Related papers: Mean value of real characters using a double Diric…

In the paper we obtain new estimates for binary and ternary sums of multiplicative characters with additive convolutions of characteristic functions of sets, having small additive doubling. In particular, we improve a result of M.-C. Chang.…

A method of estimating sums of multiplicative functions braided with Dirichlet characters is demonstrated, leading to a taxonomy of the characters for which such sums are large.

Let $p$ be an odd prime. Using I. M. Vinogradov's bilinear estimate, we present an elementary approach to estimate nontrivially the character sum $$ \sum_{x\in H}\chi(x+a),\qquad a\in\Bbb F_p^*, $$ where $H<\Bbb F_p^*$ is a multiplicative…

The theory of summability of divergent series is a major branch of mathematical analysis that has found important applications in engineering and science. It addresses methods of assigning natural values to divergent sums, whose…

We prove that the average size of a mixed character sum $$\sum_{1\le n \le x} \chi(n) e(n\theta) w(n/x)$$ (for a suitable smooth function $w$) is on the order of $\sqrt{x}$ for all irrational real $\theta$ satisfying a weak Diophantine…

We give asymptotics for shifted convolutions of the form $$\sum_{n < X} \frac{\sigma_{2u}(n,\chi)\sigma_{2v}(n+k,\psi)}{n^{u+v}}$$ for nonzero complex numbers $u,v$ and nontrivial Dirichlet characters $\chi,\psi$. We use the technique of…

In this paper, we evaluate the sum $\sum_{m,n}\leg {m}{n}d(n)$, where $\leg {m}{n}$ is the Kronecker symbol and $d(n)$ is the divisor function.

We extend to Dirichlet L-functions associated with arbitrary primitive characters a range of objects and properties -- including Eisenstein series and period functions -- that were originally introduced and studied by Lewis and Zagier…

In this paper, we study the distribution of the sequence of integers $d(n^2)$ under the assumption of the strong Riemann hypothesis. Under this assumption, we provide a refined asymptotic formula for the sum $\displaystyle\sum_{n\leq…

We prove a generalized version of Rogers' mean value formula in the space $X_n$ of unimodular lattices in $R^n$, which gives the mean value of a multiple sum over a lattice $L$ and its dual $L^*$. As an application, we prove that for $L$…

In this paper, we investigate the distribution of the maximum of character sums over the family of primitive quadratic characters attached to fundamental discriminants $|d|\leq x$. In particular, our work improves results of Montgomery and…

The paper considers a method for converting a divergent Dirichlet series into a convergent Dirichlet series by directly converting the coefficients of the original series $1\rightarrow\delta_{n}(s)$ for the Riemann Zeta function. In the…

Let $q\ge3$ be an integer, $\chi$ denote a Dirichlet character modulo $q$, for any real number $a\ge 0$, we define the generalized Dirichlet $L$-functions $$ L(s,\chi,a)=\sum_{n=1}^{\infty}\frac{\chi(n)}{(n+a)^s}, $$ where $s=\sigma+it$…

In this paper, we study a Dirichlet series generated by powers of harmonic numbers. As an application of these functions, we derive certain series involving harmonic numbers. We also study the analytic properties of these Dirichlet series…

Inspired by a famous identity of Ramanujan, we propose a general formula linearizing the convolution of Dirichlet series as the sum of Dirichlet series with modified weights; its specialization produces new identities and recovers several…

The classical Menon's identity [7] states that \begin{equation*}\label{oldbegin1} \sum_{\substack{a\in\Bbb Z_n^\ast }}\gcd(a -1,n)=\varphi(n) \sigma_{0} (n), \end{equation*} where for a positive integer $n$, $\Bbb Z_n^\ast$ is the group of…

This paper provides asymptotics with a sharp error term for the Dirichlet summatory function of a certain class of arithmetic functions. The result applies, e.g., to the sums over r^2(n) and r(n^3), where r(m) denotes the number of ways to…

We produce a new proof of the reciprocity law for the twisted second moment of Dirichlet L-functions that was recently proved by Conrey. Our method is to analyze certain two-variable sums where the variables satisfy a linear congruence. We…

A modified Dirichlet character $f$ is a completely multiplicative function such that for some Dirichlet character $\chi$, $f(p)=\chi(p)$ for all but a finite number of primes $p\in S$, and for those exceptional primes $p\in S$, $|f(p)|\leq…

We estimate the truncated double trigonometric series $\sum_{n=0}^{N}\sum_{m=0}^{M}a_{mn} {e}^{2\pi \imath \left(m x+n y\right)}$, $a_{mn} \in\mathbb{C}$, in Lebesgue spaces with mixed norms in terms of the $p^{th}-q^{th}$ power finite…