ON THE ZEROS OF ARITHMETIC DIRICHLET SERIES WITHOUT EULER PRODUCT

1994 ◽  
Vol 43 (2) ◽  
pp. 193-203
Author(s):  
A A Karatsuba
Keyword(s):  
Dirichlet Series ◽  
Euler Product

Calculation of spectral determinants

1994 ◽  
Vol 447 (1930) ◽  
pp. 413-437 ◽  
Keyword(s):  
Periodic Orbit ◽  
Finite Number ◽  
Dirichlet Series ◽  
Energy Levels ◽  
Euler Product ◽  
Quantum Energy ◽  
Type Formula

A method for regularizing spectral determinants is developed which facilitates their computation from a finite number of eigenvalues. This is used to calcu­late the determinant ∆ for the hyperbola billiard over a range which includes 46 quantum energy levels. The result is compared with semiclassical periodic orbit evaluations of ∆ using the Dirichlet series, Euler product, and a Riemann-Siegel-type formula. It is found that the Riemann-Siegel-type expansion, which uses the least number of orbits, gives the closest approximation. This provides explicit numerical support for recent conjectures concerning the analytic proper­ties of semiclassical formulae, and in particular for the existence of resummation relations connecting long and short pseudo-orbits.

scholarly journals Wigner Distribution Analysis and the Riemann Hypothesis

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Signal Processing ◽  
Dirichlet Series ◽  
Riemann Hypothesis ◽  
Wigner Distribution ◽  
Möbius Function ◽  
Distribution Analysis ◽  
Euler Product ◽  
Instantaneous Spectrum ◽  
Mobius Function

Wigner distribution is a tool for signal processing to obtain instantaneous spectrum of a signal. From which, another representation of the Euler product can be obtained for Dirichlet series of the Mobius function, which leads to the proof of the Riemann hypothesis

scholarly journals Non-vanishing of Dirichlet series without Euler products

2018 ◽  
Vol Volume 40 ◽  
Author(s):  
William D. Banks
Keyword(s):  
Zeta Function ◽  
Dirichlet Series ◽  
Half Plane ◽  
Riemann Zeta Function ◽  
Euler Product ◽  
The Riemann Zeta Function ◽  
International Audience ◽  
Euler Products ◽  
Riemann Zeta

International audience We give a new proof that the Riemann zeta function is nonzero in the half-plane {s ∈ C : σ > 1}. A novel feature of this proof is that it makes no use of the Euler product for ζ(s).

scholarly journals Wigner Distribution Analysis Applied to Lehmer’s Conjecture on the Ramanujan tau Function

2021 ◽  
pp. 19-28
Author(s):  
Takaaki Musha
Keyword(s):  
Signal Processing ◽  
Dirichlet Series ◽  
Wigner Distribution ◽  
Distribution Analysis ◽  
Euler Product ◽  
Instantaneous Spectrum ◽  
Tau Function ◽  
Lehmer's Conjecture ◽  
Ramanujan Tau Function

Wigner distribution is a tool for signal processing to obtain instantaneous spectrum of a signal. By using Wigner distribution analysis, another representation of the Euler product can be obtained for Dirichlet series of the Ramanujan tau function. From which, it can be proved that the Ramanujan tau function never become zero for all numbers.

scholarly journals On Ramanujan and Dirichlet series with Euler products

1984 ◽  
Vol 25 (2) ◽  
pp. 203-206 ◽  
Author(s):  
S. Raghavan
Keyword(s):  
Modular Form ◽  
Dirichlet Series ◽  
Modular Forms ◽  
Hecke Operators ◽  
Euler Product ◽  
Arithmetical Functions ◽  
In Trans ◽  
Euler Products ◽  
Unpublished Manuscripts ◽  
Profound Insight

In his unpublished manuscripts (referred to by Birch [1] as Fragment V, pp. 247–249), Ramanujan [3] gave a whole list of assertions about various (transforms of) modular forms possessing naturally associated Euler products, in more or less the spirit of his extremely beautiful paper entitled “On certain arithmetical functions” (in Trans. Camb. Phil. Soc. 22 (1916)). It is simply amazing how Ramanujan could write down (with an ostensibly profound insight) a basis of eigenfunctions (of Hecke operators) whose associated Dirichlet series have Euler products, anticipating by two decades the famous work of Hecke and Petersson. That he had further realized, in the event of a modular form f not corresponding to an Euler product, the possibility of restoring the Euler product property to a suitable linear combination of modular forms of the same type as f, is evidently fantastic.

The convergence of Euler products over p-adic number fields

2009 ◽  
Vol 52 (3) ◽  
pp. 583-606 ◽  
Author(s):  
Daniel Delbourgo
Keyword(s):  
Topological Space ◽  
Zeta Function ◽  
Dirichlet Series ◽  
Riemann Zeta Function ◽  
Number Fields ◽  
Product Structure ◽  
Euler Product ◽  
Euler Products ◽  
Riemann Zeta ◽  
New Formula

AbstractWe define a topological space over the p-adic numbers, in which Euler products and Dirichlet series converge. We then show how the classical Riemann zeta function has a (p-adic) Euler product structure at the negative integers. Finally, as a corollary of these results, we derive a new formula for the non-Archimedean Euler–Mascheroni constant.

scholarly journals On a question of A. Schinzel: Omega estimates for a special type of arithmetic functions

2013 ◽  
Vol 11 (3) ◽  
Author(s):  
Manfred Kühleitner ◽  
Werner Nowak
Keyword(s):  
Dirichlet Series ◽  
Lower Bounds ◽  
Remainder Term ◽  
Arithmetic Functions ◽  
Euler Product ◽  
Arbitrary Integer

AbstractThe paper deals with lower bounds for the remainder term in asymptotics for a certain class of arithmetic functions. Typically, these are generated by a Dirichlet series of the form ζ 2(s)ζ(2s−1)ζ M(2s)H(s), where M is an arbitrary integer and H(s) has an Euler product which converges absolutely for R s > σ0, with some fixed σ0 < 1/2.

scholarly journals Quotients of L-functions

2000 ◽  
Vol 160 ◽  
pp. 143-159
Author(s):  
Bernhard E. Heim
Keyword(s):  
Modular Form ◽  
Dirichlet Series ◽  
Modular Forms ◽  
Siegel Modular Forms ◽  
Siegel Modular Form ◽  
Euler Product ◽  
Jacobi Forms ◽  
New Variant

AbstractIn this paper a certain type of Dirichlet series, attached to a pair of Jacobi forms and Siegel modular forms is studied. It is shown that this series can be analyzed by a new variant of the Rankin-Selberg method. We prove that for eigenforms the Dirichlet series have an Euler product and we calculate all the local L-factors. Globally this Euler product is essentially the quotient of the standard L-functions of the involved Jacobi- and Siegel modular form.

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