessary large collection
areas can be provided by Imaging Atmospheric Cherenkov Telescopes.
Implementation of intensity interferometers to existing telescope systems such
as VERITAS and MAGIC has proven to be successful for high-resolution imaging
of stars. In April 2022 we equipped two telescopes of the H.E.S.S. array in
Namibia with an intensity interferometry setup to measure southern sky stars
and star systems during the bright moon period. We mounted an external optical
system to the lid of the telescope cameras, which splits the incoming light
and feeds it into two photomultipliers in order to measure the zero-baseline
correlation within one telescope in addition to the cross correlation between
the telescopes. The optical elements are motorised, which enables live
correction of tracking inaccuracies of the telescopes. During the campaign we
measured the spatial correlation curves and thereby the angular diameters of
$\lambda$ Sco (Shaula) and $\sigma$ Sgr (Nunki), while we also performed
systematic studies of our interferometer using the multiple star system of
$\alpha$ Cru (Acrux).

###### keywords:

instrumentation: high angular resolution – instrumentation: interferometers –
techniques: interferometric – stars: imaging – methods: observational –
telescopes

††pubyear: 2023††pagerange: First Intensity Interferometry Measurements with
the H.E.S.S. Telescopes–First Intensity Interferometry Measurements with the
H.E.S.S. Telescopes

##  1 Introduction

Hanbury Brown and Twiss (HBT) were the pioneers of astronomical intensity
interferometry in the 1960s, when they measured the angular diameter of 32
stars with the Narrabri Stellar Intensity Interferometer (Hanbury Brown et
al., 1967, 1974). Due to the lack of fast electronics and large area
collectors, this technique was subsequently abandoned which led to the
development of stellar amplitude interferometry that is applied at
instruments, such as the VLTI (Haguenauer et al., 2010) and CHARA (ten
Brummelaar et al., 2005).
Although they are highly sensitive, the length of the baseline between the
telescopes of these experiments and therefore the optical resolution is
limited due to atmospheric turbulence. As a consequence, over the last decade
there has been renewed interest in intensity interferometry due to
improvements in technology and new large area telescopes in arrays. (Dravins
et al., 2013)

While amplitude interferometry employs first order correlation of the
electromagnetic wave field at a given spatial distance, intensity
interferometry employs second order correlation of the electromagnetic wave
field, which is the correlation of photon flux at a given spatial distance.
The advantage of this is to make kilometer-scale baselines in the optical band
accessible that are possible due to the insensitivity to atmospheric
turbulence. Such baselines give rise to angular resolutions of the instrument
on the scale of microarcseconds (Dravins, 2016).

The trade-off is a comparably small signal-to-noise for intensity
interferometry observations due to the requirement of two-photon-coincidence
measurements (Bojer et al., 2022), which is expressed by the small value of
the coherence time (usually on the order of pico- to femtoseconds). Besides
the quest for high-time-resolution detectors, one would like to collect as
many photons as possible, which requires large light buckets.

Arrays of Imaging Atmospheric Cherenkov Telescopes (IACTs), generally used as
gamma-ray observatories, are perfectly suitable for intensity interferometry
implementations: the telescopes are amongst the largest that exist on the
planet which can focus optical light, observatories typically consist of
multiple telescopes separated by baselines on the order of $100\,$m, and the
detection mode doesn’t allow gamma-ray observations during bright moon times,
so that intensity interferometry measurements can be carried out during that
time.

Two IACT observatories are already successfully performing intensity
interferometry observations. In December 2019, (Abeysekara et al., 2020)
measured the angular diameters of the two stars $\beta$ CMa and $\varepsilon$
Ori using all 4 telescopes of the VERITAS array as a combined intensity
interferometer (VSII). Recently, the VSII star survey has expanded also to
fainter stars up to a magnitude of $+4.2$, while the angular diameters of
bright stars can be measured with an accuracy of typically $3$–$4\,$per cent
and will undergo further improvements (Kieda, 2022). Further, the MAGIC
experiment started intensity interferometry observations in April 2019 with
measurements of $\varepsilon$ CMa (Adhara) and $\eta$ UMa (Benetnasch)
(Acciari et al., 2020). Due to the possibility to control individual mirror
segments ("active mirror control") the telescopes can also deliver multiple
sub-telescope baselines. An extension to the Large Sized Telescopes of the
Cherenkov Telescope Array, which is currently under construction, is in the
planning. (Cortina et al., 2022)

In this paper, we present the first intensity interferometry measurements with
the H.E.S.S. telescopes in Namibia, where we equipped two telescopes with an
intensity interferometry setup mounted to the lid of the Cherenkov cameras.
The setup features a beamsplitter at each telescope which enables measurements
of the zero-baseline correlation. Our optical pass bands, defined by
interference filters, only have a width of $2\,$nm, which is challenging for
IACT intensity interferometry. Due to the observatory being close to the
southern tropic, it provides excellent observing conditions for targets in the
southern sky. We observed Shaula ($\lambda$ Sco) and Nunki ($\sigma$ Sgr), as
well as the multiple-star-system Acrux ($\alpha$ Cru).

We introduce the concept of stellar intensity interferometry with the
important physical quantities in section 2. Section 3 describes the
experimental setup, including the motorised optical elements, which can
correct for telescope mis-pointing. We explain our measurement procedure from
the start of the night to the end in section 4, and give a detailed analysis
of the obtained data in section 5, including the final measurement results of
the angular diameters of Shaula and Nunki.

##  2 Theory

Intensity interferometry is sensitive to the second order correlation of
electromagnetic waves, i.e. photon rates $I$ at time $t$ and position R.

$g^{(2)}(\textbf{{R}},\tau)=\frac{\expectationvalue{I(\textbf{{R}},t)I(\textbf{{R}}+\textbf{{b}},t+\tau)}}{\expectationvalue{I(\textbf{{R}},t)}\expectationvalue{I(\textbf{{R}}+\textbf{{b}},t+\tau)}}\,,$
(1)

with b (= baseline) being the distance vector between two observers
(telescopes), $\tau$ the time difference where the correlation is observed,
and $\expectationvalue{}$ the time average. Since light from thermal/chaotic
light sources is known to show photon bunching (Fox, 2006), one measures
$g^{(2)}>1$ for comparably small detector separations b and time differences
$\tau$. The temporal range in which photon bunching occurs is typically
described by the coherence time $\tau_{c}$, often defined as
$\tau_{c}=\lambda_{0}^{2}/(c\Delta\lambda)$, with $\lambda_{0}$ and
$\Delta\lambda$ being the central wavelength and width of the observed
wavelength spectrum. However, we decide on a more general definition which can
be applied to arbitrary wavelength spectra.

$\tau_{c}:=\int_{-\infty}^{+\infty}\left(g^{(2)}(\tau)-1\right)\,\,d\tau$ (2)

The value of the observed coherence time at a telescope baseline
$b=|\textbf{{b}}|$ is then expected to be

$\tau_{c}(b)=0.5\times
k_{\textnormal{s}}(b)k_{\textnormal{T}}\frac{\lambda^{2}_{0}}{c\Delta\lambda}$
(3)

with $c$ being the speed of light and $k_{\textnormal{s}}(b)$ being the
spatial coherence factor, which is $1$ for $b=0$ and $0<k_{\textnormal{s}}<1$
otherwise. The factor $0.5$ stems from the fact that unpolarized light is
observed. $k_{\textnormal{T}}$ is a correction factor to the traditional
definition of the coherence time $\lambda_{0}^{2}/(c\Delta\lambda)$ taking the
exact spectral transmission profile into account. The relation between the
profile and the temporal coherence time is described via a Fourier transform
by the Wiener-Khintchine theorem (Mandel & Wolf, 1995). Numerical computations
of the coherence times using the spectral transmission profiles of the two
interference filters which are used - measured by the the manufacturer - yield
values for $k_{\textnormal{T,1}}=0.842$ and $k_{\textnormal{T,2}}=0.848$ for
perpendicular incident light.
The coherence time in Eq. 3 is inversely proportional to the bandwidth
$\Delta\lambda$ of the observed light, resulting in large coherence times for
small bandwidths. However, if the time resolution of the system is
significantly larger than the coherence time (which is true for every field
application of intensity interferometry), the signal-to-noise of a measurement
is in a first order approximation independent of the used bandwidth, since the
photon number is increasing at broader bandwidths, compensating the decreasing
value of the coherence time. Nevertheless, restricting the bandwidth by using
optical filters in the system is recommended to limit the photon flux on the
detectors and avoid systematic error contributions at very small coherence.
The spatial geometry information of the source is encoded in the spatial
coherence factor $k_{\textnormal{s}}(b)$. The degree of coherence - according
to the van Cittert-Zernike theorem - is related to the intensity profile of
the source via a Fourier transform (Mandel & Wolf, 1995),
$k_{\textnormal{s}}(b)$ can be identified with its square. Hence, significant
spatial coherence is only observed within a limited range of baselines $b$,
depending on the angular size of the observed object. Large angular diameter
stars show spatial coherence of the photons in only a very short baseline
range, whereas the spatial coherence of stars of small angular sizes extend
over a much longer baseline range.

Figure 1: Different spatial coherence curves for different (uniform disc)
angular diameters at a wavelength of $470\,$nm. $b_{\textnormal{max}}=120\,$m
indicates the maximum baseline for the combination of the two used telescopes.

Fig. 1 displays exemplary coherence curves for stars of different angular
sizes. Sampling the spatial coherence curve for a star at different baselines
allows the angular size of the star to be determined. For the given distance
$b_{\textnormal{max}}$ between two telescopes the star moving along its
trajectory in the sky is seen at a range of baselines with the maximum value
$b_{\textnormal{max}}$.

##  3 Measurement setup

###  3.1 The H.E.S.S. Telescopes

Figure 2: Photograph of the H.E.S.S. site. The inauguration of the Phase I
(the four smaller) telescopes was in 2002. Later on, in 2012, the central
telescope was added to the project. In 2022, the intensity interferometry
setups were installed on telescopes 3 and 4, which are the two left ones in
the picture. Part of the data analysis was done in the server room inside of
the control building (bottom centre). (Klepser, 2012)

The intensity interferometry measurements presented in this paper were carried
out with two of the H.E.S.S. telescopes (Phase I) located at an altitude of
1835 m.a.s.l. in the Khomas Highlands in Namibia. H.E.S.S. consists of an
array of four IACTs arranged in a square with 120 m side length. At the centre
of the array a single 600-m2 mirror area telescope was added in Phase II in
2012. The phase I telescopes have a diameter of 12 m and a focal length of
$15\,$m. The mirror area is 108 m2 consisting of 382 round mirror facets,
which are arranged in Davis-Cotton design. A sub-milliarcsecond resolution for
intensity interferometry observations is achieved by linking long telescope
baselines, large mirror area and advanced optics. (Hofmann, 2012)
A picture of the H.E.S.S. site is shown in Fig. 2.

###  3.2 Experimental Setup

Figure 3: Intensity interferometry setup mounted to the lid of the telescope
CT3. As the setup is equipped with motors, the range of motion for each
individual component is marked with red arrows. Figure 4: Schematic of the
intensity interferometer for one of the telescopes. The setup is identical for
each telescope.

Our intensity interferometry (II) setup was designed to be mounted onto the
lid of the H.E.S.S. Phase I cameras (see Fig. 3). It is a rectangular
($50\times 63$)-cm2 structure made of aluminum with bulges to reduce the total
weight to 21 kg with all equipment attached. The setup is mounted at the
centre of the lid in the focal spot of the H.E.S.S. mirrors. A schematic of
the setup can be seen in Fig. 4. The incident light beam hits the 45∘ angled
mirror (not pictured) which reflects it into the optical path. The light beam,
coming from the left side, is divided by a beamsplitter and guided onto the
two detectors. The entrance window of the optical system is equipped with a
remote controlled shutter to ensure complete darkness if needed. The data is
acquired and transmitted to the main workstation where the analysis is
performed.
The mirror measures ($41\times 25.5$) cm2 and is 1 mm thick. It is a SEA-VIS
front-surface mirror produced via CNC precision-cut by Präzisions Glas & Optik
GmbH glued to an aluminum frame for stabilisation.
The optical path is made of a 2" diameter tube system and consists of a
parallelising lens (coated concave lens from Thorlabs, Inc (2023), $F=-75$
mm), a LC-HBP465/2-50 narrowband optical (interference) filter (465 nm CWL, 2
nm width) and a converging lens (coated bi convex lens from Thorlabs, $F=100$
mm). The approach of using interference filters of only $2\,$nm bandwidth is
challenging, since the light incident on the filter needs to be perpendicular
to the filter surface for optimal controlled performance. The telescopes have
optical point spread functions on the order of $0.3\,$mrad ($0.5\,$cm)
(Bernlöhr et al., 2003). Simulations have shown that the approach works well
if the optical system can be adjusted with a precision of a few mm relative to
the focal spot position (Konrad, 2021), as shown in Fig. 5.

Figure 5: Simulated transmission spectra of the optical filter, taking into
account the optical quality of the telescope mirrors. The parallelising lens
is shifted by $1.5\,$cm along the optical axis from to the nominal position
towards the focal spot. This ensures that most of the light cone is collected
by the optical system, while the effect on the transmission spectrum of the
filter is kept small. The colors denote different displacements of the tube
system perpendicular to the optical axis w.r.t. to the focal spot position.
Image adapted from (Konrad, 2021).

After passing the interference filter, the light is split into two beams by
the beamsplitter. The beamsplitter is a 50:50 non-polarizing beamsplitter cube
from Thorlabs (BS031) with a range of 400–700 nm.
The detectors are photomultiplier tubes (PMT) by Hamamatsu of type R11265U-300
(Hamamatsu, Inc, 2023) with a peak sensitivity at 420 nm and a quantum
efficiency of 39 per cent. At our main wavelength of 465 nm the PMTs still
have a quantum efficiency of 30 per cent. The PMTs are powered by a High
Voltage (HV) power supply which is located in the back of the H.E.S.S. camera.
The first 9 dynodes are powered by the HV whereas the last 4 dynodes have an
additional booster power supply to counteract voltage drop. The PMT output is
amplified by a factor of $10$. The amplifier is placed directly behind the PMT
to minimize electronic noise. The outputs from the amplifiers are then led
through an airborne 5 UV shielded polyethylene coaxial cable. Two PMTs are
placed at each telescope to also perform zero-baseline correlation
measurements. One PMT is connected to a 10-m long signal cable, while the
other is connected to a 40-m long cable to avoid crosstalk at the sampling
interface.
A halogen lamp is installed at the second input of the beam splitter block to
function as a light source for calibration runs. The lamp is controlled
remotely and its brightness adjusted as needed.
The setup is equipped with motors to move each component individually. As
shown in Fig. 3 the wagon on the right side, where the optics and detectors
are attached, can be moved in the camera plane (x- and z-axis). The mirror can
be moved up and down in y-axis direction and also around the x- and z-axis.
This ensures that the whole light beam is captured by the optical system. When
the spot moves out of focus, the components are moved with help of the motors
towards the maximum incoming light, thereby ensuring an external online
pointing correction.
The setup is mounted to the lid of the H.E.S.S. camera as shown in Fig. 3. To
the left of the setup is a second box for placing the motor control and its
power supply, as well as the power supply of the halogen lamp. The HV cables
and signal cables are led to the back of the H.E.S.S. camera plane where the
HV power supply and the digitizer is located. The data is then transferred via
optical fibres to the workstation in the control room where data analysis is
done after the measurement.

Each of the telescope racks in the backplane of the camera is equipped with a
M4i.2211-x8 digitizer from Spectrum Instrumentation (2023). It features 8 bit
digitization on two input channels, covering possible input voltage ranges
between $\pm 40$ and $\pm 500\,$mV. The maximum sampling speed is
$1.25\,$GS/s. For our measurements we used a sampling speed of $625\,$MS/s at
a range of $\pm 200\,$mV. The digitizers are connected to the workstation’s
PCIe bus via an Adnacom R34/H18 PCIe-over-fiber extender (Adnacom, 2023).
Synchronization of the two data streams is ensured by connecting both
digitizers to the White Rabbit (WR) network of the telescopes. The digitizers
are triggered by the pulse-per-second (PPS) of the WR and start to
synchronously acquire 2 GS of data, which corresponds to $3.436\,$seconds
(note that this is independent of the incoming photon rate since we are
sampling the PMT currents). After completion the digitizers wait for the next
PPS (which arrives at the 4 seconds mark) to measure the next 2 GS. This
results in a duty cycle of $85.9\,$per cent. The procedure was chosen in order
to obtain permanent feedback of the functionality of the WR synchronization
system.
The data is stored to disk during the measurements and correlated offline
during the day and after the campaign.

##  4 Measurement procedure

Intensity interferometry observations were performed in April 2022 during the
H.E.S.S. moonlight break (full-moon/near-full-moon period) since gamma
observations are stopped when the moon has reached 60 per cent of its full
moon intensity. Available sources were chosen considering their trajectory,
magnitude and configuration. The main observed targets are the stars Shaula
($\lambda$ Sco), Nunki ($\sigma$ Sgr) and Acrux ($\alpha$ Cru). The results
are shown in section 5.
First, the telescopes are parked out and slewed to the target of interest for
the night, which is then tracked continuously until a new one is chosen or the
night is over. While the telescopes are tracking, a PMT calibration
measurement with the HV off is performed in order to remove the DC offset
later in the analysis. To determine the live photon rates via shape and height
distribution of the photon pulses, a PMT pulse calibration measurement with
the HV on and low photon rates using the halogen lamp is carried out as well.
A detailed description of the calibration procedure is given in Zmija et al.
(2021).

###  4.1 Focusing the targets

After the PMT calibration, our optics are focused on the star by optimizing
the photon rates using the motors. In a first step both the mirror and optical
system are moved in the camera plane (X, Z) and a heatmap of the corresponding
rates is produced (compare Fig. 6). The brightest area of the heatmap is then
identified as area of interest and further investigated by another (finer)
iteration of this procedure. Eventually, the final position of the mirror is
determined by fitting a 2d Gaussian to the new heatmap. Furthermore, the
optimal distance between the mirror and the optical system is determined by a
simple one-dimensional optimization to find the maximum photon rates. The
distance between the mirror and the camera plane can be adjusted manually,
even though this only yields minor improvements. A rotation of the mirror
around any axis did prove to not be necessary.

Figure 6: Pointing correction: Heatmap of the recorded photon rates from a
scan in the camera plane. The red rectangle shows the area of interest as
determined by the first scan. The Gaussian (blue contours) is fitted to the
data from the detailed scan. Its centre gives the optimal position for the
motors. The definition of the coordinate axes can be found in Fig. 3.

###  4.2 Data acquisition

After having found the optimal motor positions, we start our II measurements
while constantly checking the live rates of the stellar target. In Fig. 8 an
example plot for the rates of Shaula of CT 3 from the night of the 19th April
2022 is shown. Shaula is a star which rises during that time of the year which
is shown nicely by the rising rates at the beginning of the night, when the
star is increasing in elevation and atmospheric absorption is decreasing.
During the middle of the night there are recurring rate losses. This happens
when the optical system is not in the focus of the star beam anymore due to
the telescope position depending mis-pointing of the telescopes. If the rate
loss is on the order of 20 per cent, the mirror is refocused to get back to
the maximum rate. This procedure can be seen well at the end of the night at
3:33 UTC time. The large drops in rate at earlier times were recorded while
the mirror was refocused. During that process the mirror was driven in and out
of focus in different directions until the maximum rate was obtained again.
The measurement ends when either the star is less than $20^{\circ}$ above the
horizon which is too low for tracking or the sun is $10^{\circ}$ below the
horizon which is too bright for measurements.

###  4.3 Night Sky Background

Uncorrelated background photons from the night sky affect the measurements and
have to be taken into account, if the background rates contribute
significantly to the total photon rate. We therefore occasionally checked the
night sky background, which we expect to be dominated by the moonlight, by
slightly mis-pointing the telescopes by not more than $1\,$degree from the
target star. In none of these night sky background measurements we observed
photon rates which exceeded $10\,$MHz per PMT, which usually is $<5\,$per cent
of the photon rates on source, even when the moon was only a few degrees away
from the source. Fig. 7 shows a systematic night sky background measurement
with very small separations from the moon, which was as $66\,$per cent
brightness in that night. We concluded that night sky background is not a
major issue for the bright stars observed in this campaign, as long as the
target star is separated by more than only a few degrees from the moon. For
fainter stars effects of the relatively higher night sky background rate, such
as reduced signal-to-noise and bunching peak heights, need to be considered.

Figure 7: Night sky background at $66\%$ moon phase

###  4.4 End of observations

At the end of an observation night, the shutter to the optical system is
closed, the HV is turned off and the telescopes are parked for daytime. Since
the setup is not removed from the lid during the nights where both intensity
interferometry and gamma measurements take place, the motors are driven to
their parking position. In this position the components, especially the
mirror, are stowed in such a way that they don’t disturb the opening of the
lid for gamma observations. When the telescopes are stowed the amplifiers are
turned off and the correlation analysis of the recorded data is started.

Figure 8: Photon rates of star Shaula in the night of the 19th April 2022. The
recorded rates of both detectors of telescope 3 (channel 3A and channel 3B)
are shown. As Shaula rises in the beginning of the night the photon rates
increase. The vertical blue lines represent some of the times when the motors
are refocused to the maximum possible photon rate to counteract rate losses.
Due to imperfections in tracking there are frequently occurring dips in the
rates, especially between midnight and 2am.

##  5 Analysis and Measurement results

###  5.1 Correlation procedure

Two binary files (one from each telescope), each including the two data
acquisition channels (see section 3.2) are correlated offline while no
measurement is running. The 4 waveform channels are read from the disk and
each of the 6 possible correlation channels is calculated, using the correlate
function from the python package cupy. We correlated the measurements of each
night during the following daytime. However, since a single $3.436-$seconds
measurement takes $13\,$seconds to be analysed using the correlation mechanism
of April 2022, we could only analyse a fraction during our time on site. The
remaining data was correlated later in the lab once the computers and hard
drives arrived back in Erlangen.

###  5.2 Noise cuts

Although the PMTs are encased in aluminum tubes to reduce noise, the Fourier
transforms of recorded photomultiplier currents as well as the resulting
$g^{(2)}$ functions show remaining sharp peaks at several frequencies. Noise
peaks would also show up in measurements carried out in the lab in Erlangen
even with further aluminum shielding. Therefore, the data has to be cleaned
before further analysis is done. The first cleaning is done with a lowpass
filter with a cut-off frequency of $200\,$MHz. It is applied to all 6
correlation channels. For higher frequencies a notch filter is applied.

###  5.3 Zero-baseline correlations

In both telescopes, the zero-baseline correlation between the two channels in
one telescope is calculated. We denote this type of (single telescope)
correlation as "auto-correlation", even though this is not in accordance with
its usual meaning of correlating a single signal with itself. The purpose of
the auto-correlation is two-fold: it provides an additional data point for
baseline zero in the spatial coherence plot, and it enables monitoring
systematic effects in the telescopes that can affect the correlations. Since
the baseline between the two channels in one telescope does not change over
time, the measured coherence time should neither. Possible effects including
variation of the angle of incidence of the incoming light on the interference
filter however can result in a varying coherence time, which can also transfer
to the cross correlations between the telescopes, and should be checked
carefully.

Figure 9: Top: zero baseline ("auto-")correlations of Acrux measured in CT3.
For reasons of clarity, only every third measurement segment is plotted,
together with a Gaussian fit. The times in the legend correspond to the
central time of each segment. Bottom: integrated coherence times at zero
baseline of all measurement segments of Acrux in both CT3 and CT4. The shaded
areas show the standard deviation of the data around the mean (dashed lines).

Fig. 9 (top) shows auto-correlation measurements of the star system Acrux at
different moments in time. For a better overview the measurements are plotted
with an offset on the vertical axis with respect to each other. The photon
bunching peak at time difference $0$ is clearly visible, but also systematic
patterns, especially at negative time differences, are apparent. Systematic
oscillations of the same magnitude may also be present in the peak region and
artificially increase or decrease the found peak integrals. This results in an
additional systematic uncertainty, which is of special interest in the
investigation of the combined spatial coherence and auto-correlation (figures
13, 12, 14) of the observed targets, and is further discussed in section 5.8.

The bunching peak can be described sufficiently well by a Gaussian shape, its
integral is expected to be equal to the coherence time in Eq. 3 for $k(b=0)=1$
if the star is assumed to not be resolved by the telescope’s diameter. We
obtain

$\tau_{c}(b=0)=0.5\times 0.842\times\frac{(465\,\textnormal{nm})^{2}}{c\times
2\,\textnormal{nm}}=152\,\textnormal{fs}$ (4)

The peak integrals of every measurement interval of Acrux, ranging over three
different nights, are displayed in Fig. 9 (bottom) for both telescopes. The
measured coherence times show a clear deviation from the expected value of
$152\,$fs, which directly results in a reduced signal-to-noise. Apart from the
fact that the multiple star system might already be partly resolved by a
single telescope, this issue is further discussed in section 5.9 for Nunki and
Shaula, where the same systematic is found for the combined auto- and cross
correlation data. The measured coherence times, however, are consistent with
each other and constant over time, indicating that the interferometer’s and
telescope’s systematics are stable and do not further affect the measurement
procedure.

###  5.4 Optical path delay compensation

While the telescopes keep tracking the same target in the sky, the optical
path length of the light arriving at both telescopes is different and
continuously changes during the measurement. Fig. 10 shows two exemplary cases
where different pointing directions lead to the light arriving at different
telescopes first.

Figure 10: Different optical path delays between both telescopes at different
pointing positions. In the red scenario the light arrives at the upper
telescope first, while in the blue scenario it is the lower telescope.

The optical path delay directly relates to the time position $\tau$ in
$g^{(2)}(\tau)$ of the expected correlation peak. Since we need to accumulate
data over time spans of order 1 hour to measure significant bunching signals,
we need to continuously correct for the optical path delay within one
measurement segment to prevent washing out the signal peak over too many time
bins. The correction is applied digitally by checking the absolute time of
acquisition of every $3.4-$s file. The position of the observed object in the
sky is computed and from that the relative optical path delay and delay time
between the two telescopes are calculated. Each elementary $g^{(2)}$ function
consisting of $3.436\,$s of data is shifted by the corresponding number of
time bins before the averaging process.

###  5.5 Cable delay correction

After the optical path delay correction, the remaining electronic delay -
induced by the different cable lengths - results in bunching peaks close to
two different time delays: if two channels of similar cable lengths are
correlated, the bunching peak is close to $\tau=0$. That is true for the
correlations of CT3 channel A with CT4 channel A, which both use signal cables
of $\approx 10\,$m, and the correlation of CT3 channel B with CT4 channel B,
which both use signal cables of $\approx 40\,$m. In the case of correlating
one $10-$m channel with one $40-$m channel, the additional signal travel time
in the longer cable shifts the bunching peaks to $\tau\approx 117\,$ns,
corresponding to a signal travel speed in the cables of $0.855\,c$. This is
the case for the correlations of CT3 channel A with CT4 channel B and CT4
channel A with CT3 channel B (and also for the auto-correlations in each
telescope, which have already been cable-delay corrected in Fig. 9).

Fig. 11 (upper) shows the four cross correlation channels of the star Shaula
by accumulating data over the entire measurement time ($\approx 19\,$h). By
doing this one averages over all the different spatial coherences present
during the measurement period and thus cannot extract specific spatial
coherence information. Instead, due to the large statistics the bunching
signals are of high significance and can be used to determine the exact time
delay for each correlation channel. In Fig. 11 (upper) the two time delay
regions around $0$ and $117\,$ns are visible among the four correlations. By
applying Gaussian fits to the bunching peaks, the cable delay times can be
determined with low uncertainty. It is also apparent that with increasing
total cable lengths the signal height (and also integral) decreases, resulting
in the highest signal for the correlation of both $10-$m channels, and the
smallest signal for both $40-$m channels. This behaviour is not expected and
under investigation in the lab for future campaigns, but appears to be
consistent amongst the different observed targets and thus doesn’t falsify
later evaluations. Effects of potential changes in cable lengths due to
thermal expansions are on the order of less than $100\,$ps signal time delay
and can be neglected.

Figure 11: Top: bunching peaks of the four cross correlation channels measured
with Shaula, accumulated over $16\,$hours. Bottom: the four peaks are shifted
to zero time delay and averaged.

###  5.6 Data averaging and parameter fixing

Since each of the four cross-correlation channels is an independent
measurement, it should be possible to independently use every one of them to
observe the change in spatial coherence with changing projected telescope
baselines, and from that extract the angular size of the star. However, we
recognize that upon dividing the complete measurement time of a target into
reasonably short time segments for investigation of the change in spatial
coherence (for a detailed explanation of what is called reasonable and how it
is done see section 5.7), the signal-to-noise of the bunching peaks -
especially at large baselines and thus low spatial coherence - is not high
enough to conveniently fit a Gaussian function. Therefore we use all four
correlation channels in each time segment and average over them. In Fig. 11
(lower) this is at first done for the the total accumulated time of Shaula.
With the determined delay times from the upper plot each of the channels is
shifted in time to overlay the bunching peaks at zero time delay, and then
averaged inversely proportional to the square of the standard error of the
data points excluding the peak region. The averaged signal then has smaller
fluctuations and thus allows for improved fitting of Gaussian functions to the
bunching peaks even at shorter measurement times. However, we recognize that
it is beneficial to fix the mean and the width of the Gaussian function when
fitting into a single segment of the data, and only fit the amplitude. The
mean and the width used for any segment analysis are extracted from a Gaussian
fit to the time-accumulated bunching peak in Fig. 11 (lower).

###  5.7 Measurement segments

For a detailed spatial coherence analysis the accumulated data has to be
divided into smaller time segments in which the correlation is calculated. The
choice of the segment time length is a trade off between uncertainties in
horizontal and vertical direction in the spatial correlation plots of
coherence time versus baseline (see e.g. upper right plot in Fig. 12): the
smaller the time segment is chosen to be, the less projected baseline range is
being scanned by the telescopes during that time, and hence the smaller the
uncertainty in the baseline. However, short measurement times result in less
photon statistics, the signal-to-noise of the bunching peaks is smaller and
the uncertainties in determining the coherence times become larger. Latter
effect also depends on the photon rates, bright stars allow for shorter
measurement segments, whereas fainter stars require longer ones.

Evening | Shaula | Nunki | Acrux
---|---|---|---
04/16 | - | - |  $5\times 29.78\,$min
04/17 | - | - |  $15\times 30.52\,$min
04/18 | - |  $3\times 87.45\,$min |  $4\times 26.34\,$min
04/19 |  $4\times 91.84\,$min | - | -
04/20 |  $4\times 68.49\,$min | - | -
04/21 |  $4\times 79.78\,$min | - | -
04/22 | - |  $3\times 74.91\,$min | -
04/23 | - |  $2\times 63.68\,$min | -
Total | 16h 0min | 10h 14min | 11h 52min
Table 1: Overview of the measurement segments and segment times for the three
main targets of the campaign, Shaula, Nunki and Acrux.

Table 1 lists the separation into measurement segments for Shaula, Nunki and
Acrux in each observation night. Since the photon rates of the bright star
system Acrux ($0.76\,$mag, Corben (1966)) are significantly ($\approx$ factor
2) higher than for Shaula and Nunki, which both are similarly bright, we
divided the Acrux data into smaller measurement segments. As the total
measurement time of a target in different observation nights vary, we also
used slightly different segment times for the same target in different nights
in order to optimally use the observation time of a night without removing a
significant amount of data that didn’t fill up to a segment.

###  5.8 Spatial coherence curves

Fig. 12 (left) shows the $g^{(2)}$ functions of the different measurement
segments of Shaula. For a better visualization of single measurements, they
are separated from each other in vertical direction. The correlation of the
four single channels are displayed as thin lines with the colour code being
the same as in Fig. 11. The thick lines are the averages of theses channels,
which are used to apply Gaussian fits (black dashed lines) with the amplitude
being the only fit parameter. Mean and width are extracted from the time-
averaged signal in the lower plot in Fig. 11, we do not fit an offset to the
correlations. The integral of the fit is the coherence time as defined in
section 2. The displayed time stamps are UTC and denote the centre time of
each measurement segment.

For each measurement segment we also calculate the corresponding telescope
baseline value. Each $3.4$-second measurement file of a segment can be mapped
to a corresponding baseline. We average over all these baseline values within
a segment in order to get a single value, the $1\sigma$ fluctuation of all the
baselines in a segment is used as uncertainty on the baseline (horizontal
error).

In Fig. 12 (top right) the coherence times are plotted versus the
corresponding average baseline of the measurement segment. The black data
point close to zero baseline is the auto correlation averaged for both
telescopes over the entire measurement time. Even though it is called zero-
baseline correlation before, we plotted it at $(5.4\pm 2.5)\,$m as this is the
mean distance of two photon impact positions on the mirror of one telescope.
The auto correlation is plotted in the lower right plot. As mentioned before,
it is expected to have the highest signal-to-noise and smallest uncertainty of
the coherence time due to accumulating statistics over the whole measurement
time. However, since we find systematic noise contributions in both telescopes
affecting the auto (but not the cross) correlations, the auto correlation
suffers from a high uncertainty. To estimate the effect of systematic
oscillations in the baseline, we take the strongest up- and down oscillations
measured in the auto-correlation of Acrux (see section 5.3), averaged over the
entire measurement time, in the time range displayed in fig. 9. These
fluctuations add as an uncertainty to the measured amplitude, and therefore to
the peak integral. The systematic uncertainties are displayed as red band
around the fit in the auto-correlations (figures 13) and 12.

Fig. 13 shows cross correlations, auto correlations and the spatial coherence
values for Nunki.

###  5.9 Data analysis

For our model we assume that Nunki can be treated as a single star, since the
primary component is dominant in the system and potential companions do not
significantly contribute to the measured photon rates. It is an appropriate
approximation, since Nunki only has a mag 9.95 companion (Mason et al., 2001).
We use a uniform disc model and fit the corresponding function to the spatial
coherence data of Nunki (Davis et al., 2000):

$f(x)=a\left[\frac{2J_{1}(\pi x\varphi/\lambda)}{\pi
x\varphi/\lambda}\right]^{2}$ (5)

with $x$ being the baseline, $\lambda=465\,$nm the operating wavelength and
$J_{1}$ being the Bessel function of first kind. The two fit parameters are
the constant $a$ and the angular diameter $\varphi$.

The constant $a$ corresponds to the coherence time at zero baseline and should
thus be equal to Eq. 3 with $k_{s}=1$, resulting in
$a_{\textnormal{exp}}=152\,$fs. We however obtain
$a_{\textnormal{Nunki}}=(43.76\pm 10.40)\,$fs, indicating a coherence loss
factor of about $3.5$. The reasons for this loss are yet unclear. We are
investigating potential coherence losses due to the use of long signal cables,
as is already indicated in Fig. 11, where correlations between channels with
long cables are significantly smaller than correlations between channels with
short cables. We further identify the narrow bandpass filters as potential
source of coherence loss. If light on these interference filters comes in at
an angle, the transmitted wavelengths are shifted to smaller values. Light
with different inclination angles on the filter therefore broadens the
transmission spectrum of the filters and results in reduced temporal
coherence. For that reason future intensity interferometry observations at
H.E.S.S. will feature broader wavelength filters of $10\,$nm FWHM (see also
section 6).

The second fit parameter $\varphi$ directly yields the angular diameter of the
star, which is main subject of observation. We obtain an angular diameter of
$\varphi_{\textnormal{Nunki}}=(0.52\pm 0.07)\,$mas.

Treating the other two observed targets as single stars is not suitable.
Shaula is a triple star system (Handler & Schwarzenberg-Czerny, 2013), with
the two main components having a magnitude ratio of $0.66\pm 0.10$ (Tango et
al., 2006). The constellation of Acrux consists of many individual components
- ($\alpha^{1}$ Cru and $\alpha^{2}$ Cru can even be separated visually)
(Kołaczek-Szymański, 2020). In both cases we do not expect the data to (fully)
agree with a single star fit model. We however applied a fit according to
equation 5 (see figures 12 and 14), and conclude from the fact that the
reduced $\chi^{2}$ for Shaula ($2.42$) and Acrux ($3.01$) are significantly
larger than for Nunki ($1.03$) that the data in fact don’t align well with a
single star model. Especially for Acrux we took many data at large baselines,
which show indications of a more complex source geometry. Potential future
observations may allow for more detailed analyses treating these systems as
binary or multiple star systems and extracting both orbital and stellar
parameters.

Despite Shaula being treated best as a binary (neglecting the faint secondary
component) (Tango et al., 2006), the single star fit model does contain
information about the angular diameter of the single stars. Even though the
fit misses fast oscillations in the spatial coherence curve, induced by the
angular separation of both dominant stars, the overall trend of the curve is
related to the angular diameters of the stellar disks. Assuming both stars
have similar angular diameters, we find
$\varphi_{\textnormal{Shaula}}=(0.49\pm 0.06)\,$mas for both stars. This
value, however, should be treated with care. The assumption of similar stellar
radii as well as the arbitrary sampling of the potentially fast oscillating
spatial coherence curve are most likely causing systematic errors.

###  5.10 Data comparison

From literature one finds $\varphi_{\textnormal{Nunki}}=0.68\,$mas (Underhill
et al., 1979), computed from photometry data (Blackwell & Shallis, 1977),
which shows some deviance to our measurements of $(0.51\pm 0.05)\,$mas. For
Shaula we find $(5.2\pm 1.7)\,\textup{R}_{\odot}$ for the primary and $(5.3\pm
1.3)\,\textup{R}_{\odot}$ for the tertiary star (Handler & Schwarzenberg-
Czerny, 2013), which supports our assumption of similar radii in section 5.9,
and from the Hipparcos parallax results a distance of
$d_{\textnormal{Shaula}}=(175\pm 23)\,$pc. However, Tango et al. (2006) obtain
a distance of $d_{\textnormal{Shaula}}=(112\pm 5)\,$pc. Using the latter we
calculate an angular diameter of $\varphi_{\textnormal{Shaula}}=(0.43\pm
0.14)\,$mas, whereas (Underhill et al., 1979) obtain an angular diameter of
$0.64\,$mas computed from infrared photometry data (Blackwell & Shallis,
1977). We make clear, that due to the disagreeing set of parameters that is
found in literature, as well as our simplified description model of Shaula, we
are not making a strong statement about an agreement of our measurements with
previous ones.

Figure 12: $g^{(2)}$ cross correlation functions of each measurement segment
(left), the zero-baseline correlation (bottom right) and the corresponding
spatial correlations (top right) of Shaula Figure 13: $g^{(2)}$ cross
correlation functions of each measurement segment (left), the zero-baseline
correlation (bottom right) and the corresponding spatial correlations (top
right) of Nunki Figure 14: Spatial correlations of Acrux. The horizontal axis
is divided into three parts for better visualization. The colours mark the
three measurements nights of Acrux. The zero-baseline data point is marked in
black on the very left of the plot.

##  6 Conclusion and outlook

We equipped two of the Phase I telescopes of the H.E.S.S. gamma ray
observatory with an external intensity interferometry setup for the time of
the moonlight measurement break in April 2022. We measured the spatial
coherence curves of two stars and were able to extract angular diameters with
a precision at the $10\,$per cent scale, while combining auto-correlation
measurements in the same telescope with cross-correlation measurements between
two telescopes. We noticed that the absolute measured coherence times are
smaller than expected, which could partly be caused by imperfections in the
light inclination at the small ($2\,$nm) bandwidth interference filters, which
limits signal-to-noise of the observations.

As for the next measurement campaign in April/May 2023 some changes are made
to the setup.
First of all the bandwidth of the interference filters are broadened to
$10\,$nm instead of $2\,$nm. As described in subsection 5.9 this results in
more stable optics, and therefore potentially less spatial coherence losses
and higher signal-to-noise.
Furthermore, we are expanding our setup to two colour measurements. Each of
the two PMTs on one telescope has their own band-pass filter at a different
wavelength ($375$ and $470\,$nm) making it possible to measure with two
colours simultaneously - at the cost of the zero baseline measurement. Both
measurement modes are interchangeable however.
The third adjustment is equipping a third telescope with our setup to gain
more telescope baselines.

## Acknowledgements

We thank the H.E.S.S. collaboration for reviewing this work and for allowing
us to use their telescopes for our measurement campaign. We also thank the
H.E.S.S. local crew for their support on site and the operations team,
especially Dmitriy Kostunin and Gianluca Giavitto, for their help online and
with installing our setup. Thanks to Dmitry Malyshev for his contributions to
preparations and discussions, and to Alison Mitchell for her thoughts and
input on this work. This work was supported with a grant by the Deutsche
Forschungsgemeinschaft (‘Optical intensity interferometry with the H.E.S.S.
gamma-ray telescopes’- FU 1093/3-1).

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author. Correlation histograms are available in time-intervals
of $3.436\,$s. Due to the large size of the digitized waveforms in excess of
several TB, the raw data can not be made available online.

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