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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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(3) Littlewood, J. E., Lectures on the Theory of Functions, Oxford University Press, 1944.Google Scholar
(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