arXiv subjects

Marc Munsch

Publications and source records attributed to Marc Munsch.

At least 19 recordsLinked to original sources

arXiv2026

Small values of signed harmonic sums and logarithmic means of multiplicative functions

We construct sequences $\{a_n\}_{n\in\mathbb{N}}\in\{-1,1\}^{\mathbb{N}}$ with small values of signed harmonic sums \[ \sum_{n\in\mathcal{A}\cap[1,N]}\frac{a_n}{n}, \] for any reasonably dense subsets $\mathcal{A}\subset\mathbb{N}.$ We apply these methods to further construct completely multiplicative functions $f:\mathbb{N}\to\{-1,1\}$ with unusually small logarithmic partial sums, that is, \[ \sum_{n \leq N}\frac{f(n)}{n} \ll \exp\left(-c_0 \frac{N^{1/3}}{(\log N)^{1/3}} \right) \] holds for infinitely many $N\to\infty$. The proofs combine careful analysis of the small-scale distribution of random harmonic sums over subsets of $\mathbb{N}$, together with deterministic inductive arguments inspired by the ``anatomy" of integers.

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arXiv2026

Large sieve inequality for sums of Legendre symbols over short intervals

Using the Burgess bound and the Selberg sieve, we obtain an upper bound for the second moment of sums of Legendre symbols over intervals , with the modulus ranging over primes . The bound is nontrivial and yields a power saving in , uniformly for , provided that , where as . This may be viewed as a short-interval analogue of a result of D. R. Heath-Brown (1995) on moments of quadratic character sums over the initial interval . In particular, it implies that, for any prescribed interval of this length, the quadratic residues and non-residues are asymptotically equidistributed for almost all primes . We also establish estimates for higher moments conditionally on the Generalised Riemann Hypothesis. These bounds rely on a sharp uniform estimate for the number of tuples of integers in a shifted interval whose product is a square.

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arXiv2026

Large values of $L(\sigma,\chi)$ for subgroups of characters

We obtain (conditional and unconditional) results on large values of $L$-functions $L(s,\chi)$ in the critical strip $1/2 \leq \Re s \leq 1$ when the character $\chi$ runs through a thin subgroup of all characters modulo an integer $q$. Some of these bounds are based on new zero-density estimates on average over a subgroup of characters. These bounds follow from a mean value estimate for character sums, which is based on the work of D. R. Heath-Brown (1979). As yet another application of this mean value estimate, we obtain an unconditional version of a conditional (on the Generalised Riemann Hypothesis) result of Z. Rudnick and A. Zaharescu (2000) about gaps between primitive roots.

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arXiv2025

Bounds for moments of quadratic character sums and theta functions

In this paper, we investigate the size of moments of quadratic character sums averaged over the family of fundamental discriminants. We obtain an asymptotic formula for all integer moments in a restricted range of parameters using a multivariate tauberian theorem. As a consequence, we prove unconditional lower bounds for all even integer moments of quadratic character sums in a wide range of parameters. Moreover, assuming the Generalised Riemann Hypothesis (GRH), we prove a sharp upper bound on moments of character sums of arbitrary length. In a similar fashion, we obtain unconditional lower bounds on moments of quadratic theta functions and matching conditional upper bounds under GRH. In the case of the second moment of theta functions, we prove an optimal upper bound unconditionally improving the previous results of Louboutin and the first named author.

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arXiv2024

Sign changes of short character sums and real zeros of Fekete polynomials

We discuss a general approach producing quantitative bounds on the number of sign changes of the weighted sums $$\sum_{n\le x}f(n)w_n$$ where $f:\mathbb{N}\to \mathbb{R}$ is a family of multiplicative functions and $w_n\in\mathbb{R}$ are certain weights.As a consequence, we show that for a typical fundamental discriminant $D,$ the partial sums of the real character $\chi_D$ change sign $\gg (\log\log D)/\log\log\log \log D$ times on very short initial interval (which goes beyond the range in Vinogradov's conjecture). We also prove that the number of real zeros (localized away from $1$) of the Fekete polynomial associated to a typical fundamental discriminant $D$ is $\gg \frac{\log\log D}{\log\log\log\log D}.$ This comes close to establishing a conjecture of Baker and Montgomery which predicts $\asymp \log \log D$ real zeros. Finally, the same approach shows that almost surely for large $x\ge 1$, the partial sums $\sum_{n\le y}f(n)$ of a (Rademacher) random multiplicative function exhibit $\gg \log \log x/\log \log\log\log x$ sign changes on the interval $[1,x].$ These results rely crucially on uniform quantitative estimates for the joint distribution of $-\frac{L'}{L}(s, \chi_D)$ at several points $s$ in the vicinity of the central point $s=1/2$, as well as concentration results for $\log L(s,\chi_D)$ in the same range, which we establish. In the second part of the paper, we obtain, for large families of discriminants, ``non-trivial" upper bounds on the number of real zeros of Fekete polynomials, breaking the square root bound. Finally, we construct families of discriminants with associated Fekete polynomials having no zeros away from $1.$

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arXiv2023

Moments and non-vanishing of $L$-functions over thin subgroups

We obtain an asymptotic formula for all moments of Dirichlet $L$-functions $L(1,\chi)$ modulo $p$ when averaged over a subgroup of characters $\chi$ of size $(p-1)/d$ with $\varphi(d)=o(\log p)$. Assuming the infinitude of Mersenne primes, the range of our result is optimal and improves and generalises the previous result of S. Louboutin and M. Munsch (2022) for second moments. We also use our ideas to get an asymptotic formula for the second moment of $L(1/2,\chi)$ over subgroups of characters of similar size. This leads to non-vanishing results in this family where the proportion obtained depends on the height of the smallest rational number lying in the dual group. Additionally, we prove that, in both cases, we can take much smaller subgroups for almost all primes $p$. Our method relies on pointwise and average estimates on small solutions of linear congruences which in turn leads us to use and modify some results for product sets of Farey fractions.

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arXiv2023

$L_q$ norms and Mahler measure of Fekete polynomials

We show that the distribution of the values of Fekete polynomials $F_p$ on the unit circle is governed, as $p\to\infty$, by an explicit limiting (non-Gaussian) random point process.This allows us to prove that the Mahler measure of $F_p$ satisfies $$M_0(F_p)\sim k_0\sqrt{p},$$ as $p\to\infty$ where $k_0=0.74083\dots,$ thus solving an old open problem. Further, we obtain an asymptotic formula for all moments $\|F_p\|_q$ with $0<q<\infty,$ resolving another open problem and improving previous results of G\"{u}nther and Schmidt (who treated the case $q=2k,$ $k\in\mathbb{N}$).

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arXiv2023

Pointwise and correlation bounds on Dedekind sums over small subgroups

We obtain new bounds, pointwisely and on average, for Dedekind sums $\mathsf{s}(\lambda,p)$ modulo a prime $p$ with $\lambda$ of small multiplicative order $d$ modulo $p$. Assuming the infinitude of Mersenne primes, the range of our results is optimal. Moreover, we relate high moments of $L(1,\chi)$ over subgroups of characters to some correlations of Dedekind sums and use our recent results to study these correlations.

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arXiv2022

Mean square values of $L$-functions over subgroups for non primitive characters, Dedekind sums and bounds on relative class numbers

An explicit formula for the mean value of $\vert L(1,\chi)\vert^2$ is known, where $\chi$ runs over all odd primitive Dirichlet characters of prime conductors $p$. Bounds on the relative class number of the cyclotomic field ${\mathbb Q}(\zeta_p)$ follow. Lately the authors obtained that the mean value of $\vert L(1,\chi)\vert^2$ is asymptotic to $\pi^2/6$, where $\chi$ runs over all odd primitive Dirichlet characters of prime conductors $p\equiv 1\pmod{2d}$ which are trivial on a subgroup $H$ of odd order $d$ of the multiplicative group $({\mathbb Z}/p{\mathbb Z})^*$, provided that $d\ll\frac{\log p}{\log\log p}$. Bounds on the relative class number of the subfield of degree $\frac{p-1}{2d}$ of the cyclotomic field ${\mathbb Q}(\zeta_p)$ follow. Here, for a given integer $d_0>1$ we consider the same questions for the non-primitive odd Dirichlet characters $\chi'$ modulo $d_0p$ induced by the odd primitive characters $\chi$ modulo $p$. We obtain new estimates for Dedekind sums and deduce that the mean value of $\vert L(1,\chi')\vert^2$ is asymptotic to $\frac{\pi^2}{6}\prod_{q\mid d_0}\left (1-\frac{1}{q^2}\right )$, where $\chi$ runs over all odd primitive Dirichlet characters of prime conductors $p$ which are trivial on a subgroup $H$ of odd order $d\ll\frac{\log p}{\log\log p}$. As a consequence we improve the previous bounds on the relative class number of the subfield of degree $\frac{p-1}{2d}$ of the cyclotomic field ${\mathbb Q}(\zeta_p)$. Moreover, we give a method to obtain explicit formulas and use Mersenne primes to show that our restriction on $d$ is essentially sharp.

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arXiv2021

Large sieve estimate for multivariate polynomial moduli and applications

We prove large sieve inequalities with multivariate polynomial moduli and deduce a general Bombieri--Vinogradov type theorem for a class of polynomial moduli having a sufficient number of variables compared to its degree. This sharpens previous results of the first author in two aspects: the range of the moduli as well as the class of polynomials which can be handled. As a consequence, we deduce that there exist infinitely many primes $p$such that $p-1$ has a prime divisor of size $\gg p^{2/5+o(1)}$ that is the value of an incomplete norm form polynomial.

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arXiv2021

Difference sets and the metric theory of small gaps

Let $(a_n)_{n \geq 1}$ be a sequence of distinct positive integers. In a recent paper Rudnick established asymptotic upper bounds for the minimal gaps of $\{a_n \alpha \bmod 1, 1 \leq n \leq N\}$ as $N \to \infty$, valid for Lebesgue-almost all $\alpha$ and formulated in terms of the additive energy of $\{a_1, \dots, a_N\}$. In the present paper we argue that the metric theory of minimal gaps of such sequences is not controlled by the additive energy, but rather by the cardinality of the difference set of $\{a_1, \dots, a_N\}$. We establish a (complicated) sharp convergence/divergence test for the typical asymptotic order of the minimal gap, and prove (slightly weaker) general upper and lower bounds which allow for a direct application. A major input for these results comes from the recent proof of the Duffin--Schaeffer conjecture by Koukoulopoulos and Maynard. We show that our methods give very precise results for slowly growing sequences whose difference set has relatively high density, such as the primes or the squares. Furthermore, we improve a metric result of Blomer, Bourgain, Rudnick and Radziwill on the order of the minimal gap in the eigenvalue spectrum of a rectangular billiard.

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arXiv2021

Additive energy and a large sieve inequality for sparse sequences

We consider the large sieve inequality for sparse sequences of moduli and give a general result depending on the additive energy (both symmetric and asymmetric) of the sequence of moduli. For example, in the case of monomials $f(X) = X^k$ this allows us to improve, in some ranges of the parameters, the previous bounds of S. Baier and L. Zhao (2005), K.~Halupczok (2012, 2015, 2018) and M.~Munsch (2020). We also consider moduli defined by polynomials $f(X) \in \mathbb{Z}[X]$, Piatetski-Shapiro sequences and general convex sequences. We then apply our results to obtain a version of the Bombieri--Vinogradov theorem with Piatetski-Shapiro moduli improving the level of distribution of R.~C.~Baker (2014).

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arXiv2021

A pair correlation problem, and counting lattice points with the zeta function

The pair correlation is a localized statistic for sequences in the unit interval. Pseudo-random behavior with respect to this statistic is called Poissonian behavior. The metric theory of pair correlations of sequences of the form $(a_n α)_{n \geq 1}$ has been pioneered by Rudnick, Sarnak and Zaharescu. Here $α$ is a real parameter, and $(a_n)_{n \geq 1}$ is an integer sequence, often of arithmetic origin. Recently, a general framework was developed which gives criteria for Poissonian pair correlation of such sequences for almost every real number $α$, in terms of the additive energy of the integer sequence $(a_n)_{n \geq 1}$. In the present paper we develop a similar framework for the case when $(a_n)_{n \geq 1}$ is a sequence of reals rather than integers, thereby pursuing a line of research which was recently initiated by Rudnick and Technau. As an application of our method, we prove that for every real number $θ>1$, the sequence $(n^θα)_{n \geq 1}$ has Poissonian pair correlation for almost all $α\in \mathbb{R}$.

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arXiv2020

Small Gál sums and applications

In recent years, maximizing Gál sums regained interest due to a firm link with large values of $L$-functions. In the present paper, we initiate an investigation of small sums of Gál type, with respect to the $L^1$-norm. We also consider the intertwined question of minimizing weighted versions of the usual multiplicative energy. We apply our estimates to: (i) a logarithmic refinement of Burgess' bound on character sums, improving previous results of Kerr, Shparlinski and Yau; (ii) an improvement on earlier lower bounds by Louboutin and the second author for the number of non vanishing theta functions associated to Dirichlet characters; and (iii) new lower bounds for low moments of character sums.

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arXiv2020

Second moment of Dirichlet $L$-functions, character sums over subgroups and upper bounds on relative class numbers

We prove an asymptotic formula for the mean-square average of $L$- functions associated to subgroups of characters of sufficiently large size. Our proof relies on the study of certain character sums ${\cal A}(p,d)$ recently introduced by E. Elma. We obtain an asymptotic formula for ${\cal A}(p,d)$ which holds true for any divisor $d$ of $p-1$ removing previous restrictions on the size of $d$. This anwers a question raised in Elma's paper. Our proof relies both on estimates on the frequency of large character sums and techniques from the theory of uniform distribution. As an application we deduce the following bound $h_{p,d}^- \leq 2\left (\frac{(1+o(1))p}{24}\right )^{m/4}$ on the relative class numbers of the imaginary number fields of conductor $p\equiv 1\mod d$ and degree $m=(p-1)/d$.

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arXiv2019

The Duffin-Schaeffer conjecture with extra divergence

The Duffin-Schaeffer conjecture is a fundamental unsolved problem in metric number theory. It asserts that for every non-negative function $ψ:~\mathbb{N} \rightarrow \mathbb{R}$ for almost all reals $x$ there are infinitely many coprime solutions $(a,n)$ to the inequality $|nx - a| < ψ(n)$, provided that the series $\sum_{n=1}^\infty ψ(n) φ(n) /n$ is divergent. In the present paper we prove that the conjecture is true under the "extra divergence" assumption that divergence of the series still holds when $ψ(n)$ is replaced by $ψ(n) / (\log n)^\varepsilon$ for some $\varepsilon > 0$. This improves a result of Beresnevich, Harman, Haynes and Velani, and solves a problem posed by Haynes, Pollington and Velani.

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arXiv2019

Large sieve inequality for power moduli

In this note we give a new bound for large sieve with characters to power moduli which improves in some range of the parameters the previous bounds of Baier/Zhao and Halupczok.

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arXiv2019

Minimizing GCD sums and applications to non-vanishing of theta functions and to Burgess' inequality

In recent years the question of maximizing GCD sums regained interest due to its firm link with large values of $L$-functions. In the present paper we initiate the study of minimizing for positive weights~$w$ of normalized $L^1$- norm the sum $\sum_{m_1 , m_2 \leqslant N} w({m_1})w({m_2})\frac{(m_1,m_2)}{\sqrt{m_1m_2}} $. We consider as well the intertwined question of minimizing a weighted version of the usual multiplicative energy. We give three applications of our results. Firstly we obtain a logarithmic refinement of Burgess' bound on character sums $\displaystyle{\sum_{M 0$, there exists at least $ \gg p/(\log p)^{ δ+o(1)}$ (with $δ=1-\frac{1+\log_2 2}{\log 2} \approx 0.08607$) even characters such that $θ(x,χ) \neq 0$. Lastly we obtain lower bounds on small moments of character sums.

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