Dirichlet L-functions Research Articles

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].

Some identities involving certain Hardy sum and Kloosterman sum

TextBy using the properties of Gauss sums and the mean value theorem of the Dirichlet L-function, a hybrid mean value problem involving certain Hardy sum and Kloosterman sum is studied. Two exact computational formulae are given, through which the cancellation phenomenon is revealed. VideoFor a video summary of this paper, please visit https://youtu.be/d-X81xErU7Q.

On the reciprocity law for the twisted second moment of Dirichlet $L$-functions

We investigate the reciprocity law, studied by Conrey~\cite{Con07} and Young~\cite{You11a}, for the second moment of Dirichlet L-functions twisted by $\chi(a)$ modulo a prime $q$. We show that the error term in this reciprocity law can be extended to a continuous function of $a/q$ with respect to the real topology. Furthermore, we extend this reciprocity result, proving an exact formula involving also shifted moments. We also give an expression for the twisted second moment involving the coefficients of the continued fraction expansion of $a/q$, and, consequently, we improve upon a classical result of Selberg on the second moment of Dirichlet L-functions with two twists. Finally, we obtain a formula connecting the shifted second moment of the Dirichlet $L$-functions with the Estermann function. In particular cases, this result can be used to obtain some simple explicit exact formulae for the moments.

Analytic monoids and factorization problems

The main goal of this paper is to initiate study of analytic monoids as a general framework for quantitative theory of factorization. So far the latter subject was developed either in concrete settings, for instance in orders of number fields, or abstractly, in an axiomatic way. Some of the abstract approaches are too general to address delicate problems concerning oscillatory nature of the related counting functions, or are too restrictive in the sense that they suffer from the lack of examples except classical ones i.e. the Hilbert monoids of algebraic integers and their products. The notion of an analytic monoid is enough flexible to allow constructions of many other examples, and also ensures sufficiently rich analytic structure. In particular, we construct examples of such monoids with the associated L-functions being products of classical Dirichlet L-functions and L-functions of twisted irreducible unitary cuspidal automorphic representations of GL_d({mathbb {A}}_{mathbb {Q}}) satisfying the Ramanujan conjecture and having real coefficients. Finally, to illustrate how a typical problem from the quantitative theory of factorization can be studied in the framework of analytic monoids, we formulate several results concerning oscillations of the remainder term in the asymptotic formula for the number of irreducible elements with norms less or equal x, as x tends to infinity.

Improvements to Turing's method II

Turing's method uses explicit bounds on $|\int _{t_{1}}^{t_{2}} S(t)\, dt|$, where $\pi S(t)$ is the argument of the Riemann zeta-function. This article improves the bound on $|\int _{t_{1}}^{t_{2}} S(t)\, dt|$ given in \cite {TrudgianTuring}.

UNIVERSALITY THEOREMS FOR SOME COMPOSITE FUNCTIONS

In [5], it was proved that a collection consisting from Dirichlet L-functions and periodic Hurwitz zeta-functions is universal in the sense that the shifts of those functions approximate simultaneously a given collection of analytic functions. In the paper, we prove theorems on the universality of composite functions of the above collection.

An Explicit upper Bound of the Argument of Dirichlet <i>L</i>-functions on the Generalized Riemann Hypothesis

We prove an explicit upper bound of the function ( ) , St χ , defined by the argument of Dirichlet L-functions attached to a primitive Dirichlet character χ (mod q > 1). An explicit upper bound of the function S(t), defined by the argument of the Riemann zeta-function, have been obtained by A. Fujii (1). Our result is obtained by applying the idea of Fujii's result on S(t). The constant part of the explicit upper bound of ( ) , St χ in this paper does not depend on χ . Our proof does not cover the case q = 1 and indeed gives a better bound than the one of Fujii that covers the case q = 1.

Numerical computations concerning the GRH

We describe two new algorithms for the efficient and rigorous computation of Dirichlet L-functions and their use to verify the Generalised Riemann Hypothesis for all such L-functions associated with primitive characters of modulus q ≤ 400 000 q\leq 400\,000 . We check, to height, max ( 10 8 q , A ⋅ 10 7 q + 200 ) \textrm {max}\left (\frac {10^8}{q},\frac {A\cdot 10^7}{q}+200\right ) with A = 7.5 A=7.5 in the case of even characters and A = 3.75 A=3.75 for odd characters. In addition we confirm that no Dirichlet L-function with a modulus q ≤ 2 000 000 q\leq 2\,000\,000 vanishes at its central point.

Other Demostrative Perspective of How to See Dirichlet’s Theorem

The Dirichlet’s theorem (1837), initially guessed by Gauss, is a result of analytic number theory. Dirichlet, demonstrated that: For any two positive coprime integers and , there are infinite primes of the form , where is a non-negative integer ( 1, 2, ... ). In other words, there are infinite primes which are congruent to mod b. The numbers of the form is an arithmetic progression. Actually, Dirichlet checks a result somewhat more interesting than the previous claim, since he demonstrated that: ln → ∞ Which implies that there are infinite primes, ≡ . The proof of the theorem uses the properties of certain Dirichlet L-functions and some results on arithmetic of complex numbers, and it is sufficiently complex that some texts about numbers theory excluded it. Here is a simple proof by reductio ad absurdum which does not require extensive mathematical knowledge. Original Research Article Ferreira and Guardo; JSRR, 10(3): 1-7, 2016; Article no.JSRR.24470 2

Multiple Solutions of Riemann Type of Functional Equations

Linearly independent Dirichlet L-functions satisfying the same Riemann-type of functional equation have been supposed for long time to possess off critical line non trivial zeros. We are taking a closer look into this problem and into its connection with the Generalized Riemann Hypothesis

The families of L-series associated with decomposition of the generating functions

By using periodic functions from the nonnegative integers to the complex numbers, we generalize the generating function of the q-Apostol type Eulerian polynomials and numbers attached the character defined in [1]. Then using this generating function, we a construct new L-type series. By using periodic functions, we derive decomposition of the generating functions for the q-Euler numbers and polynomials. Applying the Mellin transformation to the decomposition of the generating functions, we introduce and investigate the various properties of a certain new family of the Dirichlet type L-series and the Dirichlet L-function. Finally, we derive many potentially useful results involving these functions polynomials and numbers.

Zero distribution of Dirichlet L-functions

The series Lχ converges absolutely and uniformly in the region Re(s) ≥ 1 + e, for any e > 0. It therefore represents a holomorphic function on the half-plane Re(s) > 1, which further extends to a meromorphic function in the complex plane C. In particular, for the principal character χ = 1, we get back the Riemann zeta function ζ(s). For a ∈ C, the zeros of L χ − a which we denote by ρa = βa + iγa, are called the a-points of Lχ, and their distribution has long been an object of study (see [17, 18, 20]). The function Lχ has only real zeros in the half plane Re(s) < 0, these zeros are called the trivial zeros. If χ(−1) = 1, the trivial zeros of Lχ are s = −2n for all non-negative integers n. If χ(−1) = −1, the trivial zeros of Lχ are s = −2n − 1 for all non-negative integers n. Beside the trivial zeros of Lχ, there are infinitely many non-trivial zeros lying in the strip 0 < Re(s) < 1. For a 6= 0, it can be shown that there is always a a-point in some neighbourhood of any trivial zero of Lχ with sufficiently large negative real part, and with finitely many exceptions there are no other in the left half-plane, thus the number of these a-points having real part in [−R, 0] is asymptotically 1 2 R. The remaining a-points all lie in a strip 0 < Re(s) < A, where A depends on a, and we call these non-trivial a-points (see [16, 19]). For a positive number T , let N χ(T ) denote the number of non-trivial a-points ρa = βa+ iγa of Lχ with |γa| ≤ T . We have the following formula

A hybrid mean value involving Cochrane sums and a new sum analogous to Kloosterman sums*

In this paper, we use the analytic method, the properties of the Dedekind sums, and the mean value theorem of Dirichlet L-function to study the mean value properties of the Cochrane sums and give an interesting mean square value formula for it.

The hybrid mean value of Dedekind sums and two-term exponential sums

Abstract In this paper, we use the mean value theorem of Dirichlet L-functions, the properties of Gauss sums and Dedekind sums to study the hybrid mean value problem involving Dedekind sums and the two-term exponential sums, and give an interesting identity and asymptotic formula for it.

Rudnick and Soundararajan's theorem for function fields

In this paper we prove a function field version of a theorem by Rudnick and Soundararajan about lower bounds for moments of quadratic Dirichlet L-functions. We establish lower bounds for the moments of quadratic Dirichlet L-functions associated to hyperelliptic curves of genus g over a fixed finite field Fq in the large genus g limit.

Riesz-type criteria and theta transformation analogues

We give character analogues of a generalization of a result due to Ramanujan, Hardy and Littlewood, and provide Riesz-type criteria for Riemann Hypotheses for the Riemann zeta function and Dirichlet L-functions. We also provide analogues of the general theta transformation formula and of recent generalizations of the transformation formulas of W.L. Ferrar and G.H. Hardy for real primitive Dirichlet characters.

On the asymptotic formulae for some multiplicative functions in short intervals

We are going to study the mean values of some multiplicative functions connected with the divisor function in short interval of summation. The asymptotics for such mean values will be proved. Considering instead of well-known multiplicative functions, their inverses lead to very weak results of application of standard methods of complex integration. In order to get better estimations, we propose another method which uses as its main tools the density estimates and zero-free region for Riemann ζ-function and Dirichlet L-functions.

On the general trigonometric sums weighted by character sums

The main purpose of this paper is using the estimates for character sums and the mean value properties of Dirichlet L-function to study the hybrid mean value properties of the general trigonometric sums, which is weighted by the 2kth and first power mean of character sums in [1,p4). The two sharp asymptotic formulae are derived and some interesting connections between these sums are established.

Selberg’s method in the problem about the zeros of linear combinations of L-functions on the critical line

In the late 1990s, Atle Selberg invented a new method, which had allowed him to prove that if a linear combination of Dirichlet L-functions satisfies a functional equation, then a positive proportion of its zeros lie on the critical line. The paper considers this method in detail for the general case of a linear combination of L-functions from the Selberg class and applies it to a linear combination of L-functions from the Selberg class of degree 2.

Functional Equations associated with Generalized Bernoulli Numbers and Polynomials

In this paper, we investigate the functional equations of the multiple Dirichlet and Hurwitz L-functions associated with Bernoulli numbers and polynomials attached to Dirichlet character.