Abstract: In order to make progress in number theory, it is sometimes necessary to use techniques from other areas of mathematics, such as algebra, analysis or geometry. In this chapter we give some number-theoretic applications of the theory of infinite series. These are based on the properties of the Riemann zeta function ς(s), which provides a link between number theory and real and complex analysis. Some of the results we obtain have probabilistic interpretations in terms of random integers. For the background on convergence of infinite series, see Appendix C. For a detailed treatment of ς(s), see Titchmarsh (1951).

Abstract: We investigate relations between elliptic multiple zeta values and describe a method to derive the number of indecomposable elements of given weight and length. Our method is based on representing elliptic multiple zeta values as iterated integrals over Eisenstein series and exploiting the connection with a special derivation algebra. Its commutator relations give rise to constraints on the iterated integrals over Eisenstein series relevant for elliptic multiple zeta values and thereby allow to count the indecomposable representatives. Conversely, the above connection suggests apparently new relations in the derivation algebra. Under https://tools.aei.mpg.de/emzv we provide relations for elliptic multiple zeta values over a wide range of weights and lengths. ar X iv :1 50 7. 02 25 4v 1 [ he pth ] 8 J ul 2 01 5

Abstract: This work lies at an intersection of three subjects: quantum field theory, algebraic geometry and number theory, in a situation where dialogue between practitioners has revealed rich structure. It contains a theorem and 7 conjectures, tested deeply by 3 optimized algorithms, on relations between Feynman integrals and L-series defined by products, over the primes, of data determined by moments of Kloosterman sums in finite fields. There is an extended introduction, for readers who may not be familiar with all three of these subjects. Notable new results include conjectural evaluations of non-critical L-series of modular forms of weights 3, 4 and 6, by determinants of Feynman integrals, an evaluation for the weight 5 problem, at a critical integer, and formulas for determinants of arbitrary size, tested up to 30 loops. It is shown that the functional equation for Kloosterman moments determines much but not all of the structure of the L-series. In particular, for problems with odd numbers of Bessel functions, it misses a crucial feature captured in this work by novel and intensively tested conjectures. For the 9-Bessel problem, these lead to an astounding compression of data at the primes.

Abstract: In the present paper, we show that under the Riemann hypothesis, and for fixed $h, \epsilon > 0$, the supremum of the real and the imaginary parts of $\log \zeta (1/2 + it)$ for $t \in [UT -h, UT + h]$ are in the interval $[(1-\epsilon) \log \log T, (1+ \epsilon) \log \log T]$ with probability tending to $1$ when $T$ goes to infinity, if $U$ is uniformly distributed in $[0,1]$. This proves a weak version of a conjecture by Fyodorov, Hiary and Keating, which has recently been intensively studied in the setting of random matrices. We also unconditionally show that the supremum of $\Re \log \zeta(1/2 + it)$ is at most $\log \log T + g(T)$ with probability tending to $1$, $g$ being any function tending to infinity at infinity.

Abstract: We prove that if $\omega$ is uniformly distributed on $[0,1]$, then as $T\to\infty$, $t\mapsto \zeta(i\omega T+it+1/2)$ converges to a non-trivial random generalized function, which in turn is identified as a product of a very well behaved random smooth function and a random generalized function known as a complex Gaussian multiplicative chaos distribution. This demonstrates a novel rigorous connection between number theory and the theory of multiplicative chaos -- the latter is known to be connected to many other areas of mathematics. We also investigate the statistical behavior of the zeta function on the mesoscopic scale. We prove that if we let $\delta_T$ approach zero slowly enough as $T\to\infty$, then $t\mapsto \zeta(1/2+i\delta_T t+i\omega T)$ is asymptotically a product of a divergent scalar quantity suggested by Selberg's central limit theorem and a strictly Gaussian multiplicative chaos. We also prove a similar result for the characteristic polynomial of a Haar distributed random unitary matrix, where the scalar quantity is slightly different but the multiplicative chaos part is identical. This essentially says that up to scalar multiples, the zeta function and the characteristic polynomial of a Haar distributed random unitary matrix have an identical distribution on the mesoscopic scale.

Abstract: We prove the leading order of a conjecture by Fyodorov, Hiary and Keating, about the maximum of the Riemann zeta function on random intervals along the critical line. More precisely, as $T \rightarrow \infty$ for a set of $t \in [T, 2T]$ of measure $(1 - o(1)) T$, we have $$ \max_{|t-u|\leq 1}\log\left|\zeta\left(\tfrac{1}{2}+i u\right)\right|=(1 + o(1))\log\log T . $$

Abstract: These notes were written for the mini-course Extrema of log-correlated ran- dom variables: Principles and Examples at the Introductory School held in January 2015 at the Centre International de Rencontres Math ematiques in Marseille. There have been many advances in the understanding of the high values of log-correlated ran- dom elds from the physics and mathematics perspectives in recent years. These elds admit correlations that decay approximately like the logarithm of the inverse of the distance between index points. Examples include branching random walks and the two- dimensional Gaussian free eld. In this paper, we review the properties of such elds and survey the progress in describing the statistics of their extremes. The branching random walk is used as a guiding example to prove the correct leading and subleading order of the maximum following the multiscale renement of the second moment method of Kistler. The approach sheds light on a conjecture of Fyodorov, Hiary & Keating on the maximum of the Riemann zeta function on an interval of the critical line and of the characteristic polynomial of random unitary matrices.

Abstract: We show that the Bernstein–Sato polynomial (that is, the b-function) of a hyperplane arrangement with a reduced equation is calculable by combining a generalization of Malgrange’s formula with the theory of Aomoto complexes due to Esnault, Schechtman, Terao, Varchenko, and Viehweg in certain cases. We prove in general that the roots are greater than $$-2$$ and the multiplicity of the root $$-1$$ is equal to the (effective) dimension of the ambient space. We also give an estimate of the multiplicities of the roots in terms of the multiplicities of the arrangement at the dense edges, and provide a method to calculate the Bernstein–Sato polynomial at least in the case of 3 variables with degree at most 7 and generic multiplicities at most 3. Using our argument, we can terminate the proof of a conjecture of Denef and Loeser on the relation between the topological zeta function and the Bernstein–Sato polynomial of a reduced hyperplane arrangement in the 3 variable case.

Abstract: In the paper, we develop an approach to evaluation of Euler sums that involve harmonic numbers and alternating harmonic numbers. We give explicit formulae for several classes of Euler sums in terms of Riemann zeta values and prove that the quadratic sums ${S_{{l^2},l}}$ and cubic sums ${S_{{l^3},l}}$ reduce to linear sums and polynomials in zeta values. The approach is based on constructive Power series and Cauchy product computations.

Abstract: We provide several properties of the geometric polynomials discussed in earlier works of the authors. Further, the geometric polynomials are used to obtain a closed form evaluation of certain series involving Riemann’s zeta function.

Abstract: We prove that there is a prime between $n^3$ and $(n+1)^3$ for all $n \geq \exp(\exp(33.3))$. This is done by first deriving the Riemann--von Mangoldt explicit formula for the Riemann zeta-function with explicit bounds on the error term. We use this along with other recent explicit estimates regarding the zeroes of the Riemann zeta-function to obtain the result. Furthermore, we show that there is a prime between any two consecutive $m$th powers for $m \geq 5 \times 10^9$. Notably, many of the explicit estimates developed in this paper can also find utility elsewhere in the theory of numbers.

Abstract: We prove the equality of the analytic torsion and the value at zero of a Ruelle dynamical zeta function associated with an acyclic unitarily flat vector bundle on a closed locally symmetric reductive manifold. This solves a conjecture of Fried. This article should be read in conjunction with an earlier paper by Moscovici and Stanton.

Abstract: Cotangent sums are associated to the zeros of the Estermann zeta function. They have also proven to be of importance in the Nyman–Beurling criterion for the Riemann Hypothesis. The main result of the paper is the proof of the existence of a unique positive measure μ on ℝ, with respect to which certain normalized cotangent sums are equidistributed. Improvements as well as further generalizations of asymptotic formulas regarding the relevant cotangent sums are obtained. We also prove an asymptotic formula for a more general cotangent sum as well as asymptotic results for the moments of the cotangent sums under consideration. We also give an estimate for the rate of growth of the moments of order 2k, as a function of k.

Abstract: We exploit transformations relating generalized q-series, infinite products, sums over integer partitions, and continued fractions, to find partition-theoretic formulas to compute the values of constants such as π, and to connect sums over partitions to the Riemann zeta function, multiple zeta values, and other number-theoretic objects.

Abstract: We present a new "integral=series" type identity of multiple zeta values, and show that this is equivalent in a suitable sense to the fundamental theorem of regularization. We conjecture that this identity is enough to describe all linear relations of multiple zeta values over Q. We also establish the regularization theorem for multiple zeta-star values, which too is equivalent to our new identity. A connection to Kawashima's relation is discussed as well.

Abstract: In this paper, we obtain some new discrete universality theorems on the approximation of analytic functions by shifts of the Riemann zeta-function. The novelty in formulation is that it involves shifts not by an arithmetical progression as before but by a more general sequence that is uniformly distributed modulo 1.

Abstract: The zeta function of a K3 surface over a finite field satisfies a number of obvious (archimedean and l -adic) and a number of less obvious ( p -adic) constraints. We consider the converse question, in the style of Honda–Tate: given a function Z satisfying all these constraints, does there exist a K3 surface whose zeta-function equals Z ? Assuming semistable reduction, we show that the answer is yes if we allow a finite extension of the finite field. An important ingredient in the proof is the construction of complex projective K3 surfaces with complex multiplication by a given CM field.

Abstract: The main goal of this paper is the presentation of an elementary analytic technique which enables the evaluation of the so-called restricted sum formulas involving multiple zeta values with even arguments, i.e. $$E(2c,K):=\sum_{\substack{\sum_{j=1}^{K}c_{j}=c\\{c}_{j}\in\mathbb{N}}} \zeta(2c_1,\ldots ,2c_K),$$ where c and K are arbitrary positive integers with $${c\ge K}$$ . Though the young and general theory of the multiple Riemann zeta function with a rich application potential may be rather complicated, our contribution makes the evaluation of the term E(2c,K) intelligible to a broad mathematical audience.

Abstract: We consider the Tate cohomology of the circle group acting on the topological Hochschild homology of schemes. We show that in the case of a scheme smooth and proper over a finite field, this cohomology theory naturally gives rise to the cohomological interpretation of the Hasse-Weil zeta function by regularized determinants envisioned by Deninger. In this case, the periodicity of the zeta function is reflected by the periodicity of said cohomology theory, whereas neither is periodic in general.

Abstract: We propose a definition of periodic topological cyclic homology and show that, for schemes smooth and proper over a finite field, the infinite dimensional cohomology theory that results provides a natural vessel for Deninger's cohomological interpretation of the Hasse-Weil zeta function by regularized determinants. In this way, the theory may be seen as a non-archimedean analogue of the cohomological interpretation of the zeta function in the archimedean case in terms of cyclic homology given recently by Connes and Consani.

Abstract: Given a sextic CM field $K$ , we give an explicit method for finding all genus- $3$ hyperelliptic curves defined over $\mathbb{C}$ whose Jacobians are simple and have complex multiplication by the maximal order of this field, via an approximation of their Rosenhain invariants. Building on the work of Weng [ J. Ramanujan Math. Soc. 16 (2001) no. 4, 339–372], we give an algorithm which works in complete generality, for any CM sextic field $K$ , and computes minimal polynomials of the Rosenhain invariants for any period matrix of the Jacobian. This algorithm can be used to generate genus-3 hyperelliptic curves over a finite field $\mathbb{F}_{p}$ with a given zeta function by finding roots of the Rosenhain minimal polynomials modulo $p$ .

Abstract: Given an abstract simplicial complex G, the connection graph G' of G has as vertex set the faces of the complex and connects two if they intersect. If A is the adjacency matrix of that connection graph, we prove that the Fredholm characteristic det(1+A) takes values in {-1,1} and is equal to the Fermi characteristic, which is the product of the w(x), where w(x)=(-1)^dim(x). The Fredholm characteristic is a special value of the Bowen-Lanford zeta function and has various combinatorial interpretations. The unimodularity theorem proven here shows that it is a cousin of the Euler characteristic as the later is the sum of the w(x). Unimodularity implies that the matrix 1+A has an inverse which takes integer values. Experiments suggest the conjecture that the range of the Green function values, the union of the entries of the inverse of 1+A form a combinatorial invariant of the simplicial complex and do not change under Barycentric or edge refinements.