Zeta regularized products

Abstract

If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda Subscript k"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>λ<!-- λ --></mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\lambda _k}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a sequence of nonzero complex numbers, then we define the zeta regularized product of these numbers, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="product Underscript k Endscripts lamda Subscript k"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mo movablelimits="false">∏<!-- ∏ --></mml:mo> <mml:mi>k</mml:mi> </mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>λ<!-- λ --></mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\prod \nolimits _k {{\lambda _k}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, to be <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="exp left-parenthesis negative upper Z prime left-parenthesis 0 right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>exp</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mi>Z</mml:mi> <mml:mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\exp ( - Z\prime (0))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z left-parenthesis s right-parenthesis equals sigma-summation Underscript k equals 0 Overscript normal infinity Endscripts lamda Subscript k Superscript negative s"> <mml:semantics> <mml:mrow> <mml:mi>Z</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mo movablelimits="false">∑<!-- ∑ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msubsup> <mml:mi>λ<!-- λ --></mml:mi> <mml:mi>k</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−<!-- − --></mml:mo> <mml:mi>s</mml:mi> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">Z(s) = \sum \nolimits _{k = 0}^\infty {\lambda _k^{ - s}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We assume that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z left-parenthesis s right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>Z</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Z(s)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has analytic continuation to a neighborhood of the origin. If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda Subscript k"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>λ<!-- λ --></mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\lambda _k}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the sequence of positive eigenvalues of the Laplacian on a manifold, then the zeta regularized product is known as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="det prime normal upper Delta"> <mml:semantics> <mml:mrow> <mml:mo movablelimits="true" form="prefix">det</mml:mo> <mml:mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mml:mi> <mml:mi mathvariant="normal">Δ<!-- Δ --></mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\det \prime \Delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the determinant of the Laplacian, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="product Underscript k Endscripts left-parenthesis lamda Subscript k Baseline minus lamda right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mo movablelimits="false">∏<!-- ∏ --></mml:mo> <mml:mi>k</mml:mi> </mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>λ<!-- λ --></mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\prod \nolimits _k {({\lambda _k} - \lambda )}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is known as the functional determinant. The purpose of this paper is to discuss properties of the determinant and functional determinant for general sequences of complex numbers. We discuss asymptotic expansions of the functional determinant as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda right-arrow negative normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo stretchy="false">→<!-- → --></mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\lambda \to - \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and its relationship to the Weierstrass product. We give some applications to the theory of Barnes’ multiple gamma functions and elliptic functions. A new proof is given for Kronecker’s limit formula and the product expansion for Barnes’ double Stirling modular constant.

Locations

• Transactions of the American Mathematical Society - View - PDF