Bilateral zeta functions and their applications (Q2864709)

For a nonnegative integer \(r\) and complex numbers \(\omega_0,\omega_1,\ldots,\omega_r\), the Barnes zeta function \(\zeta_{r+1}(s,z\mid \omega_0, \pmb{\omega})\) with \(\pmb{\omega}=(\omega_1,\ldots,\omega_r)\) is defined by NEWLINE\[NEWLINE\zeta_{r+1}(s,z\mid \omega_0, \pmb{\omega})=\sum\limits_{m_0,\ldots,m_r=0}^\infty\frac{1}{(z+m_0\omega_0+m_1\omega_1+\cdots+m_r\omega_r)^s}.NEWLINE\]NEWLINE Under certain condition for \(z,\omega_0\) and \(\pmb{\omega}\), the Barnes zeta function \(\zeta_{r+1}(s,z\mid \omega_0, \pmb{\omega})\) converges absolutely for \(\Re(s)>r+1\), and can be continued meromorphically to the whole \(s\)-plane. NEWLINENEWLINEIn the paper under review, the bilateral zeta function NEWLINE\[NEWLINE\xi_{r+1}(s,z\mid\omega_0;\pmb{\omega})=\zeta_{r+1}(s,z+\omega_0\mid \omega_0,\pmb{\omega})+\zeta_{r+1}(s,z\mid -\omega_0,\pmb{\omega})NEWLINE\]NEWLINE is introduced. The author shows that under some conditions for \(z,\omega_0\) and \(\pmb{\omega}\), the series expression of the bilateral zeta function \(\xi_{r+1}(s,z\mid\omega_0;\pmb{\omega})\) converges absolutely for \(\Re(s)>r+1\), and is continued analytically to the whole \(s\)-plane. The following Fourier series expansion is proved: NEWLINE\[NEWLINE\zeta_{r+1}(s,z\mid e^{\pi i};\pmb{\omega})=\frac{e^{-\frac{\pi is}{2}}(2\pi)^s}{\Gamma(s)}\sum\limits_{n=1}^\infty\frac{n^{s-1}e^{2\pi inz}}{(1-e^{2\pi in\omega_1})\cdots(1-e^{2\pi in\omega_r})}.NEWLINE\]NEWLINE The main result of this paper states that the Barnes zeta function can be expressed as a finite sum of bilateral zeta functions. Using the properties of the bilateral zeta function, simple proofs of the reflection formula for the multiple gamma function, the inversion formula for the Dedekind \(\eta\)-function, Ramanujan's formula for Riemann's zeta values, and the inversion formula for the Lambert series are given. Furthermore, the Fourier expansion of the Barnes zeta function and a multiple analogue of Iseki's formula are obtained [\textit{S. Iseki}, Duke Math. J. 24, 653--662 (1957; Zbl 0093.25903)], among which the latter one is new.