ﻻ يوجد ملخص باللغة العربية
We define generalised zeta functions associated to indefinite quadratic forms of signature (g-1,1) -- and more generally, to complex symmetric matrices whose imaginary part has signature (g-1,1) -- and we investigate their properties. These indefinite zeta functions are defined as Mellin transforms of indefinite theta functions in the sense of Zwegers, which are in turn generalised to the Siegel modular setting. We prove an analytic continuation and functional equation for indefinite zeta functions. We also show that indefinite zeta functions in dimension 2 specialise to differences of ray class zeta functions of real quadratic fields, whose leading Taylor coefficients at s=0 are predicted to be logarithms of algebraic units by the Stark conjectures.
We prove an analogue of Kroneckers second limit formula for a continuous family of indefinite zeta functions. Indefinite zeta functions were introduced in the authors previous paper as Mellin transforms of indefinite theta functions, as defined by Zw
We present several formulae for the large $t$ asymptotics of the Riemann zeta function $zeta(s)$, $s=sigma+i t$, $0leq sigma leq 1$, $t>0$, which are valid to all orders. A particular case of these results coincides with the classical results of Sieg
We examine partition zeta functions analogous to the Riemann zeta function but summed over subsets of integer partitions. We prove an explicit formula for a family of partition zeta functions already shown to have nice properties -- those summed over
The Riemann zeta function $zeta(s)$ is defined as the infinite sum $sum_{n=1}^infty n^{-s}$, which converges when ${rm Re},s>1$. The Riemann hypothesis asserts that the nontrivial zeros of $zeta(s)$ lie on the line ${rm Re},s= frac{1}{2}$. Thus, to f
The Riemann zeta identity at even integers of Lettington, along with his other Bernoulli and zeta relations, are generalized. Other corresponding recurrences and determinant relations are illustrated. Another consequence is the application to sums of