Related papers: The Dirichlet Series for the Exterior Square $L$-f…

We deal with negative square moments of Dirichlet $L$-functions. Summing over characters modulo $q$, we obtain an asymptotic formula for the negative second moment of $L(1,\chi)$ involving conductors. As an application, we give the improved…

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 $s=\sigma+it$…

Using the Kuznetsov trace formula, we prove a spectral decomposition for the sums of generalized Dirichlet $L$-functions. Among applications are an explicit formula relating norms of prime geodesics to moments of symmetric square…

Generalizing the work of A. Morra and the authors, we give explicit formulas for the Dirichlet series generating function of $D_{\ell}$-extensions of odd prime degree $\ell$ with given quadratic resolvent. Over the course of our proof, we…

An algorithm to compute Dirichlet $L$-functions for many quadratic characters is derived. The algorithm is optimal (up to logarithmic factors) provided that the conductors of the characters under consideration span a dyadic window.

In this paper, we show that the twisted partial exterior-square $L$-function has a meromorphic continuation to the whole complex plane with only two possible simple poles at $s=1$ and $s=0$. We do this by establishing the nonvanishing of…

In this paper, we give a method to construct a classical modular form from a Hilbert modular form. By applying this method, we can get linear formulas which relate the Fourier coefficients of the Hilbert and classical modular forms. The…

We formulate Guo--Jacquet type fundamental lemma conjectures and arithmetic transfer conjectures for inner forms of $GL_{2n}$. Our main results confirm these conjectures for division algebras of invariant $1/4$ and $3/4$.

In this article, we obtain the analytic continuation of the multiple shifted Lucas zeta function, multiple Lucas $L$-function associated to Dirichlet characters and additive characters. We then compute a complete list of exact singularities…

We verify the conjecture of [CFKRS] for the mean square near the critical point of Dirichlet L-functions for a composite modulus q. We also prove a kind of reciprocity formula when the second moment for a prime modulus is twisted by a…

Let $F$ be a non archimedian local field and $H_n(F)$ the Shalika subgroup of $GL_{2n}(F)$. We prove an explicit Plancherel formula for $H_n(F) \backslash GL_{2n}(F)$ using the theory of Jacquet-Shalika of zeta functions and we deduce…

We present an integral representation formula for a Dirichlet series whose coefficients are the values of the Liouville's arithmetic function.

We develop a (largely conjectural) theory of p-adic L-functions interpolating square roots of central L-values for automorphic forms on GSp(4) x GL(2) x GL(2), and a relation between these p-adic L-functions and families of Galois…

For Dirichlet characters $\chi$ mod $k$ where $k\geq 3$, we here give a computable formula for evaluating the mean square sums $\sum\limits_{\substack{\chi \text{ mod }k\\\chi(-1)=(-1)^r}}|L(r,\chi)|^2$ for any positive integer $r\geq 3$.…

The main objective of this article is to compute a first moment for product of Dirichlet and twisted self-dual $GL(3)$ $L$-functions. We discuss the possible simultaneous non vanishing at the central point. We use properties of symmetric…

In this paper we consider the q-extension of the generating function for the higher-order generalized Genocchi numbers and polynomials attached to Dirichlet's character.

In this paper, we establish the expected order of magnitude of the $k$th-moment of quadratic Dirichlet $L$-functions associated to hyperelliptic curves of genus $g$ as well as of prime moduli over a fixed finite field $\mathbb{F}_{q}$ for…

We consider a sum of the derivatives of Dirichlet $L$-functions over the zeros of Dirichlet $L$-functions. We give an asymptotic formula for the sum.

We use the theta correspondence to study the equivalence between Godement-Jacquet and Jacquet-Langlands L-functions for $\mathrm{GL}(2)$. We show that the resulting comparison is in fact an exotic symmetric monoidal structure on the…

Let $\pi$ be an irreducible unitary cuspidal representation of $GL_m({\Bbb A}_{\Bbb Q})$ and $L(s,\,\pi)$ be the global $L-$function attached to $\pi$. If ${\rm Re}(s)>1$, $L(s,\,\pi)$ has a Dirichlet series expression. When $\pi$ is…