71 research outputs found
Approximate functional equations for the Hurwitz and Lerch zeta-functions
As one of the asymptotic formulas for the zeta-function, Hardy and Littlewood
gave asymptotic formulas called the approximate functional equation. In 2003,
R. Garunk\v{s}tis, A. Laurin\v{c}ikas, and J. Steuding (in [1]) proved the
Riemann-Siegel type of the approximate functional equation for the Lerch
zeta-function . In this paper, we prove another type of
approximate functional equations for the Hurwitz and Lerch zeta-functions. R.
Garunk\v{s}tis, A. Laurin\v{c}ikas, and J. Steuding (in \cite{GLS2}) obtained
the results on the mean square values of with respect to . We obtain the main term of the mean square
values of using a simpler method than
their method in [2].Comment: 13 page
Zeta functions on tori using contour integration
A new, seemingly useful presentation of zeta functions on complex tori is
derived by using contour integration. It is shown to agree with the one
obtained by using the Chowla-Selberg series formula, for which an alternative
proof is thereby given. In addition, a new proof of the functional determinant
on the torus results, which does not use the Kronecker first limit formula nor
the functional equation of the non-holomorphic Eisenstein series. As a bonus,
several identities involving the Dedekind eta function are obtained as well
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