On the Non-Existence of Certain Euler Products

1980 ◽  
Vol 23 (3) ◽  
pp. 371-372
Author(s):  
M. V. Subbarao
Keyword(s):  
Euler Product ◽  
Product Decomposition ◽  
Euler Products ◽  
Image Position ◽  
The Right

In a paper with the above title, T. M. Apostol and S. Chowla [1] proved the following result:Theorem 1.For relatively prime integers h and k, l ≤ h ≤ k, the seriesdoes not admit of an Euler product decomposition, that is, an identity of the form1except when h = k = l; fc = 1, fc = 2. The series on the right is extended over all primes p and is assumed to be absolutely convergent forR(s)>1.

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.

Partial Euler Products on the Critical Line

2005 ◽  
Vol 57 (2) ◽  
pp. 267-297 ◽  
Author(s):  
Keith Conrad
Keyword(s):  
Asymptotic Behavior ◽  
Elliptic Curve ◽  
Riemann Hypothesis ◽  
Critical Line ◽  
Function Fields ◽  
Number Fields ◽  
Euler Product ◽  
Second Moments ◽  
Precise Sense ◽  
Euler Products

AbstractThe initial version of the Birch and Swinnerton-Dyer conjecture concerned asymptotics for partial Euler products for an elliptic curve L-function at s = 1. Goldfeld later proved that these asymptotics imply the Riemann hypothesis for the L-function and that the constant in the asymptotics has an unexpected factor of. We extend Goldfeld's theorem to an analysis of partial Euler products for a typical L-function along its critical line. The general phenomenon is related to second moments, while the asymptotic behavior (over number fields) is proved to be equivalent to a condition that in a precise sense seemsmuch deeper than the Riemann hypothesis. Over function fields, the Euler product asymptotics can sometimes be proved unconditionally.

scholarly journals Ramanujan congruences for p-k (mod 11')

1983 ◽  
Vol 24 (2) ◽  
pp. 107-123 ◽  
Author(s):  
Basil Gordon
Keyword(s):  
Generating Function ◽  
Euler Product ◽  
Ramanujan Congruences ◽  
Image Position

Denote bythe Euler product, and bythe partition generating function. More generally, if k is any integer, putso that p(n) = p−1(n). In [3], Atkin proved the following theorem.

scholarly journals A multivariable Euler product of Igusa type and its applications

2009 ◽  
Vol 129 (8) ◽  
pp. 1919-1930 ◽  
Author(s):  
Nobushige Kurokawa ◽  
Hiroyuki Ochiai
Keyword(s):  
Euler Product

scholarly journals A generalization of z!

1964 ◽  
Vol 4 (3) ◽  
pp. 327-341 ◽  
Author(s):  
W. B. White-Smith ◽  
V. T. Buchwald
Keyword(s):  
Infinite Product ◽  
Euler Product ◽  
Factorial Function

SummaryA generalised factorial function (z: k)! is defined as an infinite product similar to the Euler product for z!, but with the sequences of integers replaced by the roots of F(z) = sin πz+kπz. It is proved that, apart from poles in (z) < 0, (z: k)! is analytic in both variables, and that F(z) may be expressed in the form F(z) = πz/(z: k)!(—z: k)!

Sign in / Sign up

Export Citation Format

Share Document