Abstract: A recent conjecture of Fyodorov--Hiary--Keating states that the maximum of the absolute value of the Riemann zeta function on a typical bounded interval of the critical line is $\exp\{\log \log T -\frac{3}{4}\log \log \log T+O(1)\}$, for an interval at (large) height $T$. In this paper, we verify the first two terms in the exponential for a model of the zeta function, which is essentially a randomized Euler product. The critical element of the proof is the identification of an approximate tree structure, present also in the actual zeta function, which allows us to relate the maximum to that of a branching random walk.

Abstract: Dirichlet's L-functions are natural extensions of the Riemann zeta function. In this paper we rst give a brief survey of Ap ery-like series for some special values of the zeta function and certain L-functions. Then, we establish two theorems on transfor- mations of certain kinds of congruences. Motivated by the results and based on our computation, we pose 48 new conjectural series (most of which involve harmonic numbers) for such special values and related constants. For example, we conjecture that 1

Abstract: We establish a new formula for the heat kernel on regular trees in terms of classical $$I$$ -Bessel functions. Although the formula is explicit, and a proof is given through direct computation, we also provide a conceptual viewpoint using the horocyclic transform on regular trees. From periodization, we then obtain a heat kernel expression on any regular graph. From spectral theory, one has another expression for the heat kernel as an integral transform of the spectral measure. By equating these two formulas and taking a certain integral transform, we obtain as application several generalized versions of the determinant formula for the Ihara zeta function associated to finite or infinite regular graphs. Our approach to the Ihara zeta function and determinant formula through heat kernel analysis follows a similar methodology which exists for quotients of rank one symmetric spaces.

Abstract: We study the use of the Euler-Maclaurin formula to numerically evaluate the Hurwitz zeta function ?(s, a) for s,a??$s, a \in \mathbb {C}$, along with an arbitrary number of derivatives with respect to s, to arbitrary precision with rigorous error bounds. Techniques that lead to a fast implementation are discussed. We present new record computations of Stieltjes constants, Keiper-Li coefficients and the first nontrivial zero of the Riemann zeta function, obtained using an open source implementation of the algorithms described in this paper.

Abstract: We develop techniques for computing zeta functions associated with nilpotent groups, not necessarily associative algebras, and modules, as well as Igusa-type zeta functions. At the heart of our method lies an explicit convex-geometric formula for a class of $p$-adic integrals under non-degeneracy conditions with respect to associated Newton polytopes. Our techniques prove to be especially useful for the computation of topological zeta functions associated with algebras, resulting in the first systematic investigation of their properties.

Abstract: The purpose of this article is consider $|\zeta'(\sigma + it)/\zeta(\sigma + it)|$ and $|\zeta(\sigma + it)|^{-1}$ when $\sigma$ is close to unity. We prove that $|\zeta'(\sigma + it)/\zeta(\sigma + it)| \leq 87\log t$ and $|\zeta(\sigma + it)|^{-1} \leq 6.9\times 10^{6} \log t$ for $\sigma \geq 1-1/(8 \log t)$ and $t\geq 45$.

Abstract: Stimulated by earlier work by Moll and his coworkers (Amdeberhan et al., Proc. Am. Math. Soc., 139(2):535–545, 2010), we evaluate various basic log Gamma integrals in terms of partial derivatives of Tornheim–Witten zeta functions and their extensions arising from evaluations of Fourier series. In particular, we fully evaluate $$\mathcal{LG}_n=\int_0^1 \log^n\varGamma(x) \,\mathrm{d}x $$ for 1≤n≤4 and make some comments regarding the general case. The subsidiary computational challenges are substantial, interesting and significant in their own right.

Abstract: The finite Dirichlet series of the title are defined by the condition that they vanish at as many initial zeros of the zeta function as possible. It turns out that such series can produce extremely good approximations to the values of Riemann’s zeta function inside the critical strip. In addition, the coefficients of these series have remarkable number-theoretical properties discovered in large-scale high-precision numerical experiments. So far, we have found no theoretical explanation for the observed phenomena.

Abstract: In this paper, we study the distribution of orders of bounded discriminants in number fields. We use the zeta functions introduced by Grunewald, Segal, and Smith. In order to carry out our study, we use p-adic and motivic integration techniques to analyze the zeta function. We give an asymptotic formula for the number of orders contained in the ring of integers of a quintic number field. We also obtain non-trivial bounds for higher degree number fields.

Abstract: In 2009, the first author introduced a class of zeta functions, called `distance zeta functions', which has enabled us to extend the existing theory of zeta functions of fractal strings and sprays (initiated by the first author and his collaborators in the early 1990s) to arbitrary bounded (fractal) sets in Euclidean spaces of any dimensions. A closely related tool is the class of `tube zeta functions', defined using the tube function of a fractal set. These zeta functions exhibit deep connections with Minkowski contents and upper box (or Minkowski) dimensions, as well as, more generally, with the complex dimensions of fractal sets. In particular, the abscissa of (Lebesgue, i.e., absolute) convergence of the distance zeta function coincides with the upper box dimension of a set. We also introduce a class of transcendentally quasiperiodic sets, and describe their construction based on a sequence of carefully chosen generalized Cantor sets with two auxilliary parameters. As a result, we obtain a family of "maximally hyperfractal" compact sets and relative fractal drums (i.e., such that the associated fractal zeta functions have a singularity at every point of the critical line of convergence). Finally, we discuss the general fractal tube formulas and the Minkowski measurability criterion obtained by the authors in the context of relative fractal drums (and, in particular, of bounded subsets of the N-dimensional Euclidean space).

Abstract: We study an elliptic analogue of multiple zeta values, the elliptic multiple zeta values of Enriquez, which are the coefficients of the elliptic KZB associator. Originally defined by iterated integrals on a once-punctured complex elliptic curve, it turns out that they can also be expressed as certain linear combinations of indefinite iterated integrals of Eisenstein series and multiple zeta values. In this paper, we prove that the $\mathbb{Q}$-span of these elliptic multiple zeta values forms a $\mathbb{Q}$-algebra, which is naturally filtered by the length and is conjecturally graded by the weight. Our main result is a proof of a formula for the number of $\mathbb{Q}$-linearly independent elliptic multiple zeta values of lengths one and two for arbitrary weight.

Abstract: In the present paper, we give sufficient conditions for a function f to be in the subclasses $$ \sum S_{a,s}^*(A,\,B,\,\alpha ,\,\beta ) $$ and $$ \sum {\kappa _{a,\,s}}(A,\,B,\,\alpha ,\,\beta ) $$ of the class Σ of meromorphic functions which are analytic in the punctured unit disk U*. We further investigate the ratio of a function related to the Hurwitz-Lerch zeta function and its sequence of partial sums.

Abstract: List of contributors Preface A. Raghuram 1. Special values of the Riemann zeta function: some results and conjectures A. Raghuram 2. K-theoretic background R. Sujatha 3. Values of the Riemann zeta function at the odd positive integers and Iwasawa theory John Coates 4. Explicit reciprocity law of Bloch-Kato and exponential maps Anupam Saikia 5. The norm residue theorem and the Quillen-Lichtenbaum conjecture Manfred Kolster 6. Regulators and zeta functions Stephen Lichtenbaum 7. Soule's theorem Stephen Lichtenbaum 8. Soule's regulator map Ralph Greenberg 9. On the determinantal approach to the Tamagawa number conjecture T. Nguyen Quang Do 10. Motivic polylogarithm and related classes Don Blasius 11. The comparison theorem for the Soule-Deligne classes Annette Huber 12. Eisenstein classes, elliptic Soule elements and the l-adic elliptic polylogarithm Guido Kings 13. Postscript R. Sujatha.

Abstract: Recently, T. Kim considered Euler zeta function which interpolates Euler polynomials at negative integer (see [3]). In this paper, we study degenerate Euler zeta function which is holomorphic function on complex s-plane associated with degenerate Euler polynomials at negative integers.

Abstract: We derive new results about properties of the Witten zeta function associated with the group SU(3), and use them to prove an asymptotic formula for the number ofn-dimensional representations ofSU(3) counted up to equivalence. Our analysis also relates the Witten zeta function of SU(3) to a summation identity for Bernoulli numbers discovered in 2008 by Agoh and Dilcher. We give a new proof of that identity and show that it is a special case of a stronger identity involving the Eisenstein series.

Abstract: In this paper we prove some connections between the growth of a function and its Mellin transform and apply these to study an explicit example in the theory of Beurling primes.

Abstract: In this paper, we study various twisted A-harmonic sums, named following the seminal log-algebraicity papers of G. Anderson. These objects are partial sums of new types of special zeta values introduced by the first author and linked to certain rank one Drinfeld modules over Tate algebras in positive characteristic by Angles, Tavares Ribeiro and the first author. We prove, by using techniques introduced by the second author, that various infinite families of such sums may be interpolated by polynomials, and we deduce, among several other results, properties of analogues of finite zeta values but inside the framework of the Carlitz module. In the theory of finite multi-zeta values in characteristic zero, finite zeta values are all zero. In the Carlitzian setting, there exist non-vanishing finite zeta values, and we study some of their properties in the present paper.

Abstract: We discuss a precise relation between the Veneziano amplitude of string theory, rewritten in terms of ratios of the Riemann zeta function, and two elementary criteria for the Riemann hypothesis formulated in terms of integrals of the logarithm and the argument of the zeta function. We also discuss how the integral criterion based on the argument of the Riemann zeta function relates to the Li criterion for the Riemann hypothesis. We provide a new generalization of this integral criterion. Finally, we comment on the physical interpretation of our recasting of the Riemann hypothesis in terms of the Veneziano amplitude.

Abstract: We describe new zero-free regions for the derivatives $\zetak(s)$ of the Riemann zeta function, which take the form of vertical strips in the right half-plane. We show that the zeros located in the narrow complements of these zero-free regions are simple and exhibit vertical periodicities that enable one to give exact formulas for their number.

Abstract: Earlier the authors offered an equivariant version of the classical monodromy zeta function of a \(G\)-invariant function germ with a finite group \(G\) as a power series with the coefficients from the Burnside ring of the group \(G\) tensored by the field of rational numbers One of the main ingredients of the definition was the definition of the equivariant Lefschetz number of a \(G\)-equivariant transformation given by W Luck and J Rosenberg Here we use another approach to a definition of the equivariant Lefschetz number of a transformation and describe the corresponding notions of the equivariant zeta function This zeta-function is a power series with the coefficients from the Burnside ring itself We give an A’Campo type formula for the equivariant monodromy zeta function of a function germ in terms of a resolution Finally we discuss orbifold versions of the Lefschetz number and of the monodromy zeta function corresponding to the two equivariant ones