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An identity for certain Dirichlet series†

Published online by Cambridge University Press:  18 May 2009

B. C. Berndt
B. C. Berndt
Affiliation:
The University of Glasgow and The University of Illinois
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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.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 9 , Issue 2 , July 1968 , pp. 79 - 82
Copyright
Copyright © Glasgow Mathematical Journal Trust 1968

References

REFERENCES

1.Apostol, T. M. and Sklar, Abe, The approximate functional equation of Hecke's Dirichlet series, Trans. Amer. Math. Soc. 86 (1957), 446–462.Google Scholar
2.Chandrasekharan, K. and Narasimhan, Raghavan, The approximate functional equation for a class of zeta-functions, Math. Ann. 152 (1963), 30–64.CrossRefGoogle Scholar
3.Hobson, E. W., The theory of functions of a real variable, vol. II, 2nd. ed., Cambridge University Press (Cambridge, 1926).Google Scholar
4.Titchmarsh, E. C., Theory of Fourier integrals, 2nd. ed., Clarendon Press (Oxford, 1948).Google Scholar