We examine an arithmetical function defined by recursion relations on the sequence ${\left\{f\left(p^k\right)\right\}}_{k\in \mathbb{N}}$ and obtain sufficient condition(s) for the sequence to change sign infinitely often. As an application we give criteria for infinitely many sign changes of Chebyshev polynomials and that of sequence formed by the Fourier coefficients of a cusp form.

We give a Chowla-Selberg type formula that connects a generalization of the eta-function to $GL\left(n\right)$ with multiple gamma functions. We also present some simple infinite product identities for certain special values of the multiple gamma function.

On donne une nouvelle condition suffisante pour l’existence des mesures $p$-adiques admissibles $\mu$ obtenues à partir de suites de distributions ${\Phi }_j(j\ge 0)$ à valeurs dans les espaces de formes modulaires. On utilise la projection caractéristique sur le sous-espace primaire associé à une valeur propre non nulle $\alpha$ de l’opérateur $U$ d’Atkin. Notre condition est exprimée en termes des congruences entre les coefficients de Fourier des formes modulaires ${\Phi }_j$. On montre comment vérifier ces congruences, et on traite plusieurs...

Let ${\lambda }_f\left(n\right)$ be the nth normalized Fourier coefficient of a holomorphic Hecke eigenform $f\left(z\right)\in S_k\left(\Gamma \right)$. We establish that ${\sum }_{n\le x}{\lambda }_f^2\left(n^j\right)=c_{j}x+O\left(x^{1-2/({(j+1)}^2+1)}\right)$ for j = 2,3,4, which improves the previous results. For j = 2, we even establish a better result.

Let $f={\sum }_{n=1}^{\infty }a\left(n\right)q^n\in S_{k+1/2}(N,{\chi }_0)$ be a nonzero cuspidal Hecke eigenform of weight $k+\frac{1}{2}$ and the trivial nebentypus ${\chi }_0$, where the Fourier coefficients $a\left(n\right)$ are real. Bruinier and Kohnen conjectured that the signs of $a\left(n\right)$ are equidistributed. This conjecture was proved to be true by Inam, Wiese and Arias-de-Reyna for the subfamilies ${\left\{a\left(t{n}^2\right)\right\}}_n$, where $t$ is a squarefree integer such that $a\left(t\right)\ne 0$. Let $q$ and $d$ be natural numbers such that $(d,q)=1$. In this work, we show that ${\left\{a\left(t{n}^2\right)\right\}}_n$ is equidistributed over any arithmetic progression $n\equiv dmodq$.