Abstract: The representation growth of a T -group is polynomial. We study the rate of polynomial growth and the spectrum of possible growth, showing that any rational number ? can be realized as the rate of polynomial growth of a class 2 nilpotent T -group. This is in stark contrast to the related subject of subgroup growth of T -groups where it has been shown that the set of possible growth rates is discrete in Q. We derive a formula for almost all of the local representation zeta functions of a T2-group with centre of Hirsch length 2. A consequence of this formula shows that the representation zeta function of such a group is finitely uniform. In contrast, we explicitly derive the representation zeta function of a specific T2-group with centre of Hirsch length 3 whose representation zeta function is not finitely uniform. We give formulae, in terms of Igusa's local zeta function, for the subring, left-, right- and two-sided ideal zeta function of a 2-dimensional ring. We use these formulae to compute a number of examples. In particular, we compute the subring zeta function of the ring of ?integers in a quadratic number field.

Abstract: We show that the non-commutative geometric approach to the Riemann zeta function has an algebraic geometric incarnation: the "Arithmetic Site". This site involves the tropical semiring viewed as a sheaf on the topos which is the dual of the multiplicative semigroup of positive integers. We prove that the set of points of the arithmetic site over the maximal compact subring of the tropical semifield is the non-commutative space quotient of the adele class space of Q by the action of the maximal compact subgroup of the idele class group. We realize the Frobenius correspondences in the square of the "Arithmetic Site" and compute their composition. This note provides the algebraic geometric space underlying the non-commutative approach to RH.

Abstract: A simple geometric construction on the moduli spaces $\mathcal{M}_{0,n}$ of curves of genus $0$ with $n$ ordered marked points is described which gives a common framework for many irrationality proofs for zeta values. This construction yields Ap\'ery's approximations to $\zeta(2)$ and $\zeta(3)$, and for larger $n$, an infinite family of small linear forms in multiple zeta values with an interesting algebraic structure. It also contains a generalisation of the linear forms used by Ball and Rivoal to prove that infinitely many odd zeta values are irrational.

Abstract: In the first part we present the number theoretical properties of the Riemann zeta function and formulate the Riemann Hypothesis. In the second part we review some physical problems related to this hypothesis: the links with Random Matrix Theory, relation with the Lee--Yang theorem on the zeros of the partition function, random walks, billiards etc.

Abstract: This chapter provides an overview of the theory of hyperbolic zeta function of lattices. A functional equation for the hyperbolic zeta function of Cartesian lattice is obtained. Information about the history of the theory of the hyperbolic zeta function of lattices is provided. The relations with the hyperbolic zeta function of nets and Korobov optimal coefficients are considered.

Abstract: We define the multi-poly-Bernoulli numbers slightly differently from the similar numbers given in earlier papers by Bayad, Hamahata, and Masubuchi, and study their basic properties. Our motivation for the new definition is the connection to “finite multiple zeta values”, which have been studied by Hoffman and Zhao, among others, and are recast in a recent work by Zagier and the second author. We write the finite multiple zeta value in terms of our new multi-poly-Bernoulli numbers.

Abstract: This is the first part of a study of relations satisfied by finite multiple zeta values and their analogues, by means of the motivic fundamental group of the moduli spaces M_{0,4} and M_{0,5}. We define notions of double shuffle relations and associator relations for finite multiple zeta values. We give new proofs to two results of "depth drop" phenomena of multiple zeta values. We prove an explicit $p$-adic lift of a family of congruences among finite multiple zeta values, which has applications to $p$-adic zeta values.

Abstract: Abstract Our purpose in this paper is to consider a generalized form of the extended Hurwitz-Lerch Zeta function. For this extended Hurwitz-Lerch Zeta function, we obtain some classical properties which includes various integral representations, a differential formula, Mellin transforms and certain generating relations. We further consider an application to probability distributions and also point out some important special cases of the main results.

Abstract: We prove that the $\mathbb{Q}$-vector space generated by the multiple zeta values is generated by the finite real multiple zeta values introduced by Kaneko and Zagier.

Abstract: As usual let s = σ+ it. For any fixed value t = t0 with |t0| ≥ 8, and for σ ≤ 0, we show that |ζ(s)| is strictly monotone decreasing in σ, with the same result also holding for the related functions ξ of Riemann and η of Euler. The following inequality relating the monotonicity of all three functions is proved:

Abstract: We compute explicitly Dirichlet generating functions enumerating finite-dimensional irreducible complex representations of various $p$-adic analytic and adelic profinite groups of type $\mathsf{A}_2$. This has consequences for the representation zeta functions of arithmetic groups $\Gamma \subset \mathbf{H}(k)$, where $k$ is a number field and $\mathbf{H}$ a $k$-form of $\mathsf{SL}_3$: assuming that $\Gamma$ possesses the strong Congruence Subgroup Property, we obtain precise, uniform estimates for the representation growth of $\Gamma$. Our results are based on explicit, uniform formulae for the representation zeta functions of the $p$-adic analytic groups $\mathsf{SL}_3(\mathfrak{o})$ and $\mathsf{SU}_3(\mathfrak{o})$, where $\mathfrak{o}$ is a compact discrete valuation ring of characteristic $0$. These formulae build on our classification of similarity classes of integral $\mathfrak{p}$-adic $3\times3$ matrices in $\mathfrak{gl}_3(\mathfrak{o})$ and $\mathfrak{gu}_3(\mathfrak{o})$, where $\mathfrak{o}$ is a compact discrete valuation ring of arbitrary characteristic. Organising the similarity classes by invariants which we call their shadows allows us to combine the Kirillov orbit method with Clifford theory to obtain explicit formulae for representation zeta functions. In a different direction we introduce and compute certain similarity class zeta functions. Our methods also yield formulae for representation zeta functions of various finite subquotients of groups of the form $\mathsf{SL}_3(\mathfrak{o})$, $\mathsf{SU}_3(\mathfrak{o})$, $\mathsf{GL}_3(\mathfrak{o})$, and $\mathsf{GU}_3(\mathfrak{o})$, arising from the respective congruence filtrations; these formulae are valid in case that the characteristic of $\mathfrak{o}$ is either $0$ or sufficiently large. Analysis of some of these formulae leads us to observe $p$-adic analogues of `Ennola duality'.

Abstract: By using methods in the theory of majorization, a double inequality for the gamma function is extended to the k-gamma function and the k-Riemann zeta function.

Abstract: If dn = lcm (1, 2, . . . , n) then the irrationality of ζ(3) was obtained by showing a) For any nonnegative integer n we have dnJn = An −Bnζ(3) where An, Bn ∈ Z. b) 0 0 and α are constants with 0 2 and integer n> 0 we define a multiple integral over Ik

Abstract: Cohomological Theory of Crystals over Function Fields and Applications.- On Geometric Iwasawa Theory and Special Values of Zeta Functions.- The Ongoing Binomial Revolution.- Arithmetic of Gamma, Zeta and Multizeta Values for Function Fields.- Curves and Jacobians over Function Fields.

Abstract: One of the most famous problems in mathematics is the Riemann hypothesis: that the nontrivial zeros of the Riemann zeta function lie on a line in the complex plane. One way to prove the hypothesis would be to identify the zeros as eigenvalues of a Hermitian operator, many of whose properties can be derived through the analogy to quantum chaos. Using this, we construct a set of quantum graphs that have the same oscillating part of the density of states as the Riemann zeros, offering an explanation of the overall minus sign. The smooth part is completely different, and hence also the spectrum, but the graphs pick out the low-lying zeros.

Abstract: We lift the splicing formula of Nemethi and Veys, which deals with polynomials in two variables, to the motivic level. After defining the motivic zeta function and the monodromic motivic zeta function with respect to a differential form, we prove a splicing formula for them, which specializes to this formula of Nemethi and Veys. We also show that we cannot introduce a monodromic motivic zeta functions in terms of a (splice) diagram since it does not contain all the necessary information. In the last part we discuss the generalized monodromy conjecture of Nemethi and Veys. The statement also holds for motivic zeta functions but it turns out that the analogous statement for monodromic motivic zeta functions is not correct. We show some examples illustrating this.

Abstract: In this paper, we establish a Voronoi formula for the spinor zeta function of a Siegel cusp form of genus 2. We deduce from this formula quantitative results on the number of its positive (respectively, negative) coefficients in some short intervals.

Abstract: In this paper, we present several new expansion formulas for a class of generalized Hurwitz-Lerch zeta functions which were introduced by Raina and Chhajed [R. K. Raina and P. K. Chhajed, “Certain Results Involving a Class of Functions Associated with the Hurwitz Zeta Function,” Acta Math. Univ. Comenian. 73, 89–100 (2004)] and (more recently) by Srivastava et al. [H. M. Srivastava, M.-J. Luo, and R. K. Raina, “New Results Involving a Class of Generalized Hurwitz-Lerch Zeta Functions and Their Applications,” Turkish J. Anal. Number Theory 1, 26–35 (2013)]. These expansion formulas are obtained with the help of some fractional calculus theorems such as the generalized Leibniz rules, the Taylorlike expansions in terms of different functions and the generalized chain rule. Several (known or new) special cases are also considered.

Abstract: Let $K$ be a number field with ring of integers $\mathcal O$. After introducing a suitable notion of density for subsets of $\mathcal O$, generalizing that of natural density for subsets of $\mathbb Z$, we show that the density of the set of coprime $m$-tuples of algebraic integers is ${1/\zeta_K(m)}$, where $\zeta_K$ is the Dedekind zeta function of $K$.

Abstract: We compute the Lefschetz zeta function for quasi-unipotent maps on the n-dimensional torus, using arithmetical properties of the number n. In particular we compute the Lefschetz zeta function for quasi-unipotent maps, such that the characteristic polynomial of the induced map on the first homology group is the th cyclotomic polynomial, when is an odd prime. These computations involve fine combinatorial properties of roots of unity. We also show that the Lefchetz zeta functions for quasi-unipotent maps on are rational functions of total degree zero. We use these results in order to characterized the minimal set of Lefschetz periods for quasi-unipotent maps on , having finitely many periodic points all of them hyperbolic. Among this class of maps are the Morse–Smale diffeomorphisms of .

Abstract: In this paper we establish the joint universality theorems for pairs consisting of the Riemann zeta function, automorphic \(L\)-functions associated with holomorphic Hecke eigen cusp forms, Rankin–Selberg \(L\)-functions, and symmetric square \(L\)-functions.

Abstract: In this paper, we obtain a series of new conditional lower bounds for the modulus and the argument of the Riemann zeta function on very short segments of the critical line, based on the Riemann hypothesis. In particular, the conditional solution of one problem of A.A.Karatsuba is given. Some typos of the previous versions are corrected (in particular, the important remark of Prof. Yan Fyodorov is taken into account). The reference to the grant of Russian Scientific Fund is also added.

Abstract: We unconditionally prove a central limit theorem for linear statistics of the zeros of the Riemann zeta function with diverging variance. Previously, theorems of this sort have been proved under the assumption of the Riemann hypothesis. The result mirrors central limit theorems in random matrix theory that have been proved by Szeg\H{o}, Spohn, and Soshnikov among others, and therefore provides support for the view that the zeros of the zeta function are distributed like the eigenvalues of a random matrix. A key ingredient in our proof is a simple bootstrapping of classical zero density estimates of Selberg and Jutila for the zeta function, which may be of independent interest.