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Register a new userEulerian series, zeta functions and the arithmetic of partitions
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Robert Schneider
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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 number theory. We present a natural multiplicative theory of integer partitions (which are usually considered in terms of addition), and explore new classes of partition-theoretic zeta functions and Dirichlet series -- as well as Eulerian $q$-hypergeometric series -- enjoying many interesting relations. We find a number of theorems of classical number theory and analysis arise as particular cases of extremely general combinatorial structure laws. Among our applications, we prove explicit formulas for the coefficients of the $q$-bracket of Bloch-Okounkov, a partition-theoretic operator from statistical physics related to quasi-modular forms; we prove partition formulas for arithmetic densities of certain subsets of the integers, giving $q$-series formulas to evaluate the Riemann zeta function; we study $q$-hypergeometric series related to quantum modular forms and the strange function of Kontsevich; and we show how Ramanujans odd-order mock theta functions (and, more generally, the universal mock theta function $g_3$ of Gordon-McIntosh) arise from the reciprocal of the Jacobi triple product via the $q$-bracket operator, connecting also to unimodal sequences in combinatorics and quantum modular-like phenomena.
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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 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 study a Dirichlet series in two variables which counts primitive three-term arithmetic progressions of squares. We show that this multiple Dirichlet series has meromorphic continuation to $mathbb{C}^2$ and use Tauberian methods to obtain counts for arithmetic progressions of squares and rational points on $x^2+y^2=2$.
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 partitions of fixed length -- which yields complete information about analytic continuation, poles and trivial roots of the zeta functions in the family. Then we present a combinatorial proof of the explicit formula, which shows it to be a zeta function analog of MacMahons partial fraction decomposition of the generating function for partitions of fixed length.
Let $K$ be a local field and $f(x)in K[x]$ be a non-constant polynomial. The local zeta function $Z_f(s, chi)$ was first introduced by Weil, then studied in detail by Igusa. When ${rm char}(K)=0$, Igusa proved that $Z_f(s, chi)$ is a rational function of $q^{-s}$ by using the resolution of singularities. Later on, Denef gave another proof of this remarkable result. However, if ${rm char}(K)>0$, the question of rationality of $Z_f(s, chi)$ is still kept open. Actually, there are only a few known results so far. In this paper, we investigate the local zeta functions of two-variable polynomial $g(x, y)$, where $g(x, y)=0$ is the superelliptic curve with coefficients in a non-archimedean local field of positive characteristic. By using the notable Igusas stationary phase formula and with the help of some results due to Denef and Z${rm acute{u}}$${rmtilde{n}}$iga-Galindo, and developing a detailed analysis, we prove the rationality of these local zeta functions and also describe explicitly all their candidate poles.
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