An attempt to interpret the Weil explicit formula from Beurling's spectral theory

A. Beurling introduced the celebrated problem of spectral synthesis. Roughly speaking, it is a problem whether functions belonging to a certain Banach space have a possibility to be approximated by trigonometric polynomials in the appropriate topology. For this problem Beurling introduced the concept of spectral sets whose elements are regarded as exponents of trigonometric polynomials. In the Weil explicit formula we can see a certain phenomenon which may be related to Beurling's spectral sets. The purpose of this paper is to study the phenomenon.     

A formula for the central value of certain Hecke L-functions

Let be the negative of a prime, and O_K its ring of integers. Let D be a prime ideal in O_K of prime norm congruent to . Under these assumptions, there exists Hecke characters ψ_D of K with conductor (D) and infinite type (1,0). Their L-series L(ψ_D,s) are associated to a CM elliptic curve A(N,D) defined over the Hilbert class field of K. We will prove a Waldspurger-type formula for L(ψ_D,s) of the form L(ψ_D,1)=Ω∑_[A],Ir(D,[A],I)m_[A],I([D]) where the sum is over class ideal representatives I of a maximal order in the quaternion algebra ramified at |N| and infinity and [A] are class group representatives of K. An application of this formula for the case N=-7 will allow us to prove the non-vanishing of a family of L-series of level 7|D| over K.     

Dirichlet series associated with polynomials and applications

We consider the possibility of the analytic continuation of the Dirichlet series SP;Z(s) associated with a polynomial P(x) and auxiliary series Z(s). In fact, we derive a certain criterion for the analytic continuation for some class of polynomials and give examples such that SP;Z(s) cannot be continued meromorphically to the whole plane C. We also study the asymptotic behaviors of the sum MP(x)=_∑P(n₁,…,nk)?xΛ(n₁)?Λ(nk) considered first by M. Peter. Generalizations of this sum are also considered.     

Some properties on the integral of the product of several Euler polynomials

Abstract

In this paper, we study the formula for a product of two Euler poly-nomials. From this study, we derive some formulae for the integral of the product of two or more Euler polynomials.     

The Fourier inversion and the Riemann functional equation

We prove that the Riemann functional equation can be recovered by the Mellin transforms of essentially all the absolutely integrable functions. The present analysis shows also that the Riemann functional equation is equivalent to the Fourier inversion formula. We introduce the notion of a λ-pair of absolutely integrable functions and show that the components of the λ-pair satisfy an identity involving convolution type products.     

Müntz formula and zero free regions for the Riemann zeta function

A formula first derived by Müntz which relates the Riemann zeta function ζ times the Mellin transform of a test function f and the Mellin transform of the theta transform of f is exploited, together with other analytic techniques, to construct zero free regions for ζ(s) with s in the critical strip. Among these are regions with a shape independent of Res.     

On Euler numbers, polynomials and related p-adic integrals

In this note we give a new proof of Witt's formula for Euler numbers, which are related to some known or new identities involving the Euler numbers. We also obtain a brief proof of a classical result on Euler numbers modulo of two due to M.A. Stern using the approach of p-adic integration, which was recently proved by G. Liu, and Z.-W. Sun. Finally some explicit formulas for Genocchi numbers are proved and applications are given.     

Some inequalities for fourier coefficients in inner product spaces

Some inequalities for Fourier coefficients in inner product spaces and related results are given. These inequalities complement, in a sense, some of the results in a recent book due to Mitrinovi, Peari and Fink.This research was supported by a grant from La Trobe University     

An explicit formula for the linearization coefficients of Bessel polynomials

We prove a single sum formula for the linearization coefficients of the Bessel polynomials. In two special cases we show that our formula reduces indeed to Berg and Vignat??s formulas in their proof of the positivity results about these coefficients (Constr. Approx. 27:15?C32, 2008). As a bonus we also obtain a generalization of an integral formula of Boros and Moll (J. Comput. Appl. Math. 106:361?C368, 1999).     

On Li's criterion for the Riemann hypothesis for the Selberg class

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In this paper, we shall prove a generalization of Li's positivity criterion for the Riemann hypothesis for the extended Selberg class with an Euler sum. We shall also obtain two arithmetic expressions for Li's constants , where the sum is taken over all non-trivial zeros of the function F and the indicates that the sum is taken in the sense of the limit as T→∞ of the sum over ρ with |Imρ|?T. The first expression of λF(n), for functions in the extended Selberg class, having an Euler sum is given terms of analogues of Stieltjes constants (up to some gamma factors). The second expression, for functions in the Selberg class, non-vanishing on the line , is given in terms of a certain limit of the sum over primes.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=EwDtXrkuwxA.     

Euler numbers congruences and Dirichlet L-functions

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In this paper we apply Yamamoto's Theorem [Y. Yamamoto, Dirichlet series with periodic coefficients, in: Proc. Intern. Sympos. “Algebraic Number Theory”, Kyoto, 1976, JSPS, Tokyo, 1977, pp. 275-289] to find the residue modulo a prime power of the linear combination of Dirichlet L-function values L(s,χ) at positive integral arguments s such that s and χ are of the same parity, in terms of Euler numbers, whereby we obtain the finite expressions for short interval character sums. The results obtained generalize the previous results pertaining to the congruences modulo a prime power of the class numbers as the special case of s=1.

Video

For a video summary of this paper, please visit http://www.youtube.com/watch?v=_KAv4FCdVUs.     

A note on Mertens' formula for arithmetic progressions

We study the Mertens product over primes in arithmetic progressions, and find a uniform version of previous results.     

On the coefficients of ternary cyclotomic polynomials

We study coefficients of ternary cyclotomic polynomials Φ_pqr(z)=∏_ρ(z−ρ), where p, q, and r are distinct odd primes and the product is taken over all primitive pqrth roots of unity ρ.     

Effective Estimates for the Distribution of Values of Euler Products

We prove effective upper bounds for the almost periodicity of polynomial Euler products in the half-plane of absolute convergence. From this we deduce estimates for the roots of the equation , where c is any non-zero complex number which is attained by . The method relies mainly on effective diophantine approximation.The first author was supported by a grant of the Humboldt Foundation.     

Hecke actions on certain strongly modular genera of lattices

We calculate the action of some Hecke operators on spaces of modular forms spanned by the Siegel theta-series of certain genera of strongly modular lattices closely related to the Leech lattice. Their eigenforms provide explicit examples of Siegel cusp forms.Received: 10 September 2004     

Exceptional congruences for the coefficients of certain eta-product newforms

Let F(z)=∑_n=1^∞a(n)qⁿ denote the unique weight 16 normalized cuspidal eigenform on . In the early 1970s, Serre and Swinnerton-Dyer conjectured that

     

An inequality for means with applications

We show that an almost trivial inequality between the first and second moment and the maximal value of a random variable can be used to slightly improve deep theorems. Received: 25 November 2006     

Large spaces between the zeros of the Riemann zeta-function and random matrix theory

On the hypothesis that the 2k-th mixed moments of Hardy's Z-function and its derivative are correctly predicted by random matrix theory, it is established that large gaps (depending on, and apparently increasing with k) exist between the zeta zeros. The case k=3 has been worked out in an earlier paper (in this journal) and the cases k=4,5,6 are considered here. When k=6 the gaps obtained have >4 times the average gap length. This depends on calculations involving Jacobi-Schur functions and formulae for these functions due to Jacobi, Trudi and Aitken in the classical theory of equations.