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Flat space cosmologies (FSCs) are time dependent solutions of three-dimensional (3D) gravity with a vanishing cosmological constant. They can be constructed from a discrete quotient of empty 3D flat spacetime and are also called shifted-boost orbifolds. Using this quotient structure, we build a new and generalized Selberg zeta function for FSCs, and show that it is directly related to the scalar 1-loop partition function. We then propose an extension of this formalism applicable to more general quotient manifolds M /ℤ, based on representation theory of fields propagating on this background. Our prescription constitutes a novel and expedient method for calculating regularized 1-loop determinants, without resorting to the heat kernel. We compute quasinormal modes in the FSC using the zeroes of a Selberg zeta function, and match them to known results. [ABSTRACT FROM AUTHOR]

We show that the Generalized Riemann Hypothesis for Dirichlet L-functions is a consequence of certain conjectural properties of the zeros of the Riemann zeta function. Conversely, we prove that the zeros of ζ (s) satisfy those properties under GRH. [ABSTRACT FROM AUTHOR]

For any real β 0 ∈ [ 1 2 , 1) , let GRH [ β 0 ] be the assertion that for every Dirichlet character χ and all zeros ρ = β + i γ of L (s , χ) , one has β ⩽ β 0 (in particular, GRH [ 1 2 ] is the Generalized Riemann Hypothesis). In this paper, we show that the validity of GRH [ 9 10 ] depends only on certain distributional properties of the zeros of the Riemann zeta function ζ (s) . No conditions are imposed on the zeros of nonprincipal Dirichlet L-functions. [ABSTRACT FROM AUTHOR]

The Hankel matrices considered in this article arose in one reformulation of the Riemann hypothesis proposed earlier by the author. Computer calculations showed that, in the case of the Riemann zeta function, the eigenvalues and the eigenvectors of such matrices have an interesting structure. The article studies a model situation when the zeta function is replaced by a function having a single zero. For this case, we indicate the first terms of the asymptotic expansions of the smallest and largest (in absolute value) eigenvalues and the corresponding eigenvectors. [ABSTRACT FROM AUTHOR]

The paper studies the zeta function of the monoid generated by prime numbers of the form 3n + 2. The main monoid and the main set are defined. For the corresponding zeta functions, explicit finite formulas are found that give their analytic continuations to the entire complex plane, except for a countable set of poles. Inverse series for these zeta functions are found, and functional equations are derived. Three new types of monoids of natural numbers with a unique prime factorization are defined, namely, monoids of powers, Euler monoids modulo q, and unit monoids modulo q. Expressions for their zeta functions in terms of the Euler product are given. The Davenport–Heilbronn effect for zeta functions of monoids of natural numbers is discussed, which means the appearance of zeros of the zeta functions for terms obtained by decomposition into residue classes to some modulus. For monoids with an exponential sequence of primes, the barrier series hypothesis is proved and it is shown that the holomorphic domain of the zeta function of such a monoid is the complex half-plane to the right of the imaginary axis. In conclusion, topical problems with zeta functions of monoids of natural numbers that require further investigation are considered. [ABSTRACT FROM AUTHOR]

A formal weight enumerator is a homogeneous polynomial in two variables which behaves like the Hamming weight enumerator of a self-dual linear code except that the coefficients are not necessarily nonnegative integers. The notion of formal weight enumerator was first introduced by Ozeki in connection with modular forms, and a systematic investigation of formal weight enumerators has been conducted by Chinen in connection with zeta functions and Riemann hypothesis for linear codes. In this paper, we establish a relation between formal weight enumerators and Chebyshev polynomials. Specifically, the condition for the existence of formal weight enumerators with prescribed parameters (n , ε , q) is given in terms of Chebyshev polynomials. According to the parity of n and the sign ε , the four kinds of Chebyshev polynomials appear in the statement of the result. Further, we obtain explicit expressions of formal weight enumerators in the case where n is odd or ε = - 1 using Dickson polynomials, which generalize Chebyshev polynomials. We also state a conjecture from a viewpoint of binomial moments, and see that the results in this paper partially support the conjecture. [ABSTRACT FROM AUTHOR]

We study the Markov chain x n + 1 = a x n + b n on a finite field F p , where a ∈ F p × is fixed and b n are independent and identically distributed random variables in F p . Conditionally on the Riemann hypothesis for all Dedekind zeta functions, we show that the chain exhibits a cut-off phenomenon for most primes p and most values of a ∈ F p × . We also obtain weaker, but unconditional, upper bounds for the mixing time. [ABSTRACT FROM AUTHOR]

In this short note, we prove the following result: If a completely multiplicative function f : ℕ → [−1, 1] is small on average in the sense that ∑n ≤ xf(n) ≪ x1 − δ for some δ > 0, and if the Dirichlet series of f, say F(s), is such that F(1) = 0, then for any ϵ > 0, ∑p ≤ x(1 + f(p)) log p ≪ x1 − δ + ϵ. Moreover, a necessary condition for the existence of such f is that the Riemann zeta function ζ(s) has no zeros in the half-plane Re(s) > 1 − δ. [ABSTRACT FROM AUTHOR]

Let be the class of Dirichlet series introduced by Selberg and modified by Steuding, and let be the sequence of the imaginary parts of the nontrivial zeros of the Riemann zeta-function. Using the modified Montgomery's pair correlation conjecture, we prove a universality theorem for a function in on approximation of analytic functions by the shifts , . [ABSTRACT FROM AUTHOR]

Gábor Halász and Pál Turán were the first who proved unconditionally the Density Hypothesis for Riemann's zeta function in a fixed strip c 0 < Re s < 1 . They also showed that the Lindelöf Hypothesis implies a surprisingly strong bound on the number of zeros with Re s ≥ c 1 > 3 / 4 . In the present work we use an alternative approach to prove their result which does not use either Turán's power sum method or the large sieve. [ABSTRACT FROM AUTHOR]

The Fourier transform associated with the normalized logarithm of the modulus of the Riemann Zeta Function is considered. The formulas linking the Fourier transform and the zeros of the Riemann function are established that lead to the necessary and sufficient condition of satisfaction of the Riemann hypothesis. [ABSTRACT FROM AUTHOR]

We prove a new effective Chebotarev density theorem for Galois extensions L / Q that allows one to count small primes (even as small as an arbitrarily small power of the discriminant of L); this theorem holds for the Galois closures of "almost all" number fields that lie in an appropriate family of field extensions. Previously, applying Chebotarev in such small ranges required assuming the Generalized Riemann Hypothesis. The error term in this new Chebotarev density theorem also avoids the effect of an exceptional zero of the Dedekind zeta function of L, without assuming GRH. We give many different "appropriate families," including families of arbitrarily large degree. To do this, we first prove a new effective Chebotarev density theorem that requires a zero-free region of the Dedekind zeta function. Then we prove that almost all number fields in our families yield such a zero-free region. The innovation that allows us to achieve this is a delicate new method for controlling zeroes of certain families of non-cuspidalL-functions. This builds on, and greatly generalizes the applicability of, work of Kowalski and Michel on the average density of zeroes of a family of cuspidalL-functions. A surprising feature of this new method, which we expect will have independent interest, is that we control the number of zeroes in the family of L-functions by bounding the number of certain associated fields with fixed discriminant. As an application of the new Chebotarev density theorem, we prove the first nontrivial upper bounds for ℓ -torsion in class groups, for all integers ℓ ≥ 1 , applicable to infinite families of fields of arbitrarily large degree. [ABSTRACT FROM AUTHOR]

We give new proofs of some known results on the values of the Riemann zeta function at positive integers and obtain some new theorems related to these values. Considering even zeta values as ζ(2n) = ηnπ2n, we obtain the generating functions of the sequences ηn and (−1)nηn. Using the Riemann–Lebesgue lemma, we give recurrence relations for ζ(2n) and ζ(2n + 1). Furthermore, we prove some series equations for ∑ k = 1 ∞ − 1 k − 1 ζ p + k / k. [ABSTRACT FROM AUTHOR]

It was found that, in addition to trivial zeros in points (z = − 2N, N = 1, 2..., natural numbers), the Riemann's zeta function ζ(z) has zeros only on the line { z = 1 2 + i t 0 , t0 is real}. All zeros are numerated, and for each number, N, the positions of the non-overlap intervals with one zero inside are found. The simple equation for the determination of centers of intervals is obtained. The analytical function η(z), leading to the possibility fix the zeros of the zeta function ζ(z), was estimated. To perform the analysis, the well-known phenomenon, phase-slip events, is used. This phenomenon is the key ingredient for the investigation of dynamical processes in solid-state physics, for example, if we are trying to solve the TDGLE (time-dependent Ginzburg-Landau equation). [ABSTRACT FROM AUTHOR]

It was found that, in addition to trivial zeros in points (z = − 2N, N = 1, 2..., natural numbers), the Riemann's zeta function ζ(z) has zeros only on the line { z = 1 2 + i t 0 , t0 is real}. All zeros are numerated, and for each number, N, the positions of the non-overlap intervals with one zero inside are found. The simple equation for the determination of centers of intervals is obtained. The analytical function η(z), leading to the possibility fix the zeros of the zeta function ζ(z), was estimated. To perform the analysis, the well-known phenomenon, phase-slip events, is used. This phenomenon is the key ingredient for the investigation of dynamical processes in solid-state physics, for example, if we are trying to solve the TDGLE (time-dependent Ginzburg-Landau equation). [ABSTRACT FROM AUTHOR]

Zeta functions for linear codes were defined by Iwan Duursma in 1999 as generating functions for the Hamming weight enumerators of linear codes. They were generalized to the case of some invariant polynomials and so-called the formal weight enumerators by the present author. One of the most important problems from the applicable point of view is whether extremal weight enumerators satisfy the Riemann hypothesis. In this article, we show there exist extremal polynomials of the weight enumerator type, not being related to existing codes, which are invariant under the MacWilliams transform and do not satisfy the Riemann hypothesis. Such examples are contained in a certain invariant polynomial ring which is found by the use of the binomial moment. We determine the group which fixes the ring. To formulate the extremal property, we also establish an analog of the Mallows–Sloane bound for a certain sequence of the members in the ring. [ABSTRACT FROM AUTHOR]

The aim of this expository article is to introduce the reader to some of the fundamental milestones in the study of prime numbers across several centuries. Among the important developments in the study of prime numbers, we review the history of the prime number theorem, the Riemann zeta function (in relation to prime number theory), and some recent investigations into spacings between consecutive primes. We also present an important application of prime numbers in safe data transmission, namely the 'RSA public key cryptosystem'. [ABSTRACT FROM AUTHOR]

In one of his final research papers, Alan Turing introduced a method to certify the completeness of a purported list of zeros of the Riemann zeta-function. In this paper we consider Turing's method in the analogous setting of Selberg zeta-functions and we demonstrate that it can be carried out rigorously in the prototypical case of the modular surface. [ABSTRACT FROM AUTHOR]

In the present paper, we show that under the Riemann hypothesis, and for fixed h,ϵ>0, the supremum of the real and the imaginary parts of logζ(1/2+it) for t∈[UT-h,UT+h] are in the interval [(1-ϵ)loglogT,(1+ϵ)loglogT] 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 Rlogζ(1/2+it) is at most loglogT+g(T) with probability tending to 1, g being any function tending to infinity at infinity. [ABSTRACT FROM AUTHOR]

For most values of parameters λ and α, the zeros of the Lerch zeta-function L( λ, α, s) are distributed very chaotically. In this paper, we consider the particular case of equal parameters L( λ, λ, s) and show by calculations that the nontrivial zeros either lie extremely close to the critical line σ = 1 /2 or are distributed almost symmetrically with respect to the critical line. We also investigate this phenomenon theoretically. [ABSTRACT FROM AUTHOR]

An Erratum to this paper has been published: https://doi.org/10.1134/S1064562423220027 [ABSTRACT FROM AUTHOR]

It is shown that the deterministic infinite trigonometric products with parameters $$ p\in (0,1]\ \& \ s>\frac{1}{2}$$ , and variable $$t\in \mathbb {R}$$ , are inverse Fourier transforms of the probability distributions for certain random series $$\Omega _{p}^\zeta (s)$$ taking values in the real $$\omega $$ line; i.e. the $${\text{ Cl }_{p;s}^{}}(t)$$ are characteristic functions of the $$\Omega _{p}^\zeta (s)$$ . The special case $$p=1=s$$ yields the familiar random harmonic series, while in general $$\Omega _{p}^\zeta (s)$$ is a 'random Riemann- $$\zeta $$ function,' a notion which will be explained and illustrated-and connected to the Riemann hypothesis. It will be shown that $$\Omega _{p}^\zeta (s)$$ is a very regular random variable, having a probability density function (PDF) on the $$\omega $$ line which is a Schwartz function. More precisely, an elementary proof is given that there exists some $$K_{p;s}^{}>0$$ , and a function $$F_{p;s}^{}(|t|)$$ bounded by $$|F_{p;s}^{}(|t|)|\!\le \! \exp \big (K_{p;s}^{} |t|^{1/(s+1)})$$ , and $$C_{p;s}^{}\!:=\!-\frac{1}{s}\int _0^\infty \ln |{1-p+p\cos \xi }|\frac{1}{\xi ^{1+1/s}}\mathrm{{d}}\xi $$ , such that the regularity of $$\Omega _{p}^\zeta (s)$$ follows. Incidentally, this theorem confirms a surmise by Benoit Cloitre, that $$\ln {\text{ Cl }_{{{1}/{3}};2}^{}}(t) \sim -C\sqrt{t}\; \left( t\rightarrow \infty \right) $$ for some $$C>0$$ . Graphical evidence suggests that $${\text{ Cl }_{{{1}/{3}};2}^{}}(t)$$ is an empirically unpredictable (chaotic) function of t. This is reflected in the rich structure of the pertinent PDF (the Fourier transform of $${\text{ Cl }_{{{1}/{3}};2}^{}}$$ ), and illustrated by random sampling of the Riemann- $$\zeta $$ walks, whose branching rules allow the build-up of fractal-like structures. [ABSTRACT FROM AUTHOR]

Adolf Hurwitz's estate contains a note from the early 1880s on the converse to Riemann's proof of the functional equation for the zeta-function; this idea has later been elaborated by Hans Hamburger for a characterization of the zeta-function by its functional equation and by Eugène Cahen and Erich Hecke with respect to modular forms. In this note, we present Hurwitz's reasoning and comment on the historical context. [ABSTRACT FROM AUTHOR]

In this series we examine the calculation of the 2 kth moment and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. The present paper begins the general study of what we call Type II sums which utilize a circle method framework and a convolution of shifted convolution sums to obtain all of the lower order terms in the asymptotic formula for the mean square along [ T, 2 T] of a Dirichlet polynomial of length up to $$T^3$$ with divisor functions as coefficients. [ABSTRACT FROM AUTHOR]

Using the Davenport-Heilbronn method, we show that if $\lambda_{1},\lambda_{2},\ldots,\lambda_{9}$ are positive real numbers, at least one of the ratios $\lambda_{i}/\lambda_{j}$ ( $1\leq i< j\leq9$) is irrational, then the integer parts of $\lambda_{1}x_{1}^{3}+\lambda_{2}x_{2}^{3}+\lambda_{3}x_{3}^{4}+\lambda_{4}x_{4}^{4} +\lambda_{5}x_{5}^{5}+\cdots+\lambda_{9}x_{9}^{5}$ are prime infinitely often for natural numbers $x_{1},x_{2},\ldots,x_{9}$. [ABSTRACT FROM AUTHOR]

The article focuses on distribution of zero in horizontal manner of zeta functions. Topics discussed includes consideration of several approaches that deals with distribution of zeros horizontally, several problems associated with Riemann function due to its different properties, application of theorem for determination of arbitrary function, decrement in interval length by utilization of Riemann hypothesis and employment of omega- theorems for analysis of solution determined for functions.

The article discusses the original formulation of the famous Riemann hypothesis. It explores the assertion about the positions of the zeros of the Riemann zeta functions which can be defined by the Dirichlet series. It also presents several formulas about the Riemann hypothesis which be can be defined and expanded in the Laurent series.

We investigate the behavior of the Euler products of the Riemann zeta function and Dirichlet L-functions on the critical line. A refined version of the Riemann hypothesis, which is named 'the Deep Riemann Hypothesis', is examined. We also study various analogs for global function fields. We give an interpretation for the nontrivial zeros from the viewpoint of statistical mechanics. [ABSTRACT FROM AUTHOR]

Under the assumption of the Riemann hypothesis for the Riemann zeta function and some Dirichlet L-series we demonstrate that certain products of the corresponding zeta functions are completely monotonic. This may provide a method to disprove a certain Riemann hypothesis numerically. [ABSTRACT FROM AUTHOR]

In this paper we derive a formula of the Ihara zeta function of a cone over a regular graph that involves the spectrum of the adjacency matrix of the cone. We show that the Ihara zeta function and the spectrum of the adjacency matrix of the cone determine each other and we characterize those cones that satisfy the graph theory Riemann hypothesis. [ABSTRACT FROM AUTHOR]

Assuming the Riemann hypothesis, we establish upper bounds for discrete moments of the Riemann zeta-function and its derivatives on the critical line. Moreover, we express continuous moments of the Riemann zeta-function and its derivatives in terms of these discrete moments. This allows us to give conditional upper bounds for $ {\int_0^T {\left| {{\zeta^{(l)}}\left( {{{1} \left/ {2} \right.} + {\text{i}}t} \right)} \right|}^{2k}}{\text{d}}t $, where l and k are nonnegative integers. [ABSTRACT FROM AUTHOR]

In this section of Resonance, we invite readers to pose questions likely to be raised in a classroom situation.Wemay suggest strategies for dealingwith them, or invite responses, or both. 'Classroom' is equally a forum for raising broader issues and sharing personal experiences and viewpoints on matters related to teaching and learning science. [ABSTRACT FROM AUTHOR]

We prove a statement equivalent to the Riemann hypothesis. [ABSTRACT FROM AUTHOR]

We use an approach based on Fourier inversion formula and measure convolutions to obtain, in the setting of the Lindelöf hypothesis, new mean value results for the Riemann zeta function. [ABSTRACT FROM AUTHOR]

Under the Riemann hypothesis and the conjecture that the order of growth of the argument of ζ(1/2 + it) is bounded by $\left( {\log t} \right)^{\frac{1} {2} + o\left( 1 \right)}$ , we show that for any given α > 0 the interval $(X,X + \sqrt X (\log X)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} + o\left( 1 \right)} ]$ contains an integer having no prime factor exceeding X for all X sufficiently large. [ABSTRACT FROM AUTHOR]

We obtain criteria for the Riemann hypothesis using discrete Schwartz distributions defined by the arithmetical functions of Euler, von Mangoldt, Möbius, and Liouville. [ABSTRACT FROM AUTHOR]

We consider lower bounds for the Euler—Kronecker constant in the case of number fields and upper and lower bounds in the case of algebraic manifolds over a finite field. [ABSTRACT FROM AUTHOR]

Abstract  In this paper, we investigate the joint value-distribution for the Riemann zeta function and Hurwitz zeta function attached with a transcendental real parameter. Especially, we establish the joint universality theorem for these two zeta functions. [ABSTRACT FROM AUTHOR]

Planes are a good time to work on math, with no distractions and no excuses to do other things. And although recently it was proved that all but finitely many of the polynomials of each degree have only real roots, there doesn't seem to be a way to finish it off." "Couldn't you convolute the remaining polynomials and extract a higher-level symmetric copolynomial?" "And what would the point of that be?" "The resulting polynomials would now fall into the set of polynomials which we already knew had real roots, meaning the original polynomials all had real roots as well." I think you have just solved the biggest open problem in all of mathematics.". [Extracted from the article]

We prove that the integer part of the reciprocal of the tail of ζ (s) at a rational number s = 1 p for any integer with p ≥ 5 or s = 2 p for any odd integer with p ≥ 5 can be described essentially as the integer part of an explicit quantity corresponding to it. To deal with the case when s = 2 p , we use a result on the finiteness of integral points of certain curves over Q . [ABSTRACT FROM AUTHOR]

Shows a proof of a series with simple zeros of zeta function. Riemann conjecture; Riemann hypothesis; Holder's inequality; Proof of the theorem.

This paper outlines further properties concerning the fractional derivative of the Riemann ζ function. The functional equation, computed by the introduction of the Grünwald–Letnikov fractional derivative, is rewritten in a simplified form that reduces the computational cost. Additionally, a quasisymmetric form of the aforementioned functional equation is derived (symmetric up to one complex multiplicative constant). The second part of the paper examines the link with the distribution of prime numbers. The Dirichlet η function suggests the introduction of a complex strip as a fractional counterpart of the critical strip. Analytic properties are shown, particularly that a Dirichlet series can be linked with this strip and expressed as a sum of the fractional derivatives of ζ. Finally, Theorem 4.3 links the fractional derivative of ζ with the distribution of prime numbers in the left half-plane. [ABSTRACT FROM AUTHOR]