Coefficients Of Dirichlet Series Research Articles

Inversion formula for Dirichlet series and its application

A contour integration method, used to study the asymptotic of the sums of coefficients of Dirichlet series, is based on the Inversion formula. It allows you to express the sum of the coefficients in terms of the sum of the series. This approach gives effective estimates if the abscissa of absolute convergence 𝜎𝑎 > 1. In some cases, when studying arithmetical functions in generating Dirichlet series, this value is less than 1. As a rule, in this case, the Tauberian Delange theorem, which gives only the main term of asymptotic, is applied. However, generating Dirichlet series have better analytical properties than we need for the Delange theorem application. The contour integration method allows to count on precise results, but it need the inversion formula which is effective for series with 𝜎𝑎 < 1. In this paper the such inversion formula is presented and is proved to be an effective tool on examining the distribution of d(n) function values in the residue classes coprim with a module. W. Narkievicz used Delange theorem to obtain the main term of the asymptotic for frequency of hits of the values of function d(n) in residue classes. Application of the inversion formula allowed us to obtain more precise results.

Mean value of Dirichlet series coefficients of Rankin–Selberg L-functions

Abstract Let π and π′ be irreducible cuspidal automorphic representations of GL m (𝔸ℚ) and GL m′ (𝔸ℚ), respectively. Denote by a π×π′ (n) the nth coefficient of the Dirichlet series of the Rankin–Selberg L-function associated with π and π′. In this paper, we obtain the upper bound for the sum Σ n ≤ x a π × π'(n).

Selberg\u2019s orthonormality conjecture and joint universality of L-functions

In the paper we introduce a new method how to use only an orthonormality relation of coefficients of Dirichlet series defining given L-functions from the Selberg class to prove joint universality.

Oscillations of coefficients of Dirichlet series attached to automorphic forms

For $m\ge 2$, let $\pi$ be an irreducible cuspidal automorphic representation of $GL_m(\mathbb{A}_{\mathbb{Q}})$ with unitary central character. Let $a_\pi(n)$ be the $n^{th}$ coefficient of the $L$-function attached to $\pi$. Goldfeld and Sengupta have recently obtained a bound for $\sum_{n\le x} a_\pi(n)$ as $x \rightarrow \infty$. For $m\ge 3$ and $\pi$ not a symmetric power of a $GL_2(\mathbb{A}_{\mathbb{Q}})$-cuspidal automorphic representation with not all finite primes unramified for $\pi$, their bound is better than all previous bounds. In this paper, we further improve the bound of Golfeld and Sengupta. We also prove a quantitative result for the number of sign changes of the coefficients of certain automorphic $L$-functions, provided the coefficients are real numbers.

Estimates for coefficients of certain L-functions

Let $$\pi _{\varphi }$$ (or $$\pi _{\psi }$$ ) be an automorphic cuspidal representation of $$\text {GL}_{2} (\mathbb {A}_{\mathbb {Q}})$$ associated to a primitive Maass cusp form $$\varphi $$ (or $$\psi $$ ), and $$\mathrm{sym}^j \pi _{\varphi }$$ be the jth symmetric power lift of $$\pi _{\varphi }$$ . Let $$a_{\mathrm{sym}^j \pi _{\varphi }}(n)$$ denote the nth Dirichlet series coefficient of the principal L-function associated to $$\mathrm{sym}^j \pi _{\varphi }$$ . In this paper, we study first moments of Dirichlet series coefficients of automorphic representations $$\mathrm{sym}^3 \pi _{\varphi }$$ of $$\text {GL}_{4}(\mathbb {A}_{\mathbb {Q}})$$ , and $$\pi _{\psi }\otimes \mathrm{sym}^2 \pi _{\varphi }$$ of $$\text {GL}_{6}(\mathbb {A}_{\mathbb {Q}})$$ . For $$3 \le j \le 8$$ , estimates for $$|a_{\mathrm{sym}^j \pi _{\varphi }}(n)|$$ on average over a short interval have also been established.

Sign changes of coefficients of certain Dirichlet series

In this paper, we give criteria for infinitely many sign changes of the coefficients of any Dirichlet series if the coefficients are real numbers. We also provide examples where our criteria are applicable.

The growth of Dirichlet series

We define Knopp-Kojima maximum modulus and the Knopp-Kojima maximum term of Dirichlet series on the right half plane by the method of Knopp-Kojima, and discuss the relation between them. Then we discuss the relation between the Knopp-Kojima coefficients of Dirichlet series and its Knopp-Kojima order defined by Knopp-Kojima maximum modulus. Finally, using the above results, we obtain a relation between the coefficients of the Dirichlet series and its Ritt order. This improves one of Yu Jia-Rong’s results, published in Acta Mathematica Sinica 21 (1978), 97–118. We also give two examples to show that the condition under which the main result holds can not be weakened.

Exponential sums with coefficients of certain Dirichlet series

Under the generalized Lindelof Hypothesis in the t- and q-aspects, we bound exponential sums with coefficients of Dirichlet series belonging to a certain class. We use these estimates to establish a conditional result on squares of Hecke eigenvalues at Piatetski–Shapiro primes.

Effective multiplicity one on GL n and narrow zero-free regions for Rankin-Selberg L -functions

We establish zero-free regions tapering as an inverse power of the analytic conductor for Rankin-Selberg L -functions on GL n ×GL n' . Such zero-free regions are equivalent to commensurate lower bounds on the edge of the critical strip, and in the case of L ( s , π X), on the residue at s = 1. As an application we show that a cuspidal automorphic representation on GL n is determined by a finite number of its Dirichlet series coefficients, and that this number grows at most polynomially in the analytic conductor.

Strong Multiplicity One for the Selberg Class

AbstractWe investigate the problem of determining elements in the Selberg class by means of their Dirichlet series coefficients at primes.

The growth of Dirichlet series of zero order

This paper first studies the relations between the growth and the coefficients of Dirichlet series of zero order in the whole plane, and further proves that the growth of random entire functions defined by random Dirichlet series of zero order in every horizotal straight lines is almost surely equal to the growth of entire functions defined by their corresponding Dirichlet series.

Poisson formulae of Hecke type

In his lectures on "Forms of higher degree ", Igusa [10] gave a simple proof of a Poisson formula due to Yamazaki [19] associated with coefficients of Dirichlet series with functional equations involving a single /'-factor, by introducing an operator of order 2 in a function space, via the Mellin transform. Our object here is to show that this Poisson formula can be generalised a little, so as to bring within its ambit, functional equations c,f Dirichlet series involving multiple Ffactors (and, in particular, those associated with Mellin transforms of non-analytic automorphic functions). We also indicate an adelic interpretation of certain Poisson formulae above just to highlight the fact that such a h~rmula in adelic form constitutes an important step in the proof of theorem 5 in w 3 of [7] on a global representation of (G/-~a.

Simple Analytic Proof of the Prime Number Theorem

(1980). Simple Analytic Proof of the Prime Number Theorem. The American Mathematical Monthly: Vol. 87, No. 9, pp. 693-696.

Asymptotic properties of the coefficients of Dirichlet series representing entire functions

Asymptotic properties of the coefficients of Dirichlet series representing entire functions