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

A description of the properties of \L with complex characters is given. By using these, together with the more familiar \L with real characters, it is shown how certain two dimensional lattice sums, which previously could not be put into…

In this paper, We use the Fourier series expansion of real variables function, We give a formula to calculate the Dirichlet character sum, and four special examples are given.

We investigate various mean value problems involving order three primitive Dirichlet characters. In particular, we obtain an asymptotic formula for the first moment of central values of the Dirichlet L-functions associated to this family,…

We study the average of the product of the central values of two $L$-functions of modular forms $f$ and $g$ twisted by Dirichlet characters to a large prime modulus $q$. As our principal tools, we use spectral theory to develop bounds on…

The Postnikov character formula is used to express large portions of a Dirichlet character sum in terms of quadratic exponential sums. The quadratic sums are then computed using an analytic algorithm previously derived by the author. This…

We evaluate the character sum sigma_m sigma_n (m/n) where the two sums are of approximately the same length. The answer is surprising.

We evaluate the smoothed first moment of central values of a family of qudratic Hecke $L$-functions in the Gaussian field using the method of double Dirichlet series. The asymptotic formula we obtain has an error term of size…

In this article, we prove an asymptotic formula for the mean value of long smoothed Dirichlet polynomials with divisor coefficients. Our result has a main term that includes all lower order terms and a power saving error term. This is…

Let $d(n)$ be the Dirichlet divisor function and $\Delta(x)$ denote the error term of the sum $\sum_{n\leqslant x}d(n)$ for a large real variable $x$. In this paper we focus on the sum $\sum_{p\leqslant x}\Delta^2(p)$, where $p$ runs over…

We prove an asymptotic formula for the second moment of central values of Dirichlet $L$-functions restricted to a coset. More specifically, consider a coset of the subgroup of characters modulo $d$ inside the full group of characters modulo…

We study the triple convolution sum of the generalised divisor functions $$\sum_{n\leq x} d_k(n+h)d_l(n)d_m(n-h),$$ where $h \le x^{1-\epsilon}$ for any $\epsilon>0$ and $d_k(n)$ denotes the generalised divisor function which counts the…

The Menon-Sury's identity is as follows: \begin{equation*} \sum_{\substack{1 \leq a, b_1, b_2, \ldots, b_r \leq n\\\mathrm{gcd}(a,n)=1}} \mathrm{gcd}(a-1,b_1, b_2, \ldots, b_r,n)=\varphi(n) \sigma_r(n), \end{equation*} where $\varphi$ is…

We study a generalization of the classical Dedekind sum that incorporates two Dirichlet characters and develop properties that generalize those of the classical Dedekind sum. By calculating the Fourier transform of this generalized Dedekind…

We obtain a second moment formula for the L-series of holomorphic cusp forms, averaged over twists by Dirichlet characters modulo a fixed conductor Q. The estimate obtained has no restrictions on Q, with an error term that has a close to…

We prove an asymptotic formula for the mean-square average of $L$- functions associated to subgroups of characters of sufficiently large size. Our proof relies on the study of certain character sums ${\cal A}(p,d)$ recently introduced by E.…

We evaluate the average of cubic and quartic Dirichlet character sums with the modulus going up to a size comparable to the length of the individual sums. This generalizes a result of Conrey, Farmer and Soundararajan on quadratic Dirichlet…

Let K be a number field containing the n-th roots of unity for some n > 2. We prove a uniform subconvexity result for a family of double Dirichlet series built out of central values of Hecke L-functions of n-th order characters of K. The…

Conrey, Farmer and Zirnbauer introduced a recipe to find asymptotic formulas for the sum of ratios of products of shifted L-functions. These ratios conjectures are very powerful and can be used to determine many statistics of L-functions,…

We give explicit expressions for the Fourier coefficients of Eisenstein series twisted by Dirichlet characters and modular symbols on $\Gamma_0(N)$ in the case where $N$ is prime and equal to the conductor of the Dirichlet character. We…

Let $\Delta_{k}(x)$ be the error term in the classical asymptotic formula for the sum $\sum_{n\leq x}d_{k}(n)$, where $d_{k}(n)$ is the number of ways $n$ can be written as a product of $k$ factors. We study the analytic properties of the…