Abstract: Cyclotomic fields have always occupied a central place in number theory, and the so called "main conjecture" on cyclotomic fields is arguably the deepest and most beautiful theorem known about them. It is also the simplest example of a vast array of subsequent, unproven "main conjectures'' in modern arithmetic geometry involving the arithmetic behaviour of motives over p-adic Lie extensions of number fields. These main conjectures are concerned with what one might loosely call the exact formulae of number theory which conjecturally link the special values of zeta and L-functions to purely arithmetic expressions. Written by two leading workers in the field, this short and elegant book presents in full detail the simplest proof of the "main conjecture'' for cyclotomic fields. Its motivation stems not only from the inherent beauty of the subject, but also from the wider arithmetic interest of these questions. The masterly exposition is intended to be accessible to both graduate students and non-experts in Iwasawa theory.

Abstract: The case of t = 0, that is T (s, 0, u), is called the Euler–Zagier double zeta function [10]. The values T (a, b, c) for a, b, c ∈ N were first investigated by Tornheim [7] in 1950 and later Mordell [5] in 1958. Tornheim [7, Theorem 7] showed that T (a, b, c) can be expressed as a polynomial in {ζ(j) | 2 ≤ j ≤ a+ b+ c} with rational coefficients when a + b + c is odd, and that the same is true for T (2r, 2r, 2r) and T (2r− 1, 2r, 2r+1) [7, Theorem 8], but he did not give the coefficients. Mordell [5, Theorem III] proved that T (2r, 2r, 2r) = krπ 6r for some rational number kr. In 1985 Subbarao and Sitaramachandrarao [6, Theorem 4.1] explicitly determined T (2p, 2q, 2r)+T (2q, 2r, 2p)+T (2r, 2p, 2q) (p, q, r ∈ N). Then, by taking p = q = r, they gave an explicit formula for T (2r, 2r, 2r) (r ∈ N) [6, Remark 3.1]. In 1996 Huard, Williams and Zhang [3, Theorems 1–3] determined T (r, 0, N−r) (r ∈ N, N ∈ 2N+1, 1 ≤ r ≤ N−2), T (p, q,N − p − q) (p, q ∈ N ∪ {0}, N ∈ 2N + 1, 1 ≤ p + q ≤ N − 1, 0 ≤ p, q ≤ N − 2) and T (r, r, r) (r ∈ N). In 2002 Tsumura [8, Theorem 1]

Abstract: Random Matrices: from Physics to Number Theory.- Quantum and Arithmetical Chaos.- Notes on L-functions and Random Matrix Theory.- Energy Level Statistics, Lattice Point Problems, and Almost Modular Functions.- Arithmetic Quantum Chaos of Maass Waveforms.- Large N Expansion for Normal and Complex Matrix Ensembles.- Symmetries Arising from Free Probability Theory.- Universality and Randomness for the Graphs and Metric Spaces.- Zeta Functions.- From Physics to Number Theory Via Noncommutative Geometry.- More Zeta Functions for the Riemann Zeros.- Hilbert Spaces of Entire Functions and Dirichlet L-Functions.- Dynamical Zeta Functions and Closed Orbits for Geodesic and Hyperbolic Flows.- Dynamical Systems: Interval Exchange, Flat Surfaces, and Small Divisors.- Continued Fraction Algorithms for Interval Exchange Maps: an Introduction.- Flat Surfaces.- Brjuno Numbers and Dynamical Systems.- Some Properties of Real and Complex Brjuno Functions.

Abstract: We generalize the Ihara-Selberg zeta function to hypergraphs in a natural way. Hashimoto's factorization results for biregular bipartite graphs apply, leading to exact factorizations. For $(d,r)$-regular hypergraphs, we show that a modified Riemann hypothesis is true if and only if the hypergraph is Ramanujan in the sense of Winnie Li and Patrick Sole. Finally, we give an example to show how the generalized zeta function can be applied to graphs to distinguish non-isomorphic graphs with the same Ihara-Selberg zeta function.

Abstract: We give an anecdotal discussion of the problem of searching for polynomials with all roots on the unit circle, whose coefficients are rational numbers subject to certain congruence conditions. We illustrate with an example from a calculation in p-adic cohomology made by Abbott, Kedlaya, and Roe, in which (using an implementation developed with Andre Wibisono) we recover the zeta function of a surface over a finite field.

Abstract: We investigate the multiple Hurwitz zeta function $\zeta_n(s_1, \ldots, s_n;a)$, in particular those values at non-positive integers. Then, as an application, we give a generalization of Lerch's formula.

Abstract: We prove that the theory of the $p$-adics ${\mathbb Q}_p$ admits elimination of imaginaries provided we add a sort for ${\rm GL}_n({\mathbb Q}_p)/{\rm GL}_n({\mathbb Z}_p)$ for each $n$. We also prove that the elimination of imaginaries is uniform in $p$. Using $p$-adic and motivic integration, we deduce the uniform rationality of certain formal zeta functions arising from definable equivalence relations. This also yields analogous results for definable equivalence relations over local fields of positive characteristic. The appendix contains an alternative proof, using cell decomposition, of the rationality (for fixed $p$) of these formal zeta functions that extends to the subanalytic context. As an application, we prove rationality and uniformity results for zeta functions obtained by counting twist isomorphism classes of irreducible representations of finitely generated nilpotent groups; these are analogous to similar results of Grunewald, Segal and Smith and of du Sautoy and Grunewald for subgroup zeta functions of finitely generated nilpotent groups.

Abstract: We study multiple zeta values and their generalizations from the point of view of Rota--Baxter algebras. We obtain a general framework for this purpose and derive relations on multiple zeta values from relations in Rota--Baxter algebras.

Abstract: It is proved that the Riemann zeta function does not satisfy any nontrivial algebraic difference equation whose coefficients are meromorphic functions $\phi$ with Nevanlinna characteristic satisfying $T(r, \phi)=o(r)$ as $r\to \infty$

Abstract: Theory of zeta functions plays a central role in arithmetic. In this paper, we use a new approach to study them. More precisely, we first reveal an intrinsic relation between higher rank zeta functions and Epstein zeta functions, and expose a fundamental relation between stability of lattices and distances of the corresponding modular points to cusps. Applying to rank two, we then explicitly express the associated zeta functions in terms of Dedekind zetas. Based on such an expression, finally, we show that all zeros of rank two zetas are entirely sitting on the critical line whose real part equals to 1 2 . As such, this work is built up on classics of number theory. Many fine pieces of algebraic and analytic number theory are beautifully unified under our zetas:

Abstract: We explore three seemingly disparate but related avenues of inquiry: expanding what is known about the properties of the poles of the Ihara zeta function, determining what information about a graph is recoverable from its Ihara zeta function, and strengthening the ties between the Ihara zeta functions of graphs which are related to each other through common operations on graphs. Using the singular value decomposition of directed edge matrices, we give an alternate proof of the bounds on the poles of Ihara zeta functions. We then give an explicit formula for the inverse of directed edge matrices and use the inverse to demonstrate that the sum of the poles of an Ihara zeta function is zero. Next we discuss the information about a graph recoverable from its Ihara zeta function and prove that the girth of a graph as well as the number of cycles whose length is the girth can be read directly off of the reciprocal of the Ihara zeta function. We demonstrate that a graph's chromatic polynomial cannot in general be recovered from its Ihara zeta function and describe a method for constructing families of graphs which have the same chromatic polynomial but different Ihara zeta functions. We also show that a graph's Ihara zeta function cannot in general be recovered from its chromatic polynomial. Then we make the deletion of an edge from a graph less jarring (from the perspective of Ihara zeta functions) by viewing it as the limit as k goes to infinity of the operation of replacing the edge in the original graph we wish to delete with a walk of length k. We are able to prove that the limit of the Ihara zeta functions of the resulting graphs is in fact the Ihara zeta function of the original with the edge deleted. We also improve upon the bounds on the poles of the Ihara zeta function by considering digraphs whose adjacency matrices are directed edge matrices

Abstract: Summary This paper shows that a normalization of the Hurwitz zeta function is a characteristic function. This generalizes the 1938 result of Khinchine about the Riemann zeta function. The paper investigates the infinite divisibility of the resulting distribution.

Abstract: Let K be a p-adic field. We explore Igusa's p-adic zeta function, which is associated to a K-analytic function on an open and compact subset of K n . First we deduce a formula for an important coefficient in the Laurent series of this meromorphic function at a candidate pole. Afterwards we use this formula to determine all values less than −1/2 for n=2 and less than −1 for n=3 which occur as the real part of a pole.

Abstract: We define a sequence of real functions which coincide with Li's coefficients at one and which allow us to extend Li's criterion for the Riemann Hypothesis to yield a necessary and sufficient condition for the existence of zero-free strips inside the critical strip. We study some of the properties of these functions, including their oscillatory behaviour.

Abstract: In recent years Lichtenbaum has conjectured a description for the special values of Hasse--Weil zeta functions in terms of ``Weil-\'etale cohomology'' In earlier papers we studied a class of foliated dynamical systems which had some similarities with arithmetic schemes Assuming that certain zeta regularized determinants exist we now prove an analogue of Lichtenbaum's conjectures for a particularly simple class of such dynamical systems: Using results of \'Alvarez L\'opez and Kordyukov we express the leading coefficient $\zeta^*_R (0)$ of the Ruelle zeta function $\zeta_R (s)$ at $s = 0$ in terms of analytic torsion We then apply the Cheeger--M\"uller theorem to replace the analytic torsion by the Reidemeister torsion with respect to harmonic bases For our dynamical systems the latter can be expressed by the same recipe as the one in Lichtenbaum's conjectures

Abstract: A weak version of the Ihara formula is proved for zeta functions attached to quotients of the Bruhat–Tits building of PGL3. This formula expresses the zeta function in terms of Hecke-operators. It is the first step towards an arithmetical interpretation of the combinatorially defined zeta function.

Abstract: It is the purpose of this work to pursue a novel physical interpretation of the nontrivial Riemann zeta zeros and prove why the location of these zeros zn = 1/2+iλn corresponds physically to tachyonic-resonances/tachyonic-condensates, originating from the scattering of two on-shell tachyons in bosonic string theory. Namely, we prove that if there were nontrivial zeta zeros (violating the Riemann hypothesis) outside the critical line Realz = 1/2 (but inside the critical strip), these putative zeros do not correspond to any poles of the bosonic open string scattering (Veneziano) amplitude A(s, t, u). The physical relevance of tachyonic-resonances/tachyonic-condensates in bosonic string theory, establishes an important connection between string theory and the Riemann Hypothesis. In addition, one has also a geometrical interpretation of the zeta zeros in the critical line in terms of very special (degenerate) triangular configurations in the upper-part of the complex plane.

Abstract: already considered in [8, Sect. 4 ex. (A)], [4]. Other previous results appear in [11, 15] for the functions Z(σ, 1 4 ), in [5, 18] for the family { } and earlier [16, 12, 10] for the specific sums (n) ≡ ∑ ρ ρ −n (often denoted σn). In [20], we mainly strived at exhausting explicit results for the family (1), handling the family (2) in lesser detail. The present work will in turn provide a thoroughly explicit description for the family (2), in parallel to (1), but now based on a parametric analytical-continuation formula, (42). At the same

Abstract: We consider a topological dynamical system $(\Sigma , \sigma ) on the real line which is topologically conjugate to a topologically mixing one-sided subshift $(\Sigma_{A}^{+},\sigma_A )$ of finite type with structure matrix $A$. Moreover, we assume that it satisfies some additional conditions so that the map $\sigma$ can have an expanding extension with nice properties in a neighborhood of $\Sigma$. Let $u$ be an eventually positive Lipschitz continuous function on $\Sigma$ and $\zeta_u (s)$ the corresponding dynamical zeta function. We show that $\zeta_u (s)$ has a meromorphic extension in the half-plane $\text{Re }s > -\beta_u$ for some $\beta_u >0$ and its special value at the origin is given by $\zeta_u (0) =1/\text{det}(I-A)$. As an application, we can see that the zeta function $\zeta_Q (s)$ for the two-dimensional dispersing billiard table $Q$ without eclipse has a meromorphic extension in the half-plane $\text{Re }s > -\beta$ for some $\beta>0$ and $\zeta_Q (0)=-1/(J-2)2^{J-1}$ holds, where $J$ is the number of scatterers.

Abstract: Notions of integration of motivic type over the space of arcs factorized by the natural C*-action and over the space of nonparametrized arcs (branches) are developed. As an application, two motivic versions of the zeta function of the classical monodromy transformation of a germ of an analytic function on ℂd are given that correspond to these notions. Another key ingredient in the construction of these motivic versions of the zeta function is the use of the so-called power structure over the Grothendieck ring of varieties introduced by the authors.

Abstract: We present analytic properties and extensions of the constants ck appearing in the Baez-Duarte criterion for the Riemann hypothesis. These constants are the coefficients of Pochhammer polynomials in a series representation of the reciprocal of the Riemann zeta function. We present generalizations of this representation to the Hurwitz zeta and many other special functions. We relate the corresponding coefficients to other known constants including the Stieltjes constants and present summatory relations. In addition, we generalize the Maslanka hypergeometric-like representation for the zeta function in several ways.

Abstract: The Plancherel formula and the inversion formula for Weyl transforms on compact and Hausdorff groups are given. A formula expressing the relationships of the wavelet constant, the degree of the irreducible and unitary representation and the volume of an arbitrary compact and Hausdorff group is derived. The role of the Weyl transforms in the derivation of the formulas for the heat kernels of Laplacians on compact Lie groups is explicated. The Green functions and the Riemann zeta functions of Laplacians on compact Lie groups are constructed using the corresponding heat kernels.

Abstract: The original criteria of Riesz and of Hardy-Littlewood concerning the truth of the Riemann Hypothesis (RH) are revisited and further investigated in light of the recent formulations and results of Maslanka and of Baez-Duarte concerning a representation of the Riemann Zeta function. Then we introduce a general set of similar functions with the emergence of Poisson-like distributions and we present some numerical experiments which indicate that the RH may barely be true.

Abstract: We discuss whether finiteness properties of a profinite group G can be deduced from the probabilistic zeta function PG(s). In particular we prove that in the prosoluble case, if PG(s) is rational then G/ Frat(G) is finite.

Abstract: We define the weighted Bartholdi zeta function of a graph G, and give a determinant expression of it. Furthermore, we define a weighted L-function of G, and present a determinant expression for the weighted L-function of G. As a corollary, we show that the weighted Bartholdi zeta function of a regular covering of G is a product of weighted L-functions of G.