Searcharxiv⌕ Search

arXiv subjects

Athanasios Sourmelidis

Publications and source records attributed to Athanasios Sourmelidis.

18 recordsLinked to original sources
arXiv2026

Joint extreme values of $L$-functions on and off the critical line

It is shown that any number of distinct primitive $\mathrm{GL}(1)$ and $\mathrm{GL}(2)$ $L$-functions can simultaneously attain large values on the critical line. This is an unconditional improvement of a general result due to Heap and Li who have assumed the Riemann Hypothesis for more than three such $L$-functions. The joint distribution of $\mathrm{GL}(m)$ $L$-functions to the right of the critical line is also studied under certain zero-density estimates. In particular, we can partially recover results of Inoue and Li on Dirichlet $L$-functions and generally improve upon the work of Mahatab, Pa\'nkowski and Vatwani on the class of $L$-functions introduced by Selberg. The main machinery in both cases, on and off the critical line, is the resonance method of Soundararajan and Hilberdink/Voronin, respectively. On the critical line we additionally introduce a variation of Heath-Brown's method for the fractional moments of the Riemann zeta-function which makes it possible to avoid using any information on the zero distribution of $L$-functions whose degree is less than three.

math.NT↗
arXiv2025

The Lindel\"of Hypothesis for Zeta Zero Ordinates

We provide conditional and unconditional asymptotic formulae for the exponential sums $\sum_\gamma\,\gamma^{-i\tau}$, where the summation is over the ordinates of the nontrivial zeros $\rho=\beta+i\gamma$ of the Riemann zeta-function. In particular, the obtained results are related to the Lindel\"of Hypothesis for these ordinates (in the sense of Gonek et al. [10]).

math.NT↗
arXiv2024

An additive application of the resonance method

We improve upon an Omega result due to Soundararajan with respect to general trigonometric polynomials having positive Fourier coefficients. Instead of Dirichlet's approximation theorem we employ the resonance method and this leads to better extreme results in lattice point problems such as Dirichlet's divisor problem and Gauss' circle problem. Moreover, the present approach shows that the resonance method can also be viewed as an additive device, which has been used in multiplicative problems so far. Its extension to trigonometric polynomials with complex coefficients is also discussed and its connection to Bohr and Jessen's proof of Kronecker's theorem is highlighted.

math.NT↗
arXiv2023

Discrete $\Omega$-results for the Riemann zeta function

We study lower bounds for the Riemann zeta function $\zeta(s)$ along vertical arithmetic progressions in the right-half of the critical strip. We show that the lower bounds obtained in the discrete case coincide, up to the constants in the exponential, with the ones known for the continuous case, that is when the imaginary part of $s$ ranges on a given interval. Our methods are based on a discretization of the resonance method for estimating extremal values of $\zeta(s)$.

math.NT↗
arXiv2023

Notes on Universality in Short Intervals and Exponential Shifts

We improve a recent universality theorem for the Riemann zeta-function in short intervals due to Antanas Laurin\v{c}ikas with respect to the length of these intervals. Moreover, we prove that the shifts can even have exponential growth. This research was initiated by two questions proposed by Laurin\v cikas in a problem session of a recent workshop on universality.

math.NT↗
arXiv2023

Continuous and discrete universality of zeta-functions: Two sides of the same coin?

In 1975 Voronin proved the universality theorem for the Riemann zeta-function $\zeta(s)$ which roughly says that any admissible function $f(s)$ is approximated by $\zeta(s)$. A few years later Reich proved a discrete analogue of this result. The proofs of these theorems are almost identical but it is not known whether one of them implies the other. We will see that if we translate the question in the language of linear dynamics then there is a link which we exploit to obtain in a straightforward way a big variety of discrete universality results appearing in the literature.

math.NT↗
arXiv2023

Spirals of Riemann's Zeta-Function --Curvature, Denseness, and Universality--

This article deals with applications of Voronin's universality theorem for the Riemann zeta-function $\zeta$. Among other results we prove that every plane smooth curve appears up to a small error in the curve generated by the values $\zeta(\sigma+it)$ for real $t$ where $\sigma\in(1/2,1)$ is fixed. In this sense, the values of the zeta-function on any such vertical line provides an atlas for plane curves. In the same framework, we study the curvature of curves generated from $\zeta(\sigma+it)$ when $\sigma>1/2$ and we show that there is a connection with the zeros of $\zeta'(\sigma+it)$. Moreover, we clarify under which conditions the real and the imaginary part of the zeta-function are jointly universal.

math.NT↗
arXiv2023

On restricted averages of Dedekind sums

We investigate the averages of Dedekind sums over rational numbers in the set $$\mathscr{F}_\alpha(Q):=\{\, {v}/{w}\in \mathbb{Q}: 0<w\leq Q\,\}\cap [0, \alpha)$$ for fixed $\alpha\leq 1/2$. In previous work, we obtained asymptotics for $\alpha=1/2$, confirming a conjecture of Ito in a quantitative form. In the present article we extend our former results, first to all fixed rational $\alpha$ and then to almost all irrational $\alpha$. As an intermediate step we obtain a result quantifying the bias occurring in the second term of the asymptotic for the average running time of the \textit{by-excess} Euclidean algorithm, which is of independent interest.

math.NT↗
arXiv2022

Bias in the number of steps in the Euclidean algorithm and a conjecture of Ito on Dedekind sums

We investigate the number of steps taken by three variants of the Euclidean algorithm on average over Farey fractions. We show asymptotic formulae for these averages restricted to the interval $(0,1/2)$, establishing that they behave differently on $(0,1/2)$ than they do on $(1/2,1)$. These results are tightly linked with the distribution of lengths of certain continued fraction expansions as well as the distribution of the involved partial quotients. As an application, we prove a conjecture of Ito on the distribution of values of Dedekind sums. The main argument is based on earlier work of Zhabitskaya, Ustinov, Bykovski\u{i} and others, ultimately dating back to Heilbronn, relating the quantities in question to counting solutions to a certain system of Diophantine inequalities. The above restriction to only half of the Farey fractions introduces additional complications.

math.NT↗
arXiv2021

Pair Correlation of the Fractional Parts of $\alpha n^\theta$

Fix $\alpha,\theta >0$, and consider the sequence $(\alpha n^{\theta} \mod 1)_{n\ge 1}$. Since the seminal work of Rudnick--Sarnak (1998), and due to the Berry--Tabor conjecture in quantum chaos, the fine-scale properties of these dilated mononomial sequences have been intensively studied. In this paper we show that for $\theta \le 1/3$, and $\alpha>0$, the pair correlation function is Poissonian. While (for a given $\theta \neq 1$) this strong pseudo-randomness property has been proven for almost all values of $\alpha$, there are next-to-no instances where this has been proven for explicit $\alpha$. Our result holds for all $\alpha>0$ and relies solely on classical Fourier analytic techniques. This addresses (in the sharpest possible way) a problem posed by Aistleitner--El-Baz--Munsch (2021).

math.NT↗
arXiv2020

Riemann-Type Functional Equations -- Julia Line and Counting Formulae --

We study Riemann-type functional equations with respect to value-distribution theory and derive implications for their solutions. In particular, for a fixed complex number $a\neq0$ and a function from the Selberg class $\mathcal{L}$, we prove a Riemann-von Mangoldt formula for the number of a-points of the $\Delta$-factor of the functional equation of $\mathcal{L}$ and an analog of Landau's formula over these points. From the last formula we derive that the ordinates of these $a$-points are uniformly distributed modulo one. Lastly, we show the existence of the mean-value of the values of $\mathcal{L}(s)$ taken at these points.

math.NT↗
arXiv2020

Dirichlet Series with Periodic Coefficients and their Value-Distribution Near the Critical Line

The class of Dirichlet series associated with a periodic arithmetical function $f$ includes the Riemann zeta-function as well as Dirichlet $L$-functions to residue class characters. We study the value-distribution of these Dirichlet series $L(s;f)$, resp. their analytic continuation in the neighborhood of the critical line (which is the abscissa of symmetry of the related Riemann-type functional equation). In particular, for a fixed complex number $a\neq 0$, we prove for an even or odd periodic $f$ the number of $a$-points of the $\Delta$-factor of the functional equation, prove the existence of the mean-value of the values of $L(s;f)$ taken at these points, show that the ordinates of these $a$-points are uniformly distributed modulo one and apply this to show a discrete universality theorem.

math.NT↗
arXiv2020

On the Vertical Distribution of Values of $L$-functions in the Selberg Class

We prove explicit formulae for $\alpha$-points of $L$-functions from the Selberg class. Next we extend a theorem of Littlewood on the vertical distribution of zeros of the Riemann zeta-function $\zeta(s)$ to the case of $\alpha$-points of the aforementioned $L$-functions. This result implies the uniform distribution of subsequences of $\alpha$-points and from this a discrete universality theorem in the spirit of Voronin is derived.

math.NT↗
arXiv2018

On the Value-Distribution of Hurwitz Zeta-Functions with Algebraic Parameter

We study the value-distribution of the Hurwitz zeta-function with algebraic irrational parameter $\zeta(s;\alpha)=\sum_{n\geq_0}(n+\alpha)^{-s}$. In particular, we prove effective denseness results of the Hurwitz zeta-function and its derivatives in suitable strips containing the right boundary of the critical strip $1+i\mathbb{R}$. This may be considered as a first "weak" manifestation of universality for those zeta-functions.

math.NT↗
arXiv2018

On the meromorphic continuation of Beatty Zeta-Functions and Sturmian Dirichlet series

For a positive irrational number $\alpha,$ we study the ordinary Dirichlet series $\zeta_\alpha(s) = \sum\limits_{n\geq1} \lfloor\alpha n\rfloor^{-s}$ and $S_\alpha(s) = \sum\limits_{n\geq1} (\left\lceil\alpha n\right\rceil - \left\lceil \alpha (n-1)\right\rceil){n^{-s}}.$ We prove relations between them and $J_{\boldsymbol{\alpha}}(s)=\sum\limits_{n\geq1}\left(\lbrace\alpha n\rbrace-\frac{1}{2}\right)n^{-s}.$ Motivated by the previous work of Hardy and Littlewood, Hecke and others regarding $J_{\boldsymbol{\alpha}},$ we show that $\zeta_\alpha$ and $S_\alpha$ can be continued analytically beyond the imaginary axis except for a simple pole at $s=1.$ Based on the latter results, we also prove that the series $\zeta_{\alpha}(s;\beta)=\sum\limits_{n\geq0}\left(\lfloor\alpha n\rfloor+\beta\right)^{-s}$ can be continued analytically beyond the imaginary axis except for a simple pole at $s=1.$

math.NT↗
arXiv2017

The Values of the Riemann Zeta-Function on Discrete Sets

We study the values taken by the Riemann zeta-function $\zeta$ on discrete sets. We show that infinite vertical arithmetic progressions are uniquely determined by the values of $\zeta$ taken on this set. Moreover, we prove a joint discrete universality theorem for $\zeta$ with respect to certain permutations of the set of positive integers. Finally, we study a generalization of the classical denseness theorems for $\zeta$.

math.NT↗