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Generalised Dirichlet series and Hecke's functional equation†

Published online by Cambridge University Press:  20 January 2009

Bruce C. Berndt
Bruce C. Berndt
Affiliation:
University of Glasgow, Glasgow, W.2 and University of Illinois, Urbana, Illinois
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The generalised zeta-function ζ(s, α) is defined by

where α>0 and Res>l. Clearly, ζ(s, 1)=, where ζ(s) denotes the Riemann zeta-function. In this paper we consider a general class of Dirichlet series satisfying a functional equation similar to that of ζ(s). If ø(s) is such a series, we analogously define ø(s, α). We shall derive a representation for ø(s, α) which will be valid in the entire complex s-plane. From this representation we determine some simple properties of ø(s, α).

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 15 , Issue 4 , December 1967 , pp. 309 - 313
Copyright
Copyright © Edinburgh Mathematical Society 1967

References

REFERENCES

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(4) Titchmarsh, E. C., Theory of Fourier Integrals, 2nd ed., Clarendon Press, Oxford, 1948.Google Scholar
(5) Watson, G. N., Theory of Bessel Functions, 2nd ed., University Press, Cambridge, 1944.Google Scholar