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arXiv2009

On the distribution of imaginary parts of zeros of the Riemann zeta function, II

We continue our investigation of the distribution of the fractional parts of $a γ$, where $a$ is a fixed non-zero real number and $γ$ runs over the imaginary parts of the non-trivial zeros of the Riemann zeta function. We establish some connections to pair correlation functions and the distribution of primes in short intervals. We also discuss analogous results for a more general L-function. This is a sequel to the paper math.NT/0405459.

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arXiv2008

Examples of finite $p$-divisible sets of MHS

We provide some examples which give evidence to the conjectures contained in my paper "Finiteness of $p$-Divisible Sets of Multiple Harmonic Sums" (math.NT/0303043). All the main theoretical results can be found in that paper.

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arXiv2009

A few equalities involving integrals of the logarithm of the Riemann zeta-function and equivalent to the Riemann hypothesis II

This paper is a continuation of our recent paper with the same title, arXiv:0806.1596v1 [math.NT], where a number of integral equalities involving integrals of the logarithm of the Riemann zeta-function were introduced and it was shown that some of them are equivalent to the Riemann hypothesis. A few new equalities of this type are established; contrary to the preceding paper the focus now is on integrals involving the argument of the Riemann zeta-function (imaginary part of logarithm) rather than the logarithm of its module (real part of logarithm). Preliminary results of the numerical research performed using these equalities to test the Riemann hypothesis are presented. Our integral equalities, together with the equalities given in the previous paper, include all earlier known criteria of this kind, viz. Wang, Volchkov and Balazard-Saias-Yor criteria, which are certain particular cases of the general approach proposed.

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arXiv2012

A few equalities involving integrals of the logarithm of the Riemann zeta-function and equivalent to the Riemann hypothesis III. Exponential weight functions

This paper is a continuation of our recent papers with the same title, arXiv:0806.1596v1 [math.NT], arXiv:0904.1277v1 where a number of integral equalities involving integrals of the logarithm of the Riemann zeta-function were introduced and it was shown that some of them are equivalent to the Riemann hypothesis. A few new equalities of this type, which this time involve exponential functions, are established, and for the first time we have found equalities involving the integrals of the logarithm of the Riemann zeta-function taken exclusively along the real axis. Some of the equalities we have found are tested numerically. In particular, an integral equality involving the logarithm of abs(zeta(1/2+it)) and a weight function cosh(pi*t)^(-1) is shown numerically to be correct up to the 80 digits. For exponential weight function exp(-at), the possible contribution of the Riemann function zeroes non-lying on the critical line is rigorously estimated and shown to be extremely small, in particular, smaller than trillion of digits, 10^(-10^(13)), for a=4*pi.

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arXiv2012

An analogue of Selberg's formula for Motohashi's product

We prove an analogue of Selberg's explicit formula for Motohashi's product (see arXiv:1104.1358v3 [math.NT]). We also provide a zero-density theorem for the product, which follows from Soundararajan's theorem for moments of the Riemann zeta-function on the critical line.

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arXiv2011

Appendix: proof of the Uniformity Conjecture

This paper originated as an appendix to the paper "Topology and Geometry of the Berkovich Ramification Locus for Rational Functions, II" by Xander Faber arXiv:1104.0943v2 [math.NT]. It may however be read independently. We prove a variant of Alain Robert's p-adic Rolle theorem, via the theory of the radius of convergence of p-adic connections and the theory of semistable reduction of p-adic curves. We carefully compare the present author's notion [Inv. Math. 182 (2010)] of radius of convergence, of a connection on a p-adic curve X, normalized by the choice of a semistable model of X, with Kedlaya's intrinsic generic radius of convergence of a differential module [Def. 9.4.7 in p-adic Differential Equations, Cambridge Studies in Adv. Math., vol. 125 (2010)].

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arXiv2017

Motifs et adjoints

We show in many cases the existence of adjoints to extension of scalars on categories of motivic nature, in the framework of field extensions. This is to be contrasted with the more classical situation where one deals with a finite type morphism of schemes. Among various applications, one is a functorial construction of the "Tate-Safarevic motive" introduced in arXiv:1401.6847 [math.NT]. We also deduce a possible approach to Bloch's conjecture on surfaces, by reduction to curves.

math.AG↗
arXiv2015

On generalized Thue-Morse functions and their values

This paper naturally extends and generalizes our previous work "Thue-Morse constant is not badly approximable", arXiv:1407.3182 [math.NT]. Here we consider the Laurent series $f_d(x) = \prod_{n=0}^\infty (1 - x^{-d^n})$, $d\in\mathbb{N}$, $d\geq 2$ which generalize the generating function $f_2(x)$ of the Thue-Morse number, and study their continued fraction expansion. In particular, we show that the convergents of $x^{-d+1}f_d(x)$ have quite a regular structure. We address as well the question whether the corresponding Mahler numbers $f_d(a)\in\mathbb{R}$, $a,d\in\mathbb{N}$, $a,d\geq 2$, are badly approximable.

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arXiv2017

The divisibility by 2 of rational points on elliptic curves

We give a simple proof of the well-known divisibility by 2 condition for rational points on elliptic curves with rational 2-torsion. As an application of the explicit division by $2^n$ formulas obtained in Sec.2, we construct versal families of elliptic curves containing points of orders 4, 5, 6, and 8 from which we obtain an explicit description of elliptic curves over certain finite fields $\mathbb{F}_q$ with a prescribed (small) group $E(\mathbb{F}_q)$. In the last two sections we study 3- and 5-torsion. This paper supercedes arXiv:1605.09279 [math.NT] .

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arXiv2017

Mod $p$ Hilbert modular forms of parallel weight one: the ramified case

We generalize the main result of arXiv:1206.6631 [math.NT] to all totally real fields. In other words, for $p>2$ prime, we prove (under a mild Taylor-Wiles hypothesis) that if a modular representation is unramified and $p$-distinguished at all places above $p$, then it arises from a mod $p$ Hilbert modular form of parallel weight one. This (mostly) resolves the weight one part of Serre's conjecture for totally real fields.

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arXiv2021

On the asymptotic behavior of Sudler products along subsequences

Let $α\in (0,1)$ and irrational. We investigate the asymptotic behaviour of sequences of certain trigonometric products (Sudler products) $(P_N(α))_{N\in\mathbb{N}}$ with $$P_N(α) =\prod_{r=1}^N|2\sin(πr α)|.$$ More precisely, we are interested in the asymptotic behaviour of subsequences of the form $(P_{q_n(α)}(α))_{n\in\mathbb{N}}$, where $q_n(α)$ is the $n$th best approximation denominator of $α$. Interesting upper and lower bounds for the growth of these subsequences are given, and convergence results, obtained by Mestel and Verschueren (see arXiv:1411.2252math[DS]) and Grepstad and Neumüller (see arXiv:1801.09416[math.NT]), are generalized to the case of irrationals with bounded continued fraction coefficients.

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arXiv2021

Addendum to "Arithmetic exponent pairs of algebraic trace functions and applications"

This addendum devotes to a detailed proof for the inequality (9.14) in our joint work: Arithmetic exponent pairs for algebraic trace functions and applications, with an appendix by Will Sawin, arXiv:1603.07060 [math.NT], which will appear in Algebra and Number Theory. We do not intend to publish this addendum in any journals; arXiv should be a good place for those reader who want to find such details. The proof involves various averages of arithmetic functions.

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arXiv2022

A note on the hyper-sums of powers of integers, hyperharmonic polynomials and r-Stirling numbers of the first kind

Recently, Kargin et al. (arXiv:2008.00284 [math.NT]) obtained (among many other things) the following formula for the hyper-sums of powers of integers $S_k^{(m)}(n)$ \begin{equation*} S_k^{(m)}(n) = \frac{1}{m!} \sum_{i=0}^{m} (-1)^i \genfrac{[}{]}{0pt}{}{m+n+1}{i+n+1}_{n+1} S_{k+i}(n), \end{equation*} where $S_k^{(0)}(n) \equiv S_k(n)$ is the ordinary power sum $1^k + 2^k + \cdots + n^k$. In this note we point out that a formula equivalent to the preceding one was already established in a different form, namely, a form in which $\genfrac{[}{]}{0pt}{}{m+n+1}{i+n+1}_{n+1}$ is given explicitly as a polynomial in $n$ of degree $m-i$. We find out the connection between this polynomial and the so-called $r$-Stirling polynomials of the first kind. Furthermore, we determine the hyperharmonic polynomials and their successive derivatives in terms of the $r$-Stirling polynomials of the first kind, and show the relationship between the (exponential) complete Bell polynomials and the $r$-Stirling numbers of the first kind. Finally, we derive some identities involving the Bernoulli numbers and polynomials, the $r$-Stirling numbers of the first kind, the Stirling numbers of both kinds, and the harmonic numbers.

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arXiv2023

Large Sums of Fourier Coefficients of Cusp Forms

Let $N$ be a fixed positive integer, and let $f\in S_k(N)$ be a primitive cusp form given by the Fourier expansion $f(z)=\sum_{n=1}^{\infty} λ_f(n)n^{\frac{k-1}{2}}e(nz)$. We consider the partial sum $S(x,f)=\sum_{n\leq x}λ_f(x)$. It is conjectured that $S(x,f)=o(x\log x)$ in the range $x\geq k^ε$. Lamzouri proved in arXiv:1703.10582 [math.NT] that this is true under the assumption of the Generalized Riemann Hypothesis (GRH) for $L(s,f)$. In this paper, we prove that this conjecture holds under a weaker assumption than GRH. In particular, we prove that given $ε>(\log k)^{-\frac{1}{8}}$ and $1\leq T\leq (\log k)^{\frac{1}{200}}$, we have $S(x,f)\ll \frac{x\log x}{T}$ in the range $x\geq k^ε$ provided that $L(s,f)$ has no more than $ε^2\log k/5000$ zeros in the region $\left\{s\,:\, \Re(s)\geq \frac34, \, |\Im(s)-ϕ| \leq \frac14\right\}$ for every real number $ϕ$ with $|ϕ|\leq T$.

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arXiv2024

On the order of magnitude of certain integer sequences

Let $p$ be a prime number, and let $S$ be the numerical semigroup generated by the prime numbers not less than $p$. We compare the orders of magnitude of some invariants of $S$ with each other, e. g., the biggest atom $u$ of $S$ with $p$ itself: By Harald Helfgott (arXiv:1312.7748 [math.NT]), every odd integer $N$ greater than five can be written as the sum of three prime numbers. There is numerical evidence suggesting that the summands of $N$ always can be chosen between $\frac N6$ and $\frac N2$. This would imply that $u$ is less than $6p$.

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arXiv2025

Generic decompositions of Deligne--Lusztig representations

Let $G_0$ be a reductive group over $\mathbb{F}_p$ with simply connected derived subgroup, (geometrically) connected center and Coxeter number $h+1$. We extend Jantzen's generic decomposition pattern from $(2h-1)$-generic to $h$-generic Deligne--Lusztig representations, which is optimal. We also prove several results on the ``obvious'' Jordan--Hölder factors of general Deligne--Lusztig representations. As an application we improve the weight elimination result of arXiv:1610.04819 [math.NT]

math.RT↗
arXiv2024

New bound on small range sum polynomials of degree

The polynomials of degree $\frac{p-1}{2}$ of range sum $p$ was determined in {\tt arXiv:2311.06136 [math.NT]} for large enough primes. We extend this result by reducing the lower bound for the primes to $23$ by introducing a new and elementary way of estimating sums of Legendre symbols.

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