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Andrés Chirre

Publications and source records attributed to Andrés Chirre.

At least 19 recordsLinked to original sources
arXiv2026

Explicit conditional bounds for $ζ(s)$ at the edge of the critical strip

In this paper, we obtain explicit bounds for the real part of the logarithmic derivative of the Riemann zeta-function on the line $\re s=1$, assuming the Riemann hypothesis. The proof combines the Guinand--Weil explicit formula with extremal bandlimited majorants and minorants for the Poisson kernel. As an application, we revisit the classical estimates of Littlewood for the modulus of the Riemann zeta-function and of its reciprocal on the line $\re{s}=1$, and derive a slight refinement of the bounds of Lamzouri, Li, and Soundararajan. In addition, we establish an explicit bound for the modulus of the logarithmic derivative of the Riemann zeta-function on the line $\re{s}=1$ under the Riemann hypothesis, improving the lower-order term in a result of Chirre, Valås, and Simonič.

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arXiv2025

On Goldbach numbers in short intervals

Assuming the Riemann Hypothesis, we prove that for all $x\geq 2$, there exists at least one even integer within the interval $(x, x+123\log^2x]$, that can be expressed as the sum of two primes. This result is an improvement over the recent work of Cully-Hugill and Dudek, who obtained the constant $9696$ instead of $123$.

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arXiv2025

Optimal bounds for sums of non-negative arithmetic functions

Let $A(s) = \sum_n a_n n^{-s}$ be a Dirichlet series admitting meromorphic continuation to the complex plane. Assume we know the location of the poles of $A(s)$ with $|\Im s| \leq T$, and their residues, for some large constant $T$. It is natural to ask how such finite spectral information may be best used to estimate partial sums $\sum_{n\leq x} a_n$. Here, we prove a sharp, general result on sums $\sum_{n\leq x} a_n n^{-σ}$ for $a_n$ non-negative, giving an optimal way to use information on the poles of $A(s)$ with $|\Im s|\leq T$, with no need for zero-free regions. We give not just bounds, but an explicit formula with compact support. Our bounds on $ψ(x)-x$ are, unsurprisingly, better and often simpler than a long list of existing explicit versions of the Prime Number Theorem. We treat the case of $M(x)$ and similar functions in a companion paper. Our solution mixes a Fourier-analytic approach in the style of Wiener--Ikehara with contour-shifting, using optimal approximants of Beurling--Selberg type found in (Graham--Vaaler, 1981).

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arXiv2025

Optimal bounds for sums of bounded arithmetic functions

Let $A(s) = \sum_n a_n n^{-s}$ be a Dirichlet series with meromorphic continuation. Say we are given information on the poles of $A(s)$ with $|\Im s| \leq T$ for some large constant $T$. What is the best way to use such finite spectral data to give explicit estimates on sums $\sum_{n\leq x} a_n$? The problem of giving explicit bounds on the Mertens function $M(x) = \sum_{n\leq x} μ(n)$ illustrates how open this basic question was. Bounding $M(x)$ might seem equivalent to estimating $ψ(x) = \sum_{n\leq x} Λ(n)$ or the number of primes $\leq x$. However, we have long had fairly good explicit bounds on prime counts, while bounding $M(x)$ remained a notoriously stubborn problem. We prove a sharp, general result on sums $\sum_{n\leq x} a_n n^{-σ}$ for $a_n$ bounded, giving an optimal way to use information on the poles of $A(s)$ with $|\Im s|\leq T$ and no data on the poles above. Our bounds on $M(x)$ are stronger than previous ones by many orders of magnitude. (Similar results for $ψ(x)$ are given in a companion paper.) Using rigorous residue computations by D. Platt, we obtain, for $x\geq 1$, $$|M(x)|\leq \frac{3}{π\cdot 10^{10}}\cdot x + 11.39 \sqrt{x}.$$ This is a corollary of our main result, essentially an explicit formula with the contribution of each pole clearly stated; we shall discuss how this finer structure can be useful. Our proof mixes a Fourier-analytic approach in the style of Wiener--Ikehara with contour-shifting, using optimal approximants of Beurling--Selberg type (Carneiro--Littmann, 2013); for $σ=1$, the approximant in (Vaaler, 1985) reappears. While we proceed independently of existing explicit work on $M(x)$ and $ψ(x)$, our method has an important step in common with work on another problem by (Ramana--Ramaré, 2020).

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arXiv2023

Remarks on a Formula of Ramanujan

Assuming an averaged form of Mertens' conjecture and that the ordinates of the non-trivial zeros of the Riemann zeta function are linearly independent over the rationals, we analyze the finer structure of the terms in a well-known formula of Ramanujan.

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arXiv2023

Bounding zeta on the 1-line under the partial Riemann hypothesis

We provide explicit bounds in the theory of the Riemann zeta-function at the line $\Re{s}=1$, assuming that the Riemann hypothesis holds until the height $T$. In particular, we improve some bounds, in finite regions, for the logarithmic derivative and the reciprocal of the Riemann zeta-function.

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arXiv2023

An extremal problem and inequalities for entire functions of exponential type

We study two variations of the classical one-delta problem for entire functions of exponential type, known also as the Carathéodory--Fejér--Turán problem. The first variation imposes the additional requirement that the function is radially decreasing while the second one is a generalization which involves derivatives of the entire function. Various interesting inequalities, inspired by results due to Duffin and Schaeffer, Landau, and Hardy and Littlewood, are also established.

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arXiv2023

Conditional estimates for the logarithmic derivative of Dirichlet $L$-functions

Assuming the Generalized Riemann Hypothesis, we establish explicit bounds in the $q$-aspect for the logarithmic derivative $\left(L'/L\right)\left(σ,χ\right)$ of Dirichlet $L$-functions, where $χ$ is a primitive character modulo $q\geq 10^{30}$ and $1/2+1/\log{\log{q}}\leqσ\leq 1-1/\log\log q$. In addition, for $σ=1$ we improve upon the result by Ihara, Murty and Shimura (2009). Similar results for the logarithmic derivative of the Riemann zeta-function are given.

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arXiv2023

Point evaluation in Paley--Wiener spaces

We study the norm of point evaluation at the origin in the Paley--Wiener space $PW^p$ for $0 < p < \infty$, i. e., we search for the smallest positive constant $C$, called $\mathscr{C}_p$, such that the inequality $|f(0)|^p \leq C \|f\|_p^p$ holds for every $f$ in $PW^p$. We present evidence and prove several results supporting the following monotonicity conjecture: The function $p\mapsto \mathscr{C}_p/p$ is strictly decreasing on the half-line $(0,\infty)$. Our main result implies that $\mathscr{C}_p p/2$ for $1 \leq p < 2$. We also estimate the asymptotic behavior of $\mathscr{C}_p$ as $p \to \infty$ and as $p \to 0^+$. Our approach is based on expressing $\mathscr{C}_p$ as the solution of an extremal problem. Extremal functions exist for all $0<p<\infty$; they are real entire functions with only real zeros, and the extremal functions are known to be unique for $1\leq p < \infty$. Following work of Hörmander and Bernhardsson, we rely on certain orthogonality relations associated with the zeros of extremal functions, along with certain integral formulas representing respectively extremal functions and general functions at the origin. We also use precise numerical estimates for the largest eigenvalue of the Landau--Pollak--Slepian operator of time--frequency concentration. A number of qualitative and quantitative results on the distribution of the zeros of extremal functions are established. In the range $1<p<\infty$, the orthogonality relations associated with the zeros of the extremal function are linked to a de Branges space. We state a number of conjectures and further open problems pertaining to $\mathscr{C}_p$ and the extremal functions.

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arXiv2022

Hilbert spaces and low-lying zeros of L-functions

Generalizing previous work of Iwaniec, Luo, and Sarnak (2000), we use information from one-level density theorems to estimate the proportion of non-vanishing of $L$-functions in a family at a low-lying height on the critical line (measured by the analytic conductor). To solve the Fourier optimization problems that arise, we provide a unified framework based on the theory of reproducing kernel Hilbert spaces of entire functions (there is one such space associated to each symmetry type). Explicit expressions for the reproducing kernels are given. We also revisit the problem of estimating the height of the first low-lying zero in a family, considered by Hughes and Rudnick (2003) and Bernard (2015). We solve the associated Fourier optimization problem in this setting by establishing a connection to the theory of de Branges spaces of entire functions and using the explicit reproducing kernels. In an appendix, we study the related problem of determining the sharp embeddings between the Hilbert spaces associated to the five symmetry types and the classical Paley-Wiener space.

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arXiv2022

On Montgomery's pair correlation conjecture: a tale of three integrals

We study three integrals related to the celebrated pair correlation conjecture of H. L. Montgomery. The first is the integral of Montgomery's function $F(α, T)$ in bounded intervals, the second is an integral introduced by Selberg related to estimating the variance of primes in short intervals, and the last is the second moment of the logarithmic derivative of the Riemann zeta-function near the critical line. The conjectured asymptotic for any of these three integrals is equivalent to Montgomery's pair correlation conjecture. Assuming the Riemann hypothesis, we substantially improve the known upper and lower bounds for these integrals by introducing new connections to certain extremal problems in Fourier analysis. In an appendix, we study the intriguing problem of establishing the sharp form of an embedding between two Hilbert spaces of entire functions naturally connected to Montgomery's pair correlation conjecture.

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arXiv2022

Optimality for the two-parameter quadratic sieve

We study the two-parameter quadratic sieve for a general test function. We prove, under some very general assumptions, that the function considered by Barban and Vehov [BV68] and Graham [Gra78] for this problem is optimal up to and including the second-order term. We determine that second-order term explicitly.

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arXiv2022

A note on the mean values of the derivatives of $ζ'/ζ$

Assuming the Riemann hypothesis, we obtain a formula for the mean value of the $k$-derivative of $ζ'/ζ$, depending on the pair correlation of zeros of the Riemann zeta-function. This formula allows us to obtain new equivalences to Montgomery's pair correlation conjecture. This extends a result of Goldston, Gonek, and Montgomery where the mean value of $ζ'/ζ$ was considered.

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arXiv2021

The second moment of $S_n(t)$ on the Riemann hypothesis

Let $S(t) = \tfrac{1}π \arg ζ\big({1/2} + it \big)$ be the argument of the Riemann zeta-function at the point $\tfrac12 + it$. For $n \geq 1$ and $t>0$ define its antiderivatives as \begin{equation*} S_n(t) = \int_0^t S_{n-1}(τ) \hspace{0.08cm} \rm dτ+ δ_n, \end{equation*} where $δ_n$ is a specific constant depending on $n$ and $S_0(t) := S(t)$. In 1925, J. E. Littlewood proved, under the Riemann Hypothesis, that $$ \int_{0}^{T}|S_n(t)|^2 \hspace{0.06cm} \rm dt = O(T), $$ for $n\geq 1$. In 1946, Selberg unconditionally established the explicit asymptotic formulas for the second moments of $S(t)$ and $S_1(t)$. This was extended by Fujii for $S_n(t)$, when $n\geq 2$. Assuming the Riemann Hypothesis, we give the explicit asymptotic formula for the second moment of $S_n(t)$ up to the second-order term, for $n\geq 1$. Our result conditionally refines Selberg's and Fujii's formulas and extends previous work by Goldston in 1987, where the case $n=0$ was considered.

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arXiv2021

Fourier optimization and quadratic forms

We prove several results about integers represented by positive definite quadratic forms, using a Fourier analysis approach. In particular, for an integer $\ell\geq 1$, we improve the error term in the partial sums of the number of representations of integers that are a multiple of $\ell$. This allows us to obtain unconditional Brun-Titchmarsh-type results in short intervals, and a conditional Cramér-type result on the maximum gap between primes represented by a given positive definite quadratic form.

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arXiv2021

Large Oscillations of the Argument of the Riemann Zeta-function

Let $S(t)$ denote the argument of the Riemann zeta-function, defined as $$ S(t)=\dfrac{1}π\,\Im\logζ(1/2+it). $$ Assuming the Riemann hypothesis, we prove that $$ S(t)=Ω_{\pm}\bigg(\dfrac{\log t\log\log\log t}{\log\log t}\bigg). $$ This improves the classical omega results of Montgomery and matches with the $Ω$-result obtained by Bondarenko and Seip.

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arXiv2021

Large values of the argument of the Riemann zeta-function and its iterates

Let $S(σ,t)=\frac{1}π\argζ(σ+it)$ be the argument of the Riemann zeta-function at the point $σ+it$ in the critical strip. For $n\geq 1$ and $t>0$, we define \begin{equation*} S_{n}(σ,t) = \int_0^t S_{n-1}(σ,τ) \,dτ+ δ_{n,σ\,}, \end{equation*} where $δ_{n,σ}$ is a specific constant depending on $σ$ and $n$. Let $0\leq β<1$ be a fixed real number. Assuming the Riemann hypothesis, we establish lower bounds for the maximum of $S_n(σ,t+h)-S_n(σ,t)$ near the critical line, on the interval $T^β\leq t \leq T$ and in a small range of $h$. This improves some results of the first author and generalizes a result of the authors on $S(t)$. We also give new omega results for $S_n(t)$, improving a result by Selberg.

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