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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 find these zeros it is necessary to perform an analytic continuation to a region of complex $s$ for which the defining sum does not converge. This analytic continuation is ordinarily performed by using a functional equation. In this paper it is argued that one can investigate some properties of the Riemann zeta function in the region ${rm Re},s<1$ by allowing operator-valued zeta functions to act on test functions. As an illustration, it is shown that the locations of the trivial zeros can be determined purely from a Fourier series, without relying on an explicit analytic continuation of the functional equation satisfied by $zeta(s)$.
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
In this Ph.D. dissertation (2018, Emory University) we prove theorems at the intersection of the additive and multiplicative branches of number theory, bringing together ideas from partition theory, $q$-series, algebra, modular forms and analytic num
We introduce several notions of random positive operator valued measures (POVMs), and we prove that some of them are equivalent. We then study statistical properties of the effect operators for the canonical examples, obtaining limiting eigenvalue di
The principal aim in this paper is to employ a recently developed unified approach to the computation of traces of resolvents and $zeta$-functions to efficiently compute values of spectral $zeta$-functions at positive integers associated to regular (
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 indefinit