Related papers: Voronoi summation formula for the generalized divi…

We introduce a natural definition for sums of the form \[ \sum_{\nu=1}^x f(\nu) \] when the number of terms x is a rather arbitrary real or even complex number. The resulting theory includes the known interpolation of the factorial by the…

We introduce some new higher dimensional generalizations of the Dedekind sums associated with the Bernoulli functions and of those Hardy sums which are defined by the sawtooth function. We generalize a variant of Parseval's formula for the…

The paper proves a generalization of Wintner's theorem on the asymptotics of summation functions to the case of summation functions with nonlinear asymptotics. The class of arithmetic functions that have a logarithmic asymptotic mean is…

We prove an inversion formula for summatory arithmetic functions. As an application, we obtain an arithmetic relationship between summatory Piltz divisor functions and a sum of the M\"obius function over certain integers, denoted by…

This note gives a few rapidly convergent series representations of the sums of divisors functions. These series have various applications such as exact evaluations of some power series, computing estimates and proving the existence results…

The sum $S(h,k):=\sum_{j=1}^{k-1}(-1)^{j+1+[hj/k]}$ appears in the modular transformation formulae of the classical theta function $\vartheta_3(z)$. The double sum $S(k) := \sum_{h=1}^{k-1}S(h,k)$ has a remarkable distribution of values.…

We establish asymptotic formulae for various correlations involving general divisor functions $d_k(n)$ and partial divisor functions $d_l(n,A)=\sum_{q|n:q\leq n^A}d_{l-1}(q)$, where $A\in[0,1]$ is a parameter and $k,l\in\mathbb{N}$ are…

We define an $f$-restricted partition $p_f(n,k)$ of fixed length $k$ given by the bivariate generating series \begin{align*} Q_f(z,u) \coloneqq 1+\sum_{n=1}^{\infty}\sum_{k=1}^{\infty} p_f(n,k) u^kz^n…

Let $r_{k}(n)$ denote the number of representations of the positive integer $n$ as the sum of $k$ squares. We prove a generalization of a summation formula already proved by us [Advances in Applied Mathematics, 175 (2026) 103201], which…

The main purpose of this article is to survey on some key elements of a recent $\mathcal{H}_p$-theory of general Dirichlet series $\sum a_n e^{-\lambda_{n}s}$, which was mainly inspired by the work of Bayart and Helson on ordinary Dirichlet…

The aim of this article is to present in a self-contained way identities arising in elementary number theory, among which the following one: $$ \sum_{d\mid n}\frac{\mu^2(d)}{\varphi(d)\,d^s}=\prod_{p\mid n}\left(1+\frac{1}{(p-1)p^s}\right).…

We study the extent to which divisors of a typical integer $n$ are concentrated. In particular, defining the Erd\H{o}s-Hooley $\Delta$-function by $\Delta(n) := \max_t \# \{d | n, \log d \in [t,t+1]\}$, we show that $\Delta(n) \geq (\log…

In this paper, we investigate the combinatorial structure and asymptotic distribution of the solution set of the equation $\sigma(n+1) = k\sigma(n)$ for a given integer $k>1$. From a combinatorial perspective, the solutions to this equation…

In [arXiv:1504.07804], Keating, Rodgers, Roditty-Gershon and Rudnick established relationships of the mean-square of sums of the divisor function $d_k(f)$ over short intervals and over arithmetic progressions for the function field $\mathbb…

Dirichlet's $L$-functions are natural extensions of the Riemann zeta function. In this paper we first give a brief survey of Ap\'ery-like series for some special values of the zeta function and certain $L$-functions. Then, we establish two…

A generalization of the Zernike circle polynomials for expansion of functions vanishing outside the unit disk is given. These generalized Zernike functions have the form Zm,{\alpha} n ({\rho}, \vartheta) = Rm,{\alpha} n ({\rho})…

We present a new way to factor the dirichlet convolution for completely multiplicative functions whitch led us to constructing a ring that arise from the operations involved in the factorisation. We will conclude by some identities that was…

B\o gvad and H\"agg proved that for a rational function with simple poles, the zeros of successive derivatives accumulate on the Voronoi diagram of the pole set, and the normalized zero-counting measures converge to a canonical probability…

Inspired by results of Bayart on ordinary Dirichlet series $\sum a_n n^{-s}$, the main purpose of this article is to start an $\mathcal{H}_p$-theory of general Dirichlet series $\sum a_n e^{-\lambda_{n}s}$. Whereas the…

Let $p_{-r}(n)$ denote the $r$-coloured partition function, and $\sigma(n)=\sum_{d|n}d$ denote the sum of positive divisors of $n$. The aim of this note is to prove the following $$…