Let $A$ be a set of $N$ vectors in ${\mathbb Z}^n$ and let $v$ be a vector in ${\mathbb C}^N$ that has minimal negative support for $A$. Such a vector $v$ gives rise to a formal series solution of the $A$-hypergeometric system with parameter $\beta=Av$. If $v$ lies in ${\mathbb Q}^n$, then this series has rational coefficients. Let $p$ be a prime number. We characterize those $v$ whose coordinates are rational, $p$-integral, and lie in the closed interval $[-1,0]$ for which the corresponding normalized series solution has $p$-integral coefficients. From this we deduce further integrality results for hypergeometric series.

This note provides accurate truncated formulae with explicit error terms to compute Euler products over primes in arithmetic progressions of rational fractions. It further provides such a formula for the product of terms of the shape $F(1/p, 1/p^s)$ when $F$ is a two-variable polynomial with coefficients in $\mathbb{C}$ and satisfying some restrictive conditions.

The problem of finding a pair of distinct rational triangles (resp. cyclic quadrilaterals) with the same area and the same perimeter has been solved. Now we generalize these results and get several parametric solutions to find infinitely many pairs of rational triangles (resp. cyclic quadrilaterals) with areas and perimeters in fixed proportions $(\alpha,\beta)$ respectively, where $\alpha$ and $\beta$ are positive rational numbers.

Given a fixed Hermitian Jacobi cusp form, we define a family of linear operators between spaces of Hermitian Jacobi cusp forms using Rankin--Cohen brackets. We compute the adjoint maps of such a family with respect to the Petersson scalar product. The Fourier coefficients of the Hermitian Jacobi cusp forms constructed using this method involve special values of certain Dirichlet series associated to Hermitian Jacobi cusp forms.

For a given finitely generated multiplicative subgroup of the rationals which possibly contain negative numbers, we derive, subject to GRH, formulas for the densities of primes for which the index of the reduction group has a given value. We completely classify the cases of rank one, torsion groups for which the density vanishes and the the set of primes for which the index of the reduction group has a given value, is finite. For higher rank groups we propose some partial results. Finally, we present some computations comparing the approximated density computed with primes up to $10^{10}$ and those predicted by the Riemann Hypothesis.

We introduce some results for the Walsh transforms of quadratic trace forms with three or more terms. We also generalize some results on Kasami--Welch functions. We use Walsh transforms to obtain the number of rational points on Artin--Schreier curves over $\mathbb{F}_{2^k}.$ Later we construct some maximal Artin--Schreier curves over $\mathbb{F}_{2^k}$ for an even integer $k$.

This note answers, and generalizes, a question of Kaisa Matomäki. We show that given two cuspidal automorphic representations $\pi_1$ and $\pi_2$ of $GL(n)$ over a number field $F$ of respective conductors $N_1$, $N_2$, every character $\chi$ such that $\pi_1\otimes\chi\simeq\pi_2$ of conductor $Q$, satisfies the bound: $Q^n\mid N_1N_2$. If at every finite place $v$, $\pi_{1,v}$ is a discrete series whenever it is ramified, then $Q^n$ divides the least common multiple $[N_1,N_2].$

In a recent paper, J. Kaczorowski defined the generalized Euler totient function $\varphi(n, F)$ corresponding to a polynomial Euler product $F$. Let $E(x, F)$ denote the error term in the asymptotic formula of the summatory function of $\varphi(n, F).$ J.~Kaczorowski proved that the mean square of $E(x, F)$ has an asymptotic formula if $F$ satisfies GRH. In this paper we shall prove a sharper asymptotic formula under GRH.