In this paper, we find a functional equation that characterizes the series involved in the Fourier coefficients of generalized modular forms of large negative real weights.

The main purpose of this paper is to study the hybrid mean value of the character sums over short interval and $B(\chi)$, by using the mean value theorems of the Dirichlet L-functions, and give an interesting asymptotic formula.

In this paper we continue the investigations about unlike powers in arithmetic progression. We provide sharp upper bounds for the length of primitive non-constant arithmetic progressions consisting of squares/cubes and $n$-th powers.

We consider dilation operators $T_k:f\rightarrow f(2^k\cdot)$ in the framework of Triebel-Lizorkin spaces $F^s_{p,q}(\mathbb{R}^n)$. If $s>n\max\big(\frac 1p -1,0\big)$, $T_k$ is a bounded linear operator from $F^s_{p,q}(\mathbb{R}{^n)$ into itself and there are optimal bounds for its norm. We study the situation on the line $s=n\max\big(\frac 1p -1,0\big)$, an open problem mentioned in [ET96, 2.3.1]. It turns out that the results shed new light upon the diversity of different approaches to Triebel-Lizorkin spaces on this line, associated to definitions by differences, Fourier-analytical methods and subatomic decompositions.

We give a new proof of a recent theorem of Kaplansky and use it to revive an old, seemingly forgotten result of Glaisher.

Let $A$ be a finite subset of an abelian group $G$. For every element $b_i$ of the sumset $2A=\{b_0, b_1, ...,b_{|2A|-1}\}$ we denote by $D_i=\{a-a': a, a' \in A; a+a'=b_i\}$ and $r_i=|\{(a,a'): a+a'=b_i; a, a' \in A \}|$. After an eventual reordering of $2A$, we may assume that $r_0\geq r_1 \geq ...\geq r_{|2A|-1}.$ For every $1 \le s \le |2A|$ we define $R_s(A)=|D_0\cup D_1\cup...\cup D_{s-1}|$ and $R_s(k)=\max \{R_s(A): A\subseteq G, |A| =k\}.$ Bourgain and Katz and Tao obtained an estimate of $R_s(k)$ assuming $s$ being of order $k$. In this paper we describe the {\it structure} of $A$ assuming that\break $G=\mathbb{Z}^2, s=3$ and $R_3(A)$ is close to its maximal value, i.e. $R_3(A) = 3k-\theta \sqrt{k}$, with $\theta \le 1.8$.

Let $S(n)$ be the function which associates for each positive integer $n$ the smallest positive integer $k$ such that $n\mid k!$. In this note, we look at various inequalities involving the composition of the function $S(n)$ with other standard arithmetic functions such as the Euler function and the sum of divisors function. We also look at the values of $S(F_n)$ and $S(L_n)$, where $F_n$ and $L_n$ are the $n$th Fibonacci and Lucas numbers, respectively.

We study the solutions of Hecke's functional equation in the framework of the extended Selberg class. It turns out that, notwithstanding certain differences between the two theories, there are strong analogies in the behavior of the dimensions of the spaces of solutions. We also provide details to an old argument of Hecke, which is presented in a sketchy way both in Hecke's original works and in subsequent books on Hecke's theory.