Bessel series expansions of the Epstein zeta function and the functional equation

Abstract

For the Epstein zeta function of an <italic>n</italic>-ary positive definite quadratic form, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n minus 1"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n - 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> generalizations of the Selberg-Chowla formula (for the binary case) are obtained. Further, it is shown that these <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n minus 1"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n - 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> formulas suffice to prove the functional equation of the Epstein zeta function by mathematical induction. Finally some generalizations of Kronecker’s first limit formula are obtained.

Locations

• Transactions of the American Mathematical Society - View - PDF