ON THE APPROXIMATE FUNCTIONAL EQUATION FOR ζ2(s) AND OTHER DIRICHLET SERIES

1986 ◽  
Vol 37 (2) ◽  
pp. 193-209 ◽  
Author(s):  
M. JUTILA
Keyword(s):  
Functional Equation ◽  
Dirichlet Series ◽  
Approximate Functional Equation

scholarly journals An identity for certain Dirichlet series

1968 ◽  
Vol 9 (2) ◽  
pp. 79-82
Author(s):  
B. C. Berndt
Keyword(s):  
Functional Equation ◽  
Dirichlet Series ◽  
Short Proof ◽  
Approximate Functional Equation ◽  
Partial Sum

In deriving the approximate functional equation for certain Dirichlet series, one first establishes an identity for the function in terms of a partial sum of the series (e.g. see [1] and [2]). It is the purpose of this note to give a short proof of this identity for Hecke's Dirichlet series [1]. The proof is valid with only a few minor changes for the identity given by Chandrasekharan and Narasimhan [2, Lemma 2] for a much larger class of Dirichlet series. However, the brevity of the paper would be lost if we introduced the necessary terminology and notation.

scholarly journals Evaluating -functions with few known coefficients

2014 ◽  
Vol 17 (1) ◽  
pp. 245-258 ◽  
Author(s):  
David W. Farmer ◽  
Nathan C. Ryan
Keyword(s):  
Functional Equation ◽  
Computational Effort ◽  
Standard Approach ◽  
Approximate Functional Equation

AbstractWe address the problem of evaluating an $L$-function when only a small number of its Dirichlet coefficients are known. We use the approximate functional equation in a new way and find that it is possible to evaluate the $L$-function more precisely than one would expect from the standard approach. The method, however, requires considerably more computational effort to achieve a given accuracy than would be needed if more Dirichlet coefficients were available.

Approximate Functional Equation

2003 ◽  
pp. 53-69
Author(s):  
Antanas Laurinčikas ◽  
Ramūnas Garunkštis
Keyword(s):  
Functional Equation ◽  
Approximate Functional Equation

scholarly journals A note on a functional equation

1973 ◽  
Vol 15 (4) ◽  
pp. 385-388
Author(s):  
Chung-Ming An
Keyword(s):  
Functional Equation ◽  
Dirichlet Series ◽  
Gamma Function ◽  
Generalized Gamma ◽  
Special Cases ◽  
Generalized Gamma Function

The object of this note is to give an aspect to the problem of the functional equation of the generalized gamma function and Dirichlet series which are defined in [1]. In general, we cannot answer the problem yet. But it is worthy to attack this problem for some special cases.

SOME NUMBER THEORETIC APPLICATIONS OF A GENERAL MODULAR RELATION

2006 ◽  
Vol 02 (04) ◽  
pp. 599-615 ◽  
Author(s):  
SHIGERU KANEMITSU ◽  
YOSHIO TANIGAWA ◽  
HARUO TSUKADA
Keyword(s):  
Functional Equation ◽  
Dirichlet Series ◽  
The Other ◽  
Gamma Functions ◽  
Modular Relation

We state a form of the modular relation in which the functional equation appears in the form of an expression of one Dirichlet series in terms of the other multiplied by the quotient of gamma functions and illustrate it by some concrete examples including the results of Koshlyakov, Berndt and Wigert and Bellman.

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