Symmetric identity for polynomial sequences satisfying A′ n +1(x) = (n + 1)An (x)

Using umbral calculus, we establish a symmetric identity for any sequence of polynomials satisfying A′ n +1(x) = (n + 1)An (x) with A 0(x) a constant polynomial. This identity allows us to obtain in a simple way some known relations involving Apostol-Bernoulli polynomials, Apostol-Euler polynomials and generalized Bernoulli polynomials attached to a primitive Dirichlet character.

Complex numbers similar to the generalized Bernoulli numbers and their applications

Let χ be a primitive Dirichlet character modulo k ≥ 3. In this paper, we define complex numbers associated with χ, which we denote by Cr(χ) (r = 0, 1,…), and we discuss their properties and their relationships with the generalized Bernoulli numbers.

Large values of Dirichlet L-functions at zeros of a class of L-functions

In this paper, we are interested in obtaining large values of Dirichlet L-functions evaluated at zeros of a class of L-functions, that is, $$ \begin{align*}\max_{\substack{F(\rho)=0\\ T\leq \Im \rho \leq 2T}}L(\rho,\chi), \end{align*} $$ where $\chi $ is a primitive Dirichlet character and F belongs to a class of L-functions. The class we consider includes L-functions associated with automorphic representations of $GL(n)$ over ${\mathbb {Q}}$ .

Bounds for GL3L-functions in depth aspect

Let f be a Hecke–Maass cusp form for {\mathrm{SL}_{3}(\mathbb{Z})} and χ a primitive Dirichlet character of prime power conductor {\mathfrak{q}=p^{\kappa}}, with p prime. We prove the subconvexity boundL\Big{(}\frac{1}{2},\pi\otimes\chi\Big{)}\ll_{p,\pi,\varepsilon}\mathfrak{q}^{% 3/4-3/40+\varepsilon}for any {\varepsilon>0}, where the dependence of the implied constant on p is explicit and polynomial.

Hybrid level aspect subconvexity for GL(2) × GL(1) Rankin-Selberg L-Functions

International audience Let $M$ be a squarefree positive integer and $P$ a prime number coprime to $M$ such that $P \sim M^{\eta}$ with $0 < \eta < 2/5$. We simplify the proof of subconvexity bounds for $L(\frac{1]{2}, f \otimes \chi)$ when $f$ is a primitive holomorphic cusp form of level $P$ and $\chi$ is a primitive Dirichlet character modulo $M$. These bounds are attained through an unamplified second moment method using a modified version of the delta method due to R. Munshi. The technique is similar to that used by Duke-Friedlander-Iwaniec save for the modification of the delta method.

CONVERGENCE OF DIRICHLET SERIES AND EULER PRODUCTS

The first part of this paper deals with Dirichlet series, and convergence theorems are proved that strengthen the classical convergence theorem as found e.g. in Serre’s “A Course in Arithmetic.” The second part deals with Euler-type products. A convergence theorem is proved giving sufficient conditions for such products to converge in the half-plane having real part greater than 1/2. Numerical evidence is also presented that suggests that the Euler products corresponding to Dirichlet L-functions L(s, χ), where χ is a primitive Dirichlet character, converge in this half-plane.

ON A NEW CIRCLE PROBLEM

We attempt to discuss a new circle problem. Let $\unicode[STIX]{x1D701}(s)$ denote the Riemann zeta-function $\sum _{n=1}^{\infty }n^{-s}$ ($\text{Re}\,s>1$) and $L(s,\unicode[STIX]{x1D712}_{4})$ the Dirichlet $L$-function $\sum _{n=1}^{\infty }\unicode[STIX]{x1D712}_{4}(n)n^{-s}$ ($\text{Re}\,s>1$) with the primitive Dirichlet character mod 4. We shall define an arithmetical function $R_{(1,1)}(n)$ by the coefficient of the Dirichlet series $\unicode[STIX]{x1D701}^{\prime }(s)L^{\prime }(s,\unicode[STIX]{x1D712}_{4})=\sum _{n=1}^{\infty }R_{(1,1)}(n)n^{-s}$$(\text{Re}\,s>1)$. This is an analogue of $r(n)/4=\sum _{d|n}\unicode[STIX]{x1D712}_{4}(d)$. In the circle problem, there are many researches of estimations and related topics on the error term in the asymptotic formula for $\sum _{n\leq x}r(n)$. As a new problem, we deduce a ‘truncated Voronoï formula’ for the error term in the asymptotic formula for $\sum _{n\leq x}R_{(1,1)}(n)$. As a direct application, we show the mean square for the error term in our new problem.

An explicit upper bound for |L(1,χ)| when χ(2) = 1 and χ is even

Let [Formula: see text] be a primitive Dirichlet character of conductor [Formula: see text] and let us denote by [Formula: see text] the associated [Formula: see text]-series. In this paper, we provide an explicit upper bound for [Formula: see text] when [Formula: see text] is a primitive even Dirichlet character with [Formula: see text].

A note on q-analogues of Dirichlet L-functions

In this note, we consider the special values of [Formula: see text]-analogues of Dirichlet [Formula: see text]-functions, namely, the values of the functions [Formula: see text] at positive integers [Formula: see text], where [Formula: see text] is a primitive Dirichlet character and [Formula: see text] is a complex number such that [Formula: see text]. We prove that if [Formula: see text] and [Formula: see text] is algebraic, then [Formula: see text] is transcendental. We also prove that if [Formula: see text] and [Formula: see text] is algebraic, then there exists a transcendental number [Formula: see text] which depends only on [Formula: see text] and is [Formula: see text]-linearly independent with [Formula: see text] such that [Formula: see text] is algebraic. These results can be viewed as an analogue of the classical result of Hecke on the arithmetic nature of the special values [Formula: see text] for [Formula: see text].

NON-VANISHING OF DIRICHLET L-FUNCTIONS AT THE CENTRAL POINT

Let χ be a primitive Dirichlet character modulo q and L(s, χ) be the Dirichlet L-function associated to χ. Using a new two-piece mollifier we show that L(½, χ) ≠ 0 for at least 34% of the characters in the family.