Related papers: Sums with convolution of Dirichlet characters
Let $q\ge3$ be an integer, $\chi$ be a Dirichlet character modulo $q$, and $L(s,\chi)$ denote the Dirichlet $L$-functions corresponding to $\chi$. In this paper, we show some special function series, and give some new identities for the…
In this article we study analytic properties of the multiple Dirichlet series associated to additive and Dirichlet characters. For the multiple Dirichlet series associated to additive characters, the meromorphic continuation is established…
In this paper, we obtain a sharp upper bound for the sum of the first $k$-th eigenvalues for this Dirichlet problem of poly-Laplacian with any order, which is viewed as an extension of the result due to Cheng and Wei (Journal of…
We present various identities in the form of convolutions involving Stirling numbers of both kinds, Lah numbers, and binomial coefficients. Certain convolution polynomials are discussed also. The proofs are based on several series…
Let $[x]$ be the integral part of $x$, $n>1$ be a positive integer and $\chi_n$ denote the trivial Dirichlet character modulo $n$. In this paper, we use an identity established by Z. H. Sun to get congruences of…
The objective of this paper is to obtain asymptotic results for shifted sums of multiplicative functions of the form $g \ast 1$, where the function $g$ satisfies the Ramanujan conjecture and has conjectured upper bounds on square moments of…
This paper presents a family of rapidly convergent summation formulas for various finite sums of the form $\sum_{k=0}^{\lfloor x\rfloor}f(k)$, where $x$ is a positive real number.
The newform Dedekind sum $S_{\chi_1, \chi_2}$ associated to a pair of primitive Dirichlet characters $\chi_1$, $\chi_2$ of respective conductors $q_1$, $q_2$, is a group homomorphism from $\Gamma_1(q_1 q_2)$ into the number field…
The article presents several methods for the arithmetic of finite abelian groups. We introduce a tool - already used by Delsarte in [1] as I found out later - analogous to Dirichlet's convolution to obtain combinatorial results on these…
This paper improves the upper bound for the exceptional zeroes of Dirichlet L-functions with even characters. The result is obtained by improving on explicit estimate for $L'(\sigma;\chi)$ for $\sigma$ close to unity, using a result on the…
In this paper, we show that the regularized determinants of some Dirichlet series are multiplicative. As an application, we give generalizations of Lerch's formula for the classical gamma function and we determine the sum of some Dirichlet…
Inequalities for exponential sums are studied. Our results improve an old result of G. Halasz and a recent result of G. Kos. We prove several other essentially sharp related results in this paper.
We provide a systematic method to compute arithmetic sums including some previously computed by Alaca, Alaca, Besge, Cheng, Glaisher, Huard, Lahiri, Lemire, Melfi, Ou, Ramanujan, Spearman and Williams. Our method is based on quasimodular…
We prove a Voronoi formula for coefficients of a large class of $L$-functions including Maass cusp forms, Rankin-Selberg convolutions, and certain isobaric sums. Our proof is based on the functional equations of $L$-functions twisted by…
We study the shifted convolution sum of the divisor function and some other arithmetic functions.
We consider moments of higher powers of quadratic Dirichlet character sums. In a restricted region, we give their asymptotic behavior by using de la Bret\`{e}che's multivariable Tauberian theorem. We also give the lower bound of the…
In this paper we give a refinement of the bound of D. A. Burgess for multiplicative character sums modulo a prime number $q$. This continues a series of previous logarithmic improvements, which are mostly due to H. Iwaniec and E. Kowalski.…
We obtain new bounds, pointwisely and on average, for Dedekind sums $\mathsf{s}(\lambda,p)$ modulo a prime $p$ with $\lambda$ of small multiplicative order $d$ modulo $p$. Assuming the infinitude of Mersenne primes, the range of our results…
Some changes in a recent convolution formula are performed here in order to clean it up by using more conventional notations and by making use of more referrenced and documented components (namely Sierpi\'nski's polynomials, the Thue-Morse…
In this paper, we apply the Dirichlet convolution method to \begin{equation*} T_{k}(x)=\sum_{n \leq x} d_{k}(n), \end{equation*} for $k\ge 3$, where $d_{k}(n)$ is the number of ways to represent $n$ as a product of $k$ positive integer…