Showing papers on "Riemann zeta function published in 2017"

Hamiltonian for the Zeros of the Riemann Zeta Function

Abstract: A Hamiltonian operator H[over ^] is constructed with the property that if the eigenfunctions obey a suitable boundary condition, then the associated eigenvalues correspond to the nontrivial zeros of the Riemann zeta function. The classical limit of H[over ^] is 2xp, which is consistent with the Berry-Keating conjecture. While H[over ^] is not Hermitian in the conventional sense, iH[over ^] is PT symmetric with a broken PT symmetry, thus allowing for the possibility that all eigenvalues of H[over ^] are real. A heuristic analysis is presented for the construction of the metric operator to define an inner-product space, on which the Hamiltonian is Hermitian. If the analysis presented here can be made rigorous to show that H[over ^] is manifestly self-adjoint, then this implies that the Riemann hypothesis holds true.

p -Adic AdS/CFT

Abstract: We construct a p-adic analog to AdS/CFT, where an unramified extension of the p-adic numbers replaces Euclidean space as the boundary and a version of the Bruhat–Tits tree replaces the bulk. Correlation functions are computed in the simple case of a single massive scalar in the bulk, with results that are strikingly similar to ordinary holographic correlation functions when expressed in terms of local zeta functions. We give some brief discussion of the geometry of p-adic chordal distance and of Wilson loops. Our presentation includes an introduction to p-adic numbers.

Mean square of zeta function, circle problem and divisor problem revisited

Abstract: This paper is closely related to the recent work [BW17] of the same authors and our purpose is to elaborate more on some of the results and methods from [BW17]. More specifically our goal is two-fold. Firstly, we will indicate how a simple variant related to Section 4 in [BW17] leads to the following improvements of Theorem 3 in [BW17]

Large greatest common divisor sums and extreme values of the Riemann zeta function

Abstract: It is shown that the maximum of |ζ(1/2+it)| on the interval T1/2≤t≤T is at least exp((1/2+o(1))logTlogloglogT/loglogT). Our proof uses Soundararajan’s resonance method and a certain large greatest common divisor sum. The method of proof shows that the absolute constant A in the inequality sup 1≤n1<⋯

Maxima of a randomized Riemann zeta function, and branching random walks

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.

The semiclassical zeta function for geodesic flows on negatively curved manifolds

Abstract: We consider the semi-classical (or Gutzwiller–Voros) zeta functions for $$C^\infty $$ contact Anosov flows. Analyzing the spectra of the generators of some transfer operators associated to the flow, we prove that, for arbitrarily small $$\tau >0$$ , its zeros are contained in the union of the $$\tau $$ -neighborhood of the imaginary axis, $$|\mathfrak {R}(s)|<\tau $$ , and the half-plane $$\mathfrak {R}(s)<-\chi _0+\tau $$ , up to finitely many exceptions, where $$\chi _0>0$$ is the hyperbolicity exponent of the flow. Further we show that the density of the zeros along the imaginary axis satisfy an analogue of the Weyl law.

The circular unitary ensemble and the Riemann zeta function: the microscopic landscape and a new approach to ratios

Abstract: We show in this paper that after proper scalings, the characteristic polynomial of a random unitary matrix converges to a random analytic function whose zeros, which are on the real line, form a determinantal point process with sine kernel. Our scaling is performed at the so-called “microscopic” level, that is we consider the characteristic polynomial at points whose distance to 1 has order 1 / n. We prove that the rescaled characteristic polynomial does not even have a moment of order one, hence making the classical techniques of random matrix theory difficult to apply. In order to deal with this issue, we couple all the dimensions n on a single probability space, in such a way that almost sure convergence occurs when n goes to infinity. The strong convergence results in this setup provide us with a new approach to ratios: we are able to solve open problems about the limiting distribution of ratios of characteristic polynomials evaluated at points of the form $$\exp (2 i \pi \alpha /n)$$ and related objects (such as the logarithmic derivative). We also explicitly describe the dependence relation for the logarithm of the characteristic polynomial evaluated at several points on the microscopic scale. On the number theory side, inspired by the work by Keating and Snaith, we conjecture some new limit theorems for the value distribution of the Riemann zeta function on the critical line at the level of stochastic processes.

The mean square of the product of the Riemann zeta-function with Dirichlet polynomials

Abstract: Improving earlier work of Balasubramanian, Conrey and Heath-Brown [BCHB85], we obtain an asymptotic formula for the mean-square of the Riemann zetafunction times an arbitrary Dirichlet polynomial of length T , with δ = 0.01515 . . .. As an application we obtain an upper bound of the correct order of magnitude for the third moment of the Riemann zeta-function. We also refine previous work of Deshouillers and Iwaniec [DI84], obtaining asymptotic estimates in place of bounds. Using the work of Watt [Wat95], we compute the mean-square of the Riemann zetafunction times a Dirichlet polynomial of length going up to T 3/4 provided that the Dirichlet polynomial assumes a special shape. Finally, we exhibit a conjectural estimate for trilinear sums of Kloosterman fractions which implies the Lindelöf Hypothesis.

On multiple zeta values of even arguments

Abstract: For k ≤ n, let E(2n,k) be the sum of all multiple zeta values with even arguments whose weight is 2n and whose depth is k. Of course E(2n, 1) is the value ζ(2n) of the Riemann zeta function at 2n, and it is well known that E(2n, 2) = 3 4ζ(2n). Recently Shen and Cai gave formulas for E(2n, 3) and E(2n, 4) in terms of ζ(2n) and ζ(2)ζ(2n − 2). We give two formulas for E(2n,k), both valid for arbitrary k ≤ n, one of which generalizes the Shen–Cai results; by comparing the two we obtain a Bernoulli-number identity. We also give explicit generating functions for the numbers E(2n,k) and for the analogous numbers E⋆(2n,k) defined using multiple zeta-star values of even arguments.

Fractal Zeta Functions and Fractal Drums: Higher-Dimensional Theory of Complex Dimensions

Abstract: Recently, the first author has extended the definition of the zeta function associated with fractal strings to arbitrary bounded subsets $A$ of the $N$-dimensional Euclidean space ${; ; \mathbb R}; ; ^N$, where $N$ is any integer $\geq 1$. It is defined by $$\zeta_A(s)=int_{; ; A_{; ; \delta}; ; }; ; d(x, A)^{; ; s- N}; ; \D x, $$ where $d(x, A)$\label{; ; d(x, A)}; ; denotes the distance from $x$ to $A$ and $A_{; ; \delta}; ; $ is the $\delta$-neighborhood of~$A$. In this monograph, we investigate various properties of this ``distance zeta function". In particular, we prove that the zeta function is holomorphic in the half-plane $\{; ; {; ; \mathop{; ; \mathrm {; ; Re}; ; }; ; }; ; \, s>\overline\dim_BA\}; ; $, and that the bound $\overline\dim_BA$ is optimal. In other words, the abscissa of convergence of $\zeta_A$ is equal to $\overline\dim_BA$, which generalizes to arbitrary dimensions a well-known result for fractal strings (or equivalently, for arbitrary compact subsets of the real line $\eR$).\label{; ; eR}; ; Here, $\overline\dim_BA$ denotes the upper box (or Minkowski) dimension of~$A$. Extended to a meromorphic function $\zeta_A$, this ``distance zeta function" is shown to be an efficient tool for finding the box dimension of several new classes of subsets of ${; ; \mathbb R}; ; ^N$, like fractal nests, geometric chirps and multiple string chirps. It can also be used to develop a higher- dimensional theory of complex dimensions of arbitrary fractal sets in Euclidean spaces. For the sake of simplicity, we pay particular attention in this monograph to the principal complex dimensions of $A$, defined as the set of poles of $\zeta_A$ located on the ``critical line'' $\{; ; \mathop{; ; \mathrm{; ; Re}; ; }; ; s=\overline\dim_BA\}; ; $. We also introduce a new zeta function, denoted by $\tilde\zeta_A$ and called a ``tube zeta function", and show, in particular, how to calculate the Minkowski content of a suitable (Minkowski measurable) bounded set $A$ in ${; ; \mathbb R}; ; ^N$ in terms of the residue of $\tilde\zeta_A(s)$ at $s=\dim_BA$, the box dimension of $A$. More generally, without assuming that $A$ is Minkowski measurable, we obtain analogous results, but now expressed as inequalities involving the upper and lower Minkowski contents of $A$. In addition, we obtain a new class of harmonic functions generated by fractal sets and represented via singular integrals. Furthermore, a class of sets is constructed with unequal upper and lower box dimensions, possessing alternating zeta functions. Moreover, by using a suitable notion of equivalence between zeta functions, we simplify some aspects of the theory of geometric zeta functions attached to fractal strings. In addition, we study the problem of the existence and constructing the meromorphic extensions of zeta functions of fractals ; in particular, we provide a natural sufficient condition for the existence of such extensions. An analogous problem is studied in the context of spectral zeta functions associated with bounded open subsets in Euclidean spaces with fractal boundary. We introduce transcendentally quasiperiodic sets, and construct a class of such sets, using generalized Cantor sets with two parameters, along with the Gel'fond- Schneider theorem from the theory of transcendental numbers. With the help of this construction, we obtain an explicit example of a maximally hyperfractal set ; namely, a compact set $A\st\eR^N$ such that the associated distance and tube zeta functions have the critical line $\{; ; \re s=\ov\dim_BA\}; ; $ as a natural boundary. Actually, for this example, much more is true: every point of the critical line is a nonremovable singularity of the fractal zeta functions $\zeta_A$ and $\tilde\zeta_A$. Furthermore, we introduce the notion of relative fractal drum, which extends the notion of fractal string and of fractal drum. The associated definition of relative box dimension is such that it can achieve negative values as well. Using known results about the spectral asymptotics of fractal drums, and some of our earlier work, we recover known results about the existence of a (nontrivial) meromorphic extension of the spectral zeta function of a fractal drum. We also use some of our new results to establish the optimality of the upper bound obtained for the corresponding abscissa of meromorphic continuation of the spectral zeta function. Moreover, we develop a higher-dimensional theory of fractal tube formulas, with or without error terms, for relative fractal drums (and, in particular, for bounded sets) in $\eR^N$, for any $N\geq 1$. Such formulas, interpreted either pointwise or distributionally, enable us to express the volume of the tubular neighborhoods of the underlying fractal drums in terms of the associated complex dimensions. Therefore, they make apparent the deep connections between the theory of complex dimensions and the intrinsic oscillations of fractals. Accordingly, a geometric object is said to be ``fractal'' if it has at least one nonreal complex dimension (with positive real part) or else, the corresponding fractal zeta function has a natural boundary (along a suitable curve). We also formulate and establish a Minkowski measurability criterion for relative fractal drums (and, in particular, for bounded sets) in $\eR^N$, for any $N\geq 1$. More specifically, under suitable assumptions, a relative fractal drum (and, in particular, a bounded set) in $\eR^N$ is shown to be Minkowski measurable if and only if its only complex dimension with real part equal to its (upper) Minkowski dimension $D$ is $D$ itself, and it is simple. Throughout the book, we illustrate our results by a variety of examples, such as the Cantor set and string, the Cantor dust, a version of the Cantor graph (i.e., the ``devil's staircase''), fractal strings (including self- similar strings), fractal sprays (including self-similar sprays), the Sierpi\'nski gasket and carpet as well as their higher-dimensional counterparts, along with non self-similar examples, including fractal nests and geometric chirps. Finally, we propose a classification of bounded sets in Euclidean spaces, based on the properties of their tube functions (that is, the volume of their $\d$-neighborhoods, viewed as a function of the small positive number $\d$), and suggest various open problems concerning distance and tube zeta functions, along with their natural extensions in the context of ``relative fractal drums". Moreover, we suggest directions for future research in the higher-dimensional theory of the fractal complex dimensions of arbitrary compact subsets of Euclidean spaces (as well as more generally, of metric measure spaces). We stress that a significant advantage of the present theory of fractal zeta functions, and therefore, of the corresponding higher-dimensional theory of complex dimensions of fractal sets developed in this book, is that it is applicable to arbitrary bounded (or equivalently, compact) subsets of $\eR^N$, for any $N\ge1$. (At least in principle, it can also be extended to arbitrary compact metric measure spaces, although this is not explicitly done in this work.) In particular, no assumption of self- similarity or, more generally, of ``self- alikeness'' of any kind, is made about the underlying fractals (or, in the broader theory developed here, about the relative fractal drums under consideration).

Hilbert spaces and the pair correlationof zeros of the Riemann zeta-function

Abstract: Montgomery’s pair correlation conjecture predicts the asymptotic behavior of the function N(T, β) defined to be the number of pairs γ and γ′ of ordinates of nontrivial zeros of the Riemann zetafunction satisfying 0 0, using Montgomery’s formula and some extremal functions of exponential type. These functions are optimal in the sense that they majorize and minorize the characteristic function of the interval [−β, β] in a way to minimize the L1 ( R, { 1− ( sinπx πx )2} dx ) -error. We give a complete solution for this extremal problem using the framework of reproducing kernel Hilbert spaces of entire functions. This extends previous work of P. X. Gallagher [18] in 1985, where the case β ∈ 1 2 N was considered using non-extremal majorants and minorants.

A functional equation for the Riemann zeta fractional derivative

Abstract: In this paper a functional equation for the fractional derivative of the Riemann zeta function is presented. The fractional derivative of the zeta function is computed by a generalization of the Gr ...

C T for conformal higher spin fields from partition function on conically deformed sphere

Abstract: We consider the one-parameter generalization S 4 of 4-sphere with a conical singularity due to identification τ = τ +2πq in one isometric angle. We compute the value of the spectral zeta-function at zero $$ \widehat{\zeta}(q)=\zeta \left(0;q\right) $$ that controls the coefficient of the logarithmic UV divergence of the one-loop partition function on S 4 . While the value of the conformal anomaly a-coefficient is proportional to $$ \widehat{\zeta}(1) $$ , we argue that in general the second c ∼ C T anomaly coefficient is related to a particular combination of the second and first derivatives of $$ \widehat{\zeta}(q) $$ at q = 1. The universality of this relation for C T is supported also by examples in 6 and 2 dimensions. We use it to compute the c-coefficient for conformal higher spins finding that it coincides with the “r = −1” value of the one-parameter Ansatz suggested in arXiv:1309.0785 . Like the sums of a s and c s coefficients, the regularized sum of $$ {\widehat{\zeta}}_s(q) $$ over the whole tower of conformal higher spins s = 1, 2,… is found to vanish, implying UV finiteness on S 4 and thus also the vanishing of the associated Renyi entropy. Similar conclusions are found to apply to the standard 2-derivative massless higher spin tower. We also present an independent computation of the full set of conformal anomaly coefficients of the 6d Weyl graviton theory defined by a particular combination of the three 6d Weyl invariants that has a (2, 0) supersymmetric extension.

Lowest Landau level on a cone and zeta determinants

Abstract: We consider the integer QH state on Riemann surfaces with conical singularities, with the main objective of detecting the effect of the gravitational anomaly directly from the form of the wave function on a singular geometry. We suggest the formula expressing the normalisation factor of the holomorphic state in terms of the regularized zeta determinant on conical surfaces and check this relation for some model geometries. We also comment on possible extensions of this result to the fractional QH states.

Finite Size Effects for Spacing Distributions in Random Matrix Theory: Circular Ensembles and Riemann Zeros

Abstract: According to Dyson's threefold way, from the viewpoint of global time reversal symmetry, there are three circular ensembles of unitary random matrices relevant to the study of chaotic spectra in quantum mechanics. These are the circular orthogonal, unitary, and symplectic ensembles, denoted COE, CUE, and CSE, respectively. For each of these three ensembles and their thinned versions, whereby each eigenvalue is deleted independently with probability 1 − ξ, we take up the problem of calculating the first two terms in the scaled large N expansion of the spacing distributions. It is well known that the leading term admits a characterization in terms of both Fredholm determinants and Painleve transcendents. We show that modifications of these characterizations also remain valid for the next to leading term, and that they provide schemes for high precision numerical computations. In the case of the CUE, there is an application to the analysis of Odlyzko's data set for the Riemann zeros, and in that case, some further statistics are similarly analyzed.

Explicit evaluation of quadratic euler sums

Abstract: In this paper, we work out some explicit formulae for double nonlinear Euler sums involving harmonic numbers and alternating harmonic numbers. As applications of these formulae, we give new closed form representations of several quadratic Euler sums through Riemann zeta function and linear sums. The given representations are new.

On some combinatorial identities and harmonic sums

Abstract: For any m,n ∈ ℕ we first give new proofs for the following well-known combinatorial identities Sn(m) =∑k=1nn k (–1)k–1 km =∑n≥r1≥r2≥⋯≥rm≥1 1 r1r2⋯rm and ∑k=1n(–1)n–kn kkn = n!, and then we produce the generating function and an integral representation for Sn(m). Using them we evaluate many interesting finite and infinite harmonic sums in closed form. For example, we show that ζ(3) = 1 9∑n=1∞Hn3 + 3H nHn(2) + 2H n(3) 2n , and ζ(5) = 2 45∑n=1∞Hn4 + 6H n2H n(2) + 8H nHn(3) + 3(H n(2))2 + 6H n(4) n2n , where Hn(i) are generalized harmonic numbers defined below.

Shifted moments of L functions and moments of theta functions

Abstract: Assuming the Riemann Hypothesis, Soundararajan [Ann. of Math. (2) 170 (2009), 981–993] showed that . His method was used by Chandee [Q. J. Math. 62 (2011), 545–572] to obtain upper bounds for shifted moments of the Riemann Zeta function. Building on these ideas of Chandee and Soundararajan, we obtain, conditionally, upper bounds for shifted moments of Dirichlet -functions which allow us to derive upper bounds for moments of theta functions.

Motivic zeta functions of degenerating Calabi-Yau varieties

Abstract: We study motivic zeta functions of degenerating families of Calabi-Yau varieties. Our main result says that they satisfy an analog of Igusa's monodromy conjecture if the family has a so-called Galois-equivariant Kulikov model; we provide several classes of examples where this condition is verified. We also establish a close relation between the zeta function and the skeleton that appeared in Kontsevich and Soibelman's non-archimedean interpretation of the SYZ conjecture in mirror symmetry.

Uniform analytic properties of representation zeta functions of finitely generated nilpotent groups

Abstract: Let $G$ be a finitely generated torsion-free nilpotent group. The representation zeta function $\zeta_G(s)$ of $G$ enumerates twist isoclasses of finite-dimensional irreducible complex representations of $G$. We prove that $\zeta_G(s)$ has rational abscissa of convergence $a(G)$ and may be meromorphically continued to the left of $a(G)$ and that, on the line $\{s\in\mathbb{C} \mid \textrm{Re}(s) = a(G)\}$, the continued function is holomorphic except for a pole at $s=a(G)$. A Tauberian theorem yields a precise asymptotic result on the representation growth of $G$ in terms of the position and order of this pole. We obtain these results as a consequence of a more general result establishing uniform analytic properties of representation zeta functions of finitely generated nilpotent groups of the form $\mathbf{G}(\mathcal{O})$, where $\mathbf{G}$ is a unipotent group scheme defined in terms of a nilpotent Lie lattice over the ring $\mathcal{O}$ of integers of a number field. This allows us to show, in particular, that the abscissae of convergence of the representation zeta functions of such groups and their pole orders are invariants of $\mathbf{G}$, independent of $\mathcal{O}$.

Spectral zeta functions of graphs and the Riemann zeta function in the critical strip

Abstract: We initiate the study of spectral zeta functions $\zeta_X$ for finite and infinite graphs $X$, instead of the Ihara zeta function, with a perspective towards zeta functions from number theory and connections to hypergeometric functions. The Riemann hypothesis is shown to be equivalent to an approximate functional equation of graph zeta functions. The latter holds at all points where Riemann's zeta function $\zeta(s)$ is non-zero. This connection arises via a detailed study of the asymptotics of the spectral zeta functions of finite torus graphs in the critcal strip and estimates on the real part of the logarithmic derivative of $\zeta(s)$. We relate $\zeta_{\mathbb{Z}}$ to Euler's beta integral and show how to complete it giving the functional equation $\xi_{\mathbb{Z}}(1-s)=\xi_{\mathbb{Z}}(s)$. This function appears in the theory of Eisenstein series although presumably with this spectral intepretation unrecognized. In higher dimensions $d$ we provide a meromorphic continuation of $\zeta_{\mathbb{Z}^d}(s)$ to the whole plane and identify the poles. From our aymptotics several known special values of $\zeta(s)$ are derived as well as its non-vanishing on the line $Re(s)=1$. We determine the spectral zeta functions of regular trees and show it to be equal to a specialization of Appell's hypergeometric function $F_1$ via an Euler-type integral formula due to Picard.