Showing papers on "Riemann zeta function published in 2018"

On the extreme values of the Riemann zeta function on random intervals of the critical line

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, U being a uniform random variable 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 $$\mathfrak {R}\log \zeta (1/2 + it)$$ is at most $$\log \log T + g(T)$$ with probability tending to 1, g being any function tending to infinity at infinity.

Extreme values of the Riemann zeta function and its argument

Abstract: We combine our version of the resonance method with certain convolution formulas for $$\zeta (s)$$ and $$\log \, \zeta (s)$$ . This leads to a new $$\Omega $$ result for $$|\zeta (1/2+it)|$$ : The maximum of $$|\zeta (1/2+it)|$$ on the interval $$1 \le t \le T$$ is at least $$\exp \left( (1+o(1)) \sqrt{\log T \log \log \log T/\log \log T}\right) $$ . We also obtain conditional results for $$S(t):=1/\pi $$ times the argument of $$\zeta (1/2+it)$$ and $$S_1(t):=\int _0^t S(\tau )d\tau $$ . On the Riemann hypothesis, the maximum of |S(t)| is at least $$c \sqrt{\log T \log \log \log T/\log \log T}$$ and the maximum of $$S_1(t)$$ is at least $$c_1 \sqrt{\log T \log \log \log T/(\log \log T)^3}$$ on the interval $$T^{\beta } \le t \le T$$ whenever $$0\le \beta < 1$$ .

Analytic torsion, dynamical zeta functions, and the Fried conjecture

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.

Multi-poly-bernoulli numbers and related zeta functions

Abstract: We construct and study a certain zeta function which interpolates multi-poly-Bernoulli numbers at nonpositive integers and whose values at positive integers are linear combinations of multiple zeta values. This function can be regarded as the one to be paired up with the -function defined by Arakawa and Kaneko. We show that both are closely related to the multiple zeta functions. Further we define multi-indexed poly-Bernoulli numbers, and generalize the duality formulas for poly-Bernoulli numbers by introducing more general zeta functions.

On p-adic string amplitudes in the limit p approaches to one

Abstract: In this article we discuss the limit p approaches to one of tree-level p-adic open string amplitudes and its connections with the topological zeta functions. There is empirical evidence that p-adic strings are related to the ordinary strings in the p → 1 limit. Previously, we established that p-adic Koba-Nielsen string amplitudes are finite sums of multivariate Igusa’s local zeta functions, consequently, they are convergent integrals that admit meromorphic continuations as rational functions. The meromorphic continuation of local zeta functions has been used for several authors to regularize parametric Feynman amplitudes in field and string theories. Denef and Loeser established that the limit p → 1 of a Igusa’s local zeta function gives rise to an object called topological zeta function. By using Denef-Loeser’s theory of topological zeta functions, we show that limit p → 1 of tree-level p-adic string amplitudes give rise to certain amplitudes, that we have named Denef-Loeser string amplitudes. Gerasimov and Shatashvili showed that in limit p → 1 the well-known non-local effective Lagrangian (reproducing the tree-level p-adic string amplitudes) gives rise to a simple Lagrangian with a logarithmic potential. We show that the Feynman amplitudes of this last Lagrangian are precisely the amplitudes introduced here. Finally, the amplitudes for four and five points are computed explicitly.

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.

Two kinds of the reverse Hardy-type integral inequalities with the equivalent forms related to the extended Riemann zeta function

Abstract: Applying techniques of real analysis and weight functions, we study some equivalent conditions of two kinds of the reverse Hardy-type integral inequalities with a particular nonhomogeneous kernel. The constant factors are related to the Riemann zeta function and are proved to be best possible. In the form of applications, we deduce a few equivalent conditions of two kinds of the reverse Hardy-type integral inequalities with a particular homogeneous kernel. We also consider some corollaries as particular cases.

Zeta functions of alternate mirror Calabi-Yau families

Abstract: We prove that if two Calabi–Yau invertible pencils have the same dual weights, then they share a common factor in their zeta functions. By using Dwork cohomology, we demonstrate that this common factor is related to a hypergeometric Picard–Fuchs differential equation. The factor in the zeta function is defined over the rationals and has degree at least the order of the Picard–Fuchs equation. As an application, we relate several pencils of K3 surfaces to the Dwork pencil, obtaining new cases of arithmetic mirror symmetry.

Note on the resonance method for the Riemann zeta function

Abstract: We improve Montgomery’s Ω-results for |ζ(σ + it)| in the strip 1/2 σ 1 and give in particular lower bounds for the maximum of |ζ(σ+it)| on √ T ≤ t ≤ T that are uniform in σ. We give similar lower bounds for the maximum of |_n≤x n −1/2−it | on intervals of length much larger than x. We rely on our recent work on lower bounds for maxima of |ζ(1/2 + it)| on long intervals, as well as work of Soundararajan, G´al, and others. The paper aims at displaying and clarifying the conceptually different combinatorial arguments that show up in various parts of the proofs.

Many odd zeta values are irrational

Abstract: Building upon ideas of the second and third authors, we prove that at least $2^{(1-\varepsilon)\frac{\log s}{\log\log s}}$ values of the Riemann zeta function at odd integers between 3 and $s$ are irrational, where $\varepsilon$ is any positive real number and $s$ is large enough in terms of $\varepsilon$. This lower bound is asymptotically larger than any power of $\log s$; it improves on the bound $\frac{1-\varepsilon}{1+\log2}\log s$ that follows from the Ball--Rivoal theorem. The proof is based on construction of several linear forms in odd zeta values with related coefficients.

Solutions and conservation laws of a generalized second extended (3+1)-dimensional Jimbo-Miwa equation

Abstract: In this paper we study a nonlinear multi-dimensional partial differential equation, namely, a generalized second extended (3+1)-dimensional Jimbo-Miwa equation. We perform symmetry reductions of this equation until it reduces to a nonlinear fourth-order ordinary differential equation. The general solution of this ordinary differential equation is obtained in terms of the Weierstrass zeta function. Also travelling wave solutions are derived using the simplest equation method. Finally, the conservation laws of the underlying equation are computed by employing the conservation theorem due to Ibragimov, which include conservation of energy and conservation of momentum laws.

Entropy and the spectral action

Abstract: We compute the information theoretic von Neumann entropy of the state associated to the fermionic second quantization of a spectral triple We show that this entropy is given by the spectral action of the spectral triple for a specific universal function The main result of our paper is the surprising relation between this function and the Riemann zeta function It manifests itself in particular by the values of the coefficients $c(d)$ by which it multiplies the $d$ dimensional terms in the heat expansion of the spectral triple We find that $c(d)$ is the product of the Riemann xi function evaluated at $-d$ by an elementary expression In particular $c(4)$ is a rational multiple of $\zeta(5)$ and $c(2)$ a rational multiple of $\zeta(3)$ The functional equation gives a duality between the coefficients in positive dimension, which govern the high energy expansion, and the coefficients in negative dimension, exchanging even dimension with odd dimension

Lattice Point Visibility on Generalized Lines of Sight

Abstract: For a fixed we say that a point (r, s) in the integer lattice is b-visible from the origin if it lies on the graph of a power function f(x) = axb with and no other integer lattice point lies on thi...

Notes on a Double Inequality for Ratios of any Two Neighbouring Non-zero Bernoulli Numbers

Abstract: In the paper, the author notes on a double inequality published in “Feng Qi, A double inequality for the ratio of two non-zero neighbouring Bernoulli numbers, Journal of Computational and Applied Mathematics 351 (2019), 1-5; Available online at https://doi.org/10.1016/j.cam.2018.10.049.”

Topological Hochschild homology and the Hasse-Weil zeta function

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.

The Riemann hypothesis and universality of the Riemann zeta-function

Abstract: Abstract We prove that, under the Riemann hypothesis, a wide class of analytic functions can be approximated by shifts ζ(s + iγk), k ∈ ℕ, of the Riemann zeta-function, where γk are imaginary parts of nontrivial zeros of ζ(s).

Gaps between zeros of the Riemann zeta-function

Abstract: We prove that there exist infinitely many consecutive zeros of the Riemann zeta-function on the critical line whose gaps are greater than $3.18$ times the average spacing. Using a modification of our method, we also show that there are even larger gaps between the multiple zeros of the zeta function on the critical line (if such zeros exist).

Special values and integral representations for the Hurwitz-type Euler zeta functions

Abstract: The Hurwitz-type Euler zeta function is defined as a deformation of the Hurwitz zeta function: \begin{equation*} \zeta_E(s,x)=\sum_{n=0}^\infty\frac{(-1)^n}{(n+x)^s}. \end{equation*} In this paper, by using the method of Fourier expansions, we shall evaluate several integrals with integrands involving Hurwitz-type Euler zeta functions $\zeta_E(s,x)$. Furthermore, the relations between the values of a class of the Hurwitz-type (or Lerch-type) Euler zeta functions at rational arguments have also been given.

Derivatives and fast evaluation of the Tornheim zeta function

Abstract: We study analytic properties of the Tornheim zeta function $${\mathcal W}(r,s,t)$$ , which is also named after Mordell and Witten. In particular, we evaluate the function $${\mathcal W}(s,s,\tau s)$$ ( $$\tau >0$$ ) at $$s=0$$ and, as our main result, find the derivative of this function at $$s=0$$ . Our principal tool is an identity due to Crandall that involves a free parameter and provides an analytic continuation. Furthermore, we derive special values of a permutation sum. Throughout this paper, we show by way of examples that Crandall’s identity can be used for efficient and high-precision evaluations of the Tornheim zeta function.

New Computations of the Riemann Zeta Function on the Critical Line

Abstract: We present highlights of computations of the Riemann zeta function around large values and high zeros. The main new ingredient in these computations is an implementation of the second author’s fast algorithm for numerically evaluating quadratic exponential sums. In addition, we use a new simple multi-evaluation method to compute the zeta function in a very small range at little more than the cost of evaluation at a single point.

A modular-invariant modified Weierstrass sigma-function as a building block for lowest-Landau-level wavefunctions on the torus

Abstract: A “modified” variant of the Weierstrass sigma, zeta, and elliptic functions is proposed whereby the zeta function is redefined by ζ(z) ↦ ζ(z) ≡ ζ(z)−γ2z, where γ2 is a lattice invariant related to the almost-holomorphic modular invariant of the quasi-modular-invariant weight-2 Eisenstein series. If ωi is a primitive half-period, ζ(ωi) = πωi*/A, where A is the area of the primitive cell of the lattice. The quasiperiodicity of the modified sigma function is much simpler than that of the original, and it becomes the building-block for the modular-invariant formulation of lowest-Landau-level wavefunctions on the torus. It is suggested that the “modified” sigma function is more natural than the original Weierstrass form, which was formulated before quasi-modular forms were understood. For the high-symmetry (square and hexagonal) lattices, the modified and original sigma functions coincide.A “modified” variant of the Weierstrass sigma, zeta, and elliptic functions is proposed whereby the zeta function is redefined by ζ(z) ↦ ζ(z) ≡ ζ(z)−γ2z, where γ2 is a lattice invariant related to the almost-holomorphic modular invariant of the quasi-modular-invariant weight-2 Eisenstein series. If ωi is a primitive half-period, ζ(ωi) = πωi*/A, where A is the area of the primitive cell of the lattice. The quasiperiodicity of the modified sigma function is much simpler than that of the original, and it becomes the building-block for the modular-invariant formulation of lowest-Landau-level wavefunctions on the torus. It is suggested that the “modified” sigma function is more natural than the original Weierstrass form, which was formulated before quasi-modular forms were understood. For the high-symmetry (square and hexagonal) lattices, the modified and original sigma functions coincide.

Almost Periodic Functions in terms of Bohr's Equivalence Relation

Abstract: In this paper we introduce an equivalence relation on the classes of almost periodic functions of a real or complex variable which is used to refine Bochner’s result that characterizes these spaces of functions. In fact, with respect to the topology of uniform convergence, we prove that the limit points of the family of translates of an almost periodic function are precisely the functions which are equivalent to it, which leads us to a characterization of almost periodicity. In particular we show that any exponential sum which is equivalent to the Riemann zeta function, $$\zeta (s)$$ , can be uniformly approximated in $$\{s=\sigma +it:\sigma >1\}$$ by certain vertical translates of $$\zeta (s)$$ .

Pseudomoments of the Riemann zeta function

Abstract: The $2$kth pseudomoments of the Riemann zeta function $\zeta(s)$ are, following Conrey and Gamburd, the $2k$th integral moments of the partial sums of $\zeta(s)$ on the critical line. For fixed $k>1/2$, these moments are known to grow like $(\log N)^{k^2}$, where $N$ is the length of the partial sum, but the true order of magnitude remains unknown when $k\le 1/2$. We deduce new Hardy--Littlewood inequalities and apply one of them to improve on an earlier asymptotic estimate when $k\to\infty$. In the case $k 1$ and the question of whether the lower bound $(\log N)^{k^2\alpha^2}$ known from earlier work yields the true growth rate. Using ideas from recent work of Harper, Nikeghbali, and Radziwi{ll} and some probabilistic estimates of Harper, we obtain the somewhat unexpected result that these pseudomements are bounded below by $\log N$ to a power larger than $k^2\alpha^2$ when $k<1/e$ and $N$ is sufficiently large.

Motivic Euler products and motivic height zeta functions

Abstract: A motivic height zeta function associated to a family of varieties parametrised by a curve is the generating series of the classes, in the Grothendieck ring of varieties, of moduli spaces of sections of this family with varying degrees. This text is devoted to the study of the motivic height zeta function associated to a family of varieties with generic fiber having the structure of an equivariant compactification of a vector group. Our main theorem describes the convergence of this motivic height zeta function with respect to a topology on the Grothendieck ring of varieties coming from the theory of weights in cohomology. We deduce from it the asymptotic behaviour, as the degree goes to infinity, of a positive proportion of the coefficients of the Hodge-Deligne polynomial of the above moduli spaces: in particular, we get an estimate for their dimension and the number of components of maximal dimension. The main tools for this are a notion of motivic Euler product for series with coefficients in the Grothendieck ring of varieties, an extension of Hrushovski and Kazhdan's motivic Poisson summation formula, and a motivic measure on the Grothendieck ring of varieties with exponentials constructed using Denef and Loeser's motivic vanishing cycles.

Character integral representation of zeta function in AdS d+1 . Part II. Application to partially-massless higher-spin gravities

Abstract: We compute the one-loop free energies of the type-Al and type-Bl higher-spin gravities in (d + 1)-dimensional anti-de Sitter (AdSd+1) spacetime. For large d and l, these theories have a complicated field content, and hence it is difficult to compute their zeta functions using the usual methods. Applying the character integral representation of zeta function developed in the companion paper [ arXiv:1805.05646 ] to these theories, we show how the computation of their zeta function can be shortened considerably. We find that the results previously obtained for the massless theories (l = 1) generalize to their partially-massless counterparts (arbitrary l) in arbitrary dimensions.