We show that the Riemann Hypothesis is equivalent to the assertion $(y_m)\in\ell_2$ where $y_m$ is defined by \[ \lambda_m=\frac12(\log m+\gamma-\log(2\pi)-1)+y_m, \] and $m\lambda_m$ represents the numbers in Xian-Jin Li's criterion. This confirms and further sharpens a conjecture of J. B. Keiper. We also present some other hypotheses equivalent to the Riemann Hypothesis.

The object of this work is to present the status of art of an open problem: to provide an analogue for the cohomology of Shimura curves of the Ihara's lemma [Ihara73] which holds for modular curves. We will formulate our conjecture and locate it in the more general setting of the congruence subgroup problem. We will exploit the relationship between cohomology of Shimura curves and certain spaces of modular forms to establish some consequences of the conjecture about congruence modules of modular forms and about the problem of raising the level.

We consider the Dirichlet series associated to the number of representations of an integer as a sum of primes. Assuming certain reasonable hypotheses on the distribution of the zeros of the Riemann zeta function we obtain the domain of meromorphic continuation of this series.

Let $D\ge 3$ be an odd integer and $\ell\ge -1$ be a non zero integer such that gcd$(\ell,D)=1$. Let $f,g:\mathbb{N} \to \mathbb{N}$ be multiplicative functions such that $f(p)=D$ and $g(p)=p+\ell$ for each prime $p$. We estimate the number of positive integers $n\le x$ such that gcd$(f(n),g(n))=1$. If $D$ is a prime larger than 3, we also examine the size of the number of positive integers $n\le x$ for which $\mbox{gcd}(g(n),f(n-1))=1$.

We determine all unitary perfect polynomials with less than five distinct prime factors over a finite field with four elements.

Consider binary forms $F(x,y), G(x,y)$ with coefficients in $\mathbb{Q}[t]$, assume that $F$ is irreducible. We give effective upper bounds for the heights of the solutions and an efficient algorithm to solve \[ w\cdot F(x,y)=z\cdot G(x,y) \] \[ in x,y\in \mathbb{Q}[t], w,z\in \mathbb{Q}[t]\cap U_S, \gcd(x,y)=1, \gcd(w,z)=1, \] where $U_S$ denotes a group of $S$-units in $\mathbb{Q}(t)$. We derive that there are only finitely many solutions up to constant factors. We also show that this is not true for global function fields. This is a generalization of the well known Thue equations. Effective upper bounds for the solutions of this general equation were given over number fields but it was not yet considered over function fields. We illustrate our method with a detailed numerical example.

A uniform version of Stirling's formula, suitable for future applications, is obtained.

The purpose of the article is to estimate the mean square of a squareroot length exponential sum of Fourier coefficients of a holomorphic cusp form.

We study the connection between the projective limit $\overline{St}_\infty$ of Stark units in $\mathbb{Z}_p$-extensions and a certain Iwasawa module already appeared in the particular cases of circular units or elliptic units. This connection is used to investigate the structure of $\overline{St}_\infty$.

In this paper, we study a quantity $R_b$ which is closely related to the quality of an important subclass of digital $(t,m,s)$-nets over a finite field $\mathbb{F}_b$, namely polynomial lattices. Niederreiter has shown by an averaging argument that there always exist generators of polynomial lattices for which $R_b$ is small, establishing thereby the existence of polynomial lattices with particularly low star discrepancy. In this work, we show that this result is best possible, i.e., we prove that for all generators of polynomial lattices the quantity $R_b$ cannot go below a certain threshold.

Let $N/\mathbb{Q}$ be a real cyclic and tame extension of prime degree $l$ with $\Gamma=\mathscr{G}al(N/\mathbb{Q})$. We give the Hom description of the class of the torsion-free part of the group of units in $N$ in the class group of the order $\mathbb{Z}\Gamma/ (\sum_{\gamma \in \Gamma}\gamma )$. This representation depends only on the structure of the ideal class group of $N$ and determines the Galois module structure of the torsion-free part of the group of units in $N$ as an ideal of the $l$th cyclotomic field. Using this approach we derive necessary and sufficient conditions for all real and tame cyclic fields of prime degree to have Minkowski units. We extend also the class of known cyclic real fields with Minkowski units.