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Identified as one of the 7 Millennium Problems, the Riemann zeta hypothesis has successfully evaded mathematicians for over 100 years. Simply stated, Riemann conjectured that all of the nontrivial zeroes of his zeta function have real part equal to 1/2. This thesis attempts to explore the theory behind Riemann’s zeta function by first starting with Euler’s zeta series and building up to Riemann’s function. Along the way we will develop the math required to handle this theory in hopes that by the end the reader will have immersed themselves enough to pursue their own exploration and research into this fascinating subject.

Thesis advisor: Dubi Kelmer / In complex analysis, analytic continuation is a common and important tool to study the properties of complex functions. In this paper, we will introduce and define the global zeta function with an associated function f, where f is a polynomial with n variables with integer coefficients. With the usage of p-adic integration, we can conclude that the global zeta function is analytically continued to all s ∈ C when f is the sum of squares with n variables. / Thesis (BS) — Boston College, 2022. / Submitted to: Boston College. College of Arts and Sciences. / Discipline: Departmental Honors. / Discipline: Mathematics.

Igusa's p-adic zeta function is associated to a polynomial f in several variables over the integers and to a prime p. It is a meromorphic function which encodes for every i the number of solutions M_i of f=0 modulo p^i. The intensive study of Igusa's p-adic zeta function by using an embedded resolution of f led to the introduction of the topological zeta function. This geometric invariant of the zero locus of a polynomial f in several variables over the complex numbers was introduced in the early nineties by Denef and Loeser. It is a rational function which they obtained as a limit of Igusa's p-adic zeta functions and which is defined by using an embedded resolution.<br />I have studied the smallest poles of the topological zeta function and the smallest real parts of the poles of Igusa's p-adic zeta function. For n=2 and n=3, I obtained results by using an embedded resolution of singularities. I discovered that the smallest real part of a pole of Igusa's p-adic zeta function is related with the divisibility of the M_i by powers of p. I obtained a general theorem on the divisibility of the M_i by powers of p, which I used to obtain the optimal lower bound for the real part of a pole of Igusa's p-adic zeta function in arbitrary dimension n. I obtained this lower bound also for the topological zeta function by taking the limit.

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京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第20155号 / 理博第4240号 / 新制||理||1610(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 池田 保, 教授 雪江 明彦, 准教授 伊藤 哲史 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

In this thesis we provide a body of knowledge that concerns Riemann zeta-function and its generalizations in a cohesive manner. In particular, we have studied and mentioned some recent results regarding Hurwitz and Lerch functions, as well as Dirichlet's L-function. We have also investigated some fundamental concepts related to these functions and their universality properties. In addition, we also discuss different formulations and approaches to the proof of the Prime Number Theorem and the Riemann Hypothesis. These two topics constitute the main theme of this thesis. For the Prime Number Theorem, we provide a thorough discussion that compares and contrasts Norbert Wiener's proof with that of Newman's short proof. We have also related them to Hadamard's and de la Vallee Poussin's original proofs written in 1896. As far as the Riemann Hypothesis is concerned, we discuss some recent results related to equivalent formulations of the Riemann Hypothesis as well as the Generalized Riemann Hypothesis.

The spectral zeta function, introduced by Minakshisundaram and Pleijel in [36] and denoted by ζg(s), encodes important spectral information for the Laplacian on Riemannian manifolds. For instance, the important notions of the determinant of the Laplacian and Casimir energy are defined via the spectral zeta function. On homogeneous manifolds, it is known that the spectral zeta function is critical with respect to conformal metric perturbations, (see e.g Richardson ([47]) and Okikiolu ([41])). In this thesis, we compute a second variation formula of ζg(s) on closed homogeneous Riemannian manifolds under conformal metric perturbations. It is well known that the quadratic form corresponding to this second variation is given by a certain pseudodifferential operator that depends meromorphically on s. The symbol of this operator was analysed by Okikiolu in ([42]). We analyse it in more detail on homogeneous spaces, in particular on the spheres Sn. The case n = 3 is treated in great detail. In order to describe the second variation we introduce a certain distributional integral kernel, analyse its meromorphic properties and the pole structure. The Casimir energy defined as the finite part of ζg(-½) on the n-sphere and other points of ζg(s) are used to illustrate our results. The techniques employed are heat kernel asymptotics on Riemannian manifolds, the associated meromorphic continuation of the zeta function, harmonic analysis on spheres, and asymptotic analysis.

This thesis concerns representation growth of finitely generated torsion-free nilpotent groups. This involves counting equivalence classes of irreducible representations and embedding this counting into a zeta function. We call this the representation zeta function. We use a new, constructive method to calculate the representation zeta functions of two families of groups, namely the Heisenberg group over rings of quadratic integers and the maximal class groups. The advantage of this method is that it is able to be used to calculate the p-local representation zeta function for all primes p. The other commonly used method, known as the Kirillov orbit method, is unable to be applied to these exceptional cases. Specifically, we calculate some exceptional p-local representation zeta functions of the maximal class groups for some well behaved exceptional primes. Also, we describe the Kirillov orbit method and use it to calculate various examples of p-local representation zeta functions for almost all primes p.

acase@tulane.edu / Modular-type transformation formulas are the identities that are invariant under the transformation α → 1/α, and they can be represented as F (α) = F (β) where α β = 1. We derive a new transformation formula of the form F (α, z, w) = F (β, z, iw) that is a one-variable generalization of the well-known Ramanujan-Guinand identity of the form F (α, z) = F (β, z) and a two-variable generalization of Koshliakov’s formula of the form F (α) = F (β) where α β = 1. The formula is generated by first finding an integral J that is comprised of an invariance function Z and evaluating the integral to give F (α, z, w) mentioned above. The modified Bessel function K z (x) appearing in Ramanujan-Guinand identity is generalized to a new function, denoted as K z,w (x), that yields a pair of functions reciprocal in the Koshliakov kernel, which in turn yields the invariance function Z and hence the integral J and the new formula. The special function K z,w (x), first defined as the inverse Mellin transform of a product of two gamma functions and two confluent hypergeometric functions, is shown to exhibit a rich theory as evidenced by a number of integral and series representations as well as a differential-difference equation. The second topic of the thesis is 2-adic valuations of integer sequences associated with quadratic polynomials of the form x 2 +a. The sequence {n 2 +a : n ∈ Z} contains numbers divisible by any power of 2 if and only if a is of the form 4 m (8l+7). Applying this result to the sequences derived from the sums of four or fewer squares when one or more of the squares are kept constant leads to interesting results, that also points to an inherent connection with the functions r k (n) that count the number of ways to represent n as sums of k integer squares. Another class of sequences studied is the shifted sequences of the polygonal numbers given by the quadratic formula, for which the most common examples are the triangular numbers and the squares. / 1 / Aashita Kesarwani