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A tale of two omegasJun 07 2019We consider $\omega(n)$ and $\Omega(n)$, which respectively count the number of distinct and total prime factors of $n$. We survey a number of similarities and differences between these two functions, and study the summatory functions $L(x)=\sum_{n\leq ... More

Lower Order Terms for the One-Level Density of a Symplectic Family of Hecke L-FunctionsMay 23 2019In this paper we apply the $L$-function Ratios Conjecture to compute the one-level density for a symplectic family of $L$-functions attached to Hecke characters of infinite order. When the support of the Fourier transform of the corresponding test function ... More

Effective approximation of heat flow evolution of the Riemann $ξ$ function, and a new upper bound for the de Bruijn-Newman constantApr 29 2019For each $t \in \mathbf{R}$, define the entire function $$ H_t(z) := \int_0^\infty e^{tu^2} \Phi(u) \cos(zu)\ du$$ where $\Phi$ is the super-exponentially decaying function $$ \Phi(u) := \sum_{n=1}^\infty (2\pi^2 n^4 e^{9u} - 3\pi n^2 e^{5u} ) \exp(-\pi ... More

A note on small gaps between zeros of the Riemann zeta-functionApr 12 2019Assuming the Riemann Hypothesis, we improve on previous results by proving there are infinitely many zeros of the Riemann zeta-function whose differences are smaller than 0.50412 times the average spacing. To obtain this result, we generalize a set of ... More

Inverse Problems of Determining Parameters of the Fractional Partial Differential EquationsApr 11 2019When considering fractional diffusion equation as model equation in analyzing anomalous diffusion processes, some important parameters in the model related to orders of the fractional derivatives, are often unknown and difficult to be directly measured, ... More

Inverse Problems of Determining Sources of the Fractional Partial Differential EquationsApr 11 2019In this chapter, we mainly review theoretical results on inverse source problems for diffusion equations with the Caputo time-fractional derivatives of order $\alpha\in(0,1)$. Our survey covers the following types of inverse problems: 1. determination ... More

Zeros of the extended Selberg class zeta-functions and of their derivativesApr 05 2019Levinson and Montgomery proved that the Riemann zeta-function $\zeta(s)$ and its derivative have approximately the same number of non-real zeros left of the critical line. R. Spira showed that $\zeta'(1/2+it)=0$ implies $\zeta(1/2+it)=0$. Here we obtain ... More

Series representation of a cotangent sum related to the Estermann zeta functionMar 01 2019In this paper, we are interested by the cotangent sum c0(q/p) related to the Estermann zeta function for the special case when q = 1 and get explicit formula for its series expansion, which represents an improvement of the identity (2:1) Theorem (2:1) ... More

Euler product asymptotics on Dirichlet L-functionsFeb 12 2019We derive the asymptotic behaviour of partial Euler products for Dirichlet $L$-functions $L(s, \chi)$ in the critical strip upon assuming only the Generalised Riemann Hypothesis (GRH). Capturing the behaviour of the partial Euler products on the critical ... More

A Note on the Riemann $ξ$-FunctionJan 21 2019Jan 24 2019This note investigates several integrals of Riemann's $\xi-$function, presents a different, and more general, derivation of one of his key identities. Two forms of an integral equation for $\xi$ are derived of possible interest with respect to the distribution ... More

A Note on the Riemann $ξ$-FunctionJan 21 2019Feb 15 2019This note investigates a number of integrals of and integral equations satisfied by Riemann's $\xi-$function. A different, less restrictive, derivation of one of his key identities is provided. This work centers on the critical strip and it is argued ... More

An Integral Equation for Riemann's Zeta Function and its Approximate SolutionJan 06 2019Jan 08 2019Two obscure identities extracted from the literature are coupled to obtain an integral equation for Riemann's $\xi(s)$ function, and thus $\zeta(s)$ indirectly. The equation has a number of simple properties from which useful derivations flow, the most ... More

High order numerical schemes for solving fractional powers of elliptic operatorsJan 01 2019In many recent applications when new materials and technologies are developed it is important to describe and simulate new nonlinear and nonlocal diffusion transport processes. A general class of such models deals with nonlocal fractional power elliptic ... More

On the zeros of sum from n=1 to 00 of lambda_P(n)/n^sDec 31 2018Our main result is to answer a question of Michel Balazard by giving a Dirichlet series with only one zero in its half-plane of convergence. At the end of the paper we also give several sufficient conditions for the Generalized Riemann Hypothesis and ... More

An arithmetical function related to Báez-Duarte's criterion for the Riemann hypothesisDec 11 2018In this mainly expository article, we revisit some formal aspects of B{\'a}ez-Duarte's criterion for the Riemann hypothesis. In particular, starting from Weingartner's formulation of the criterion, we define an arithmetical function $\nu$, which is equal ... More

On differential independence of $\mathboldζ$ and $\mathboldΓ$Nov 23 2018Dec 01 2018In this note, we will prove that $\mathbold{\zeta}$ and $\mathbold{\Gamma}$ can not satisfy any differential equation generated through a family of functions continuous in $\mathbold{\zeta}$ with polynomials in $\mathbold{\Gamma}$.

Weyl Asymptotics for Perturbations of Morse Potential and Connections to the Riemann Zeta FunctionNov 12 2018Let $N(T;V)$ denote the number of eigenvalues of the Schr\"odinger operator $-y'' + Vy$ with absolute value less than $T$. This paper studies the Weyl asymptotics of perturbations of the Schr\"odinger operator $-y'' + \frac{1}{4}e^{2t}y$ on $[x_0,\infty)$. ... More

Arguments Related to the Riemann Hypothesis: New Methods and ResultsNov 12 2018May 01 2019Four propositions are considered concerning the relationship between the zeros of two combinations of the Riemann zeta function and the function itself. The first is the Riemann hypothesis, while the second relates to the zeros of a derivative function. ... More

An Argument in Confirmation of the Riemann HypothesisNov 12 2018Dec 19 2018Based on results due to Taylor, Lagarias and Suzuki, and Ki, the relationship between the zeros of two combinations of the Riemann zeta function and the function itself is considered. A correspondence is established between the three sets of zeros, giving ... More

Efficient Implementation of Finite Element Methods for Spatial Fractional PDEs in Three DimensionsNov 08 2018Jan 21 2019In this paper, we address the main challenges of the implementation of finite element methods for solving spatial fractional problems on three dimensional irregular convex regions. Different from the integer case, the non-locality of fractional derivative ... More

Non-trivial zeros of Riemann's Zeta function via revised Euler-Maclaurin-Siegel and Abel-Plana summation formulasNov 03 2018Nov 15 2018This paper addresses the revised Euler-Maclaurin-Siegel and Abel-Plana summation formulas and proves the Riemann hypothesis with the aid of the critical strip and the Todd type functions for the first time. The distribution formulae of the prime numbers ... More

Low-lying zeros of L-functions for Quaternion AlgebrasOct 30 2018The density conjecture of Katz and Sarnak predicts that, for natural families of L-functions, the distribution of zeros lying near the real axis is governed by a group of symmetry. In the case of the universal family of automorphic forms of bounded analytic ... More

Euler's Function on Products of Primes in ProgressionsOct 26 2018We study generalizations of some results of Jean-Louis Nicolas regarding the relation between small values of Euler's function $\varphi(n)$ and the Riemann Hypothesis. Among other things, we prove that for $1\leq q\leq 10$ and for $q=12, 14$, the generalized ... More

Pair Correlation Estimates for the Zeros of the Zeta-Function via Semidefinite ProgrammingOct 20 2018In this paper we study the distribution of the non-trivial zeros of the zeta-function $\zeta(s)$ (and other L-functions) under Montgomery's pair correlation approach. We use semidefinite programming to improve the asymptotic bounds for $N^*(T)$, $N_d(T)$ ... More

New arithmetical proof of the reciprocity law for Dedekind sumsOct 14 2018In this paper, for coprime numbers p and q we consider the well known Dedekind sums S(p,q) First, we give an improvement of the proof given by H. Rademacher and A. Whiteman, and we construct a new arithmetical proof for the reciprocity law

On pairs, triples and quadruples of points on a cubic surfaceOct 13 2018Apr 22 2019Let $X^{(n)}$ denote $n$-th symmetric power of a cubic surface $X$. We show that $X^{(4)}\times X$ is stably birational to $X^{(3)}\times X$, despite examples when $X^{(4)}$ is not stably birational to $X^{(3)}$.

On pairs, triples and quadruples of points on a cubic surfaceOct 13 2018Apr 15 2019Let $X^{(n)}$ denote $n$-th symmetric power of a cubic surface $X$. We show that $X^{(4)}\times X$ is stably birational to $X^{(3)}\times X$, despite examples when $X^{(4)}$ is not stably birational to $X^{(3)}$.

On pairs, triples and quadruples of points on a cubic surfaceOct 13 2018Let $X^{(n)}$ denote $n$-th symmetric power of a cubic surface $X$. We show that $X^{(4)}\times X$ is stably birational to $X^{(3)}\times X$, despite examples when $X^{(4)}$ is not stably birational to $X^{(3)}$.

Euler products of Selberg zeta functions in the critical stripSep 24 2018Oct 21 2018For any congruence subgroup of the modular group, we extend the region of convergence of the Euler products of the Selberg zeta functions beyond the boundary Re s = 1, if they are attached with a nontrivial irreducible unitary representation. The region ... More

Asymptotics of Goldbach RepresentationsSep 18 2018Oct 08 2018We present a historical account of the asymptotics of classical Goldbach representations with special reference to the equivalence with the Riemann Hypothesis. When the primes are chosen from an arithmetic progression comparable but weaker relationships ... More

Primes in prime number racesSep 09 2018Jan 05 2019Rubinstein and Sarnak have shown, conditional on the Riemann hypothesis (RH) and the linear independence hypothesis (LI) on the non-real zeros of $\zeta(s)$, that the set of real numbers $x\ge2$ for which $\pi(x)>$ li$(x)$ has a logarithmic density, which ... More

A proof of the Riemann hypothesis on zeros of $ζ-$functionAug 30 2018Sep 03 2018Applying the known Nyman--Beurling criterion, it is proved the Riemann hypothesis on zeros of $\zeta -$function.

A disproof of the Riemann hypothesis on zeros of $ζ-$functionAug 30 2018Apr 11 2019Applying the known Nyman--Beurling criterion, it is disproved the Riemann hypothesis on zeros of $\zeta -$function.

Approximating the Riemann Zeta-function by Polynomials with Restricted ZerosAug 07 2018Aug 09 2018We approximate the Riemann Zeta-Function by polynomials and Dirichlet polynomials with restricted zeros.

Inequalities For The Primes Counting FunctionAug 03 2018The prime counting function inequality $\pi(x+y) < \pi(x)+\pi(y)$, which is known as Hardy-Littlewood conjecture, has been established for a variety of cases such as $ \delta x \leq y \leq x$, where $0< \delta \leq 1$, and $x \leq y\leq x \log x \log ... More

Extreme values for $S_n(σ,t)$ near the critical lineJul 31 2018Let $S(\sigma,t)=\frac{1}{\pi}\arg\zeta(\sigma+it)$ be the argument of the Riemann zeta function at the point $\sigma+it$ of the critical strip. For $n\geq 1$ and $t>0$ we define $$ S_{n}(\sigma,t) = \int_0^t S_{n-1}(\sigma,\tau)\,d\tau\, + \delta_{n,\sigma\,}, ... More

A possible new path to proving the Riemann HypothesisJul 02 2018Sep 30 2018In the past 100 years, the research of Riemann Hypothesis meets many difficulties. Such situation may be caused by that people used to study Zeta function only regarding it as a complex function. Generally, complex functions are far more complex than ... More

Homotopy types and geometries below Spec ZJun 28 2018Aug 25 2018After the first heuristic ideas about `the field of one element' F_1 and `geometry in characteristics 1' (J.~Tits, C.~Deninger, M.~Kapranov, A.~Smirnov et al.), there were developed several general approaches to the construction of `geometries below Spec ... More

Weighted distribution of low-lying zeros of GL(2) L-functionsJun 15 2018Sep 12 2018We show that if the zeros of an automorphic $L$-function are weighted by the central value of the $L$-function or a quadratic imaginary base change, then for certain families of holomorphic GL(2) newforms, it has the effect of changing the distribution ... More

On the parabolic Harnack inequality for non-local diffusion equationsJun 12 2018We settle the open question concerning the Harnack inequality for globally positive solutions to non-local in time diffusion equations by constructing a counter-example for dimensions $d\ge\beta$, where $\beta\in(0,2]$ is the order of the equation with ... More

On Studying the Phase Behavior of the Riemann Zeta Function Along the Critical LineJun 01 2018The critical line of the Riemann zeta function is studied from a new viewpoint. It is found that the ratio between the zeta function at any zero and the corresponding one at a conjugate point has a certain phase and its absolute value is unity. This fact ... More

Generalized Volterra functions, its integral representations and applications to the Mathieu--type seriesMay 29 2018May 31 2018In this paper we introduce the new class of generalized Volterra functions. We prove some integral representations for them via Fox-Wright H-functions and Meijer G-functions. From positivity conditions on the weight in these representations, we found ... More

The Riemann-Roch strategy, Complex lift of the Scaling SiteMay 26 2018We describe the Riemann-Roch strategy which consists of adapting in characteristic zero Weil's proof, of RH in positive characteristic, following the ideas of Mattuck, Tate and Grothendieck. As a new step in this strategy we implement the technique of ... More

100% of the zeros of the Riemann zeta-function are on the critical lineMay 20 2018Jan 09 2019We consider a specific family of analytic functions $g_{\alpha,T}(s)$, satisfying certain functional equations and approximating to linear combinations of the Riemann zeta-function and its derivatives of the form $c_0\zeta(s)+c_1\frac{\zeta'(s)}{\log ... More

Coefficients and higher order derivatives of cyclotomic polynomials: old and newMay 14 2018Aug 22 2018The $n^{th}$ cyclotomic polynomial $\Phi_n(x)$ is the minimal polynomial of an $n^{th}$ primitive root of unity. Its coefficients are the subject of intensive study and some formulas are known for them. Here we are interested in formulas which are valid ... More

Some Order-Theoretic Properties of the Zeros of the Zeta FunctionMay 04 2018The (partially) ordered set of the non-trivial zeros of the zeta function with positive imaginary parts is considered. The order is the coordinatewise order inherited from $\mathbb{C}$. Some interesting properties regarding the minimal elements of this ... More

Monotonicity Properties and functional inequalities for the Volterra and incomplete Volterra functionsMay 03 2018Jun 01 2018In this paper we prove some monotonicity, log--convexity and log--concavity properties for the Volterra and incomplete Volterra functions. Moreover, as consequences of these results, we present some functional inequalities (like Tur\'an type inequalities) ... More

A note on entire $L$-functionsMay 03 2018In this paper, we exhibit upper and lower bounds with explicit constants for some objects related to entire $L$-functions in the critical strip, under the generalized Riemann hypothesis. The examples include the entire Dirichlet $L$-functions $L(s,\chi)$ ... More

An upper bound for discrete moments of the derivative of the Riemann zeta-functionApr 24 2018Assuming the Riemann hypothesis, we establish an upper bound for the $2k$-th discrete moment of the derivative of the Riemann zeta-function at nontrivial zeros, where $k$ is a positive real number. Our upper bound agrees with conjectures of Gonek and ... More

The Lerch zeta function as a fractional derivativeApr 21 2018We derive and prove a new formulation of the Lerch zeta function as a fractional derivative of an elementary function. We demonstrate how this formulation interacts very naturally with basic known properties of Lerch zeta, and use the functional equation ... More

The admissible domain of the non-trivial zeros of the Riemann zeta functionApr 04 2018Jul 15 2018The zeros of the Riemann zeta function outside the critical strip are the trivial zeros. Using the Riemann zeta functional, we obtain the constraint $\zeta(s) = \zeta(1-\bar{s})$ that we refer to as the zeta minimal constraint, where $s$ is a complex ... More

Non-polar singularities of local zeta functions in some smooth caseMar 27 2018It is known that local zeta functions associated with real analytic functions can be analytically continued as meromorphic functions to the whole complex plane. In this paper, the case of specific (non-real analytic) smooth functions is precisely investigated. ... More

Efficient high order algorithms for fractional integrals and fractional differential equationsMar 01 2018Aug 30 2018We propose an efficient algorithm for the approximation of fractional integrals by using Runge--Kutta based convolution quadrature. The algorithm is based on a novel integral representation of the convolution weights and a special quadrature for it. The ... More

Combinatorial applications of autocorrelation ratiosFeb 28 2018The second moment of the Riemann zeta-function twisted by a normalized Dirichlet polynomial with coefficients of the form $(\mu \star \Lambda_1^{\star k_1} \star \Lambda_2^{\star k_2} \star \cdots \star \Lambda_d^{\star k_d})$ is computed unconditionally ... More

The twisted mean square and critical zeros of Dirichlet $L$-functionsFeb 27 2018The asymptotic formula for mean square of the Riemann zeta-function times a Dirichlet polynomial of length $T^\theta$ is proved when $\theta<17/33$ and $\theta<4/7$ for a special form of the coefficient, while for a general Dirichlet $L$-function, it ... More

Explicit bounds for primes in arithmetic progressionsJan 31 2018We derive explicit upper bounds for various functions counting primes in arithmetic progressions. By way of example, if $q$ and $a$ are integers with $\gcd(a,q)=1$ and $3 \leq q \leq 10^5$, and $\theta(x;q,a)$ denotes the sum of the logarithms of the ... More

Explicit bounds for primes in arithmetic progressionsJan 31 2018Nov 27 2018We derive explicit upper bounds for various functions counting primes in arithmetic progressions. By way of example, if $q$ and $a$ are integers with $\gcd(a,q)=1$ and $3 \leq q \leq 10^5$, and $\theta(x;q,a)$ denotes the sum of the logarithms of the ... More

Summatory function of the number of prime factorsJan 21 2018We consider the summatory function of the number of prime factors for integers $\leq x$ over arithmetic progressions. Numerical experiments suggest that some arithmetic progressions consist more number of prime factors than others. Greg Martin conjectured ... More

Double Dirichlet series associated with arithmetic functionsJan 13 2018Oct 10 2018We consider double Dirichlet series associated with arithmetic functions such as the von Mangoldt function, the M\"obius function, and so on. We show analytic continuations of them by use of the Mellin-Barnes integral. Furthermore we observe their reverse ... More

Zeros of combinations of the Riemann $Ξ$-function and the confluent hypergeometric function on bounded vertical shiftsDec 22 2017In 1914, Hardy proved that infinitely many non-trivial zeros of the Riemann zeta function lie on the critical line using the transformation formula of the Jacobi theta function. Recently the first author obtained an integral representation involving the ... More

Riemann hypothesisNov 29 2017Feb 01 2019This work is dedicated to the promotion of the results Hadamard, Landau E., Walvis A., Estarmann T and Paul R. Chernoff for pseudo zeta functions. The properties of zeta functions are studied, these properties can lead to new regularities of zeta functions. ... More

Average Goldbach and the Quasi-Riemann HypothesisNov 17 2017We prove that a good average order on the Goldbach generating function implies that the real parts of the non-trivial zeros of the Riemann zeta function are strictly less than 1. This together with existing results establishes an equivalence between such ... More

Bandlimited approximations and estimates for the Riemann zeta-functionOct 28 2017In this paper, we provide explicit upper and lower bounds for the argument of the Riemann zeta-function and its antiderivatives in the critical strip under the assumption of the Riemann hypothesis. This extends the previously known bounds for these quantities ... More

Low-lying zeros of quadratic Dirichlet $L$-functions: A transition in the Ratios ConjectureOct 18 2017We study the $1$-level density of low-lying zeros of quadratic Dirichlet $L$-functions by applying the $L$-functions Ratios Conjecture. We observe a transition in the main term as was predicted by the Katz-Sarnak heuristic as well as in the lower order ... More

Inclusive Prime Number RacesSep 29 2017Let $\pi(x;q,a)$ denote the number of primes up to $x$ that are congruent to $a$ modulo $q$. A prime number race, for fixed modulus $q$ and residue classes $a_1, \ldots, a_r$ investigates the system of inequalities $\pi(x;q,a_1) > \pi(x;q,a_2) > \cdots ... More

Least Prime Primitive RootsSep 01 2017This note presents an upper bound for the least prime primitive roots $g^*(p)$ modulo $p$, a large prime. The current literature has several estimates of the least prime primitive root $g^*(p)$ modulo a prime $p\geq 2$ such as $g^*(p)\ll p^c, c>2.8$. ... More

Fourier optimization and prime gapsAug 14 2017Sep 14 2018We investigate some extremal problems in Fourier analysis and their connection to a problem in prime number theory. In particular, we improve the current bounds for the largest possible gap between consecutive primes assuming the Riemann hypothesis.

Variants of the Riemann zeta functionAug 08 2017Aug 11 2017We construct variants of the Riemann zeta function with convenient properties and make conjectures about their dynamics; some of the conjectures are based on an analogy with the dynamical system of zeta. More specifically, we study the family of functions ... More

Towards a spectral proof of Riemann's hypothesisAug 01 2017The paper presents evidence that Riemann's xi function evaluated at 2 sqrt(E) could be the characteristic function P(E) for the magnetic Laplacian minus 85/16 on a surface of curvature -1 with magnetic field 9/4, a cusp of width 1, a DIrichlet condition ... More

On $r$-gaps between zeros of the Riemann zeta-functionJul 31 2017Dec 28 2017Under the Riemann Hypothesis, we prove for any natural number $r$ there exist infinitely many large natural numbers $n$ such that $(\gamma_{n+r}-\gamma_n)/(2\pi /\log \gamma_n) > r + \Theta\sqrt{r}$ and $(\gamma_{n+r}-\gamma_n)/(2\pi /\log \gamma_n) < ... More

A new reason for doubting the Riemann hypothesisJun 29 2017Jul 06 2017We make plausible the existence of counterexamples to the Riemann hypothesis located in the neighbourhood of unusually large peaks of $\vert \zeta \vert$. The main ingredient in our argument is an identity which links the zeros of a function $f$ defined ... More

Estimates of sums related to the Nyman-Beurling criterion for the Riemann HypothesisMay 28 2017We give an estimate for sums appearing in the Nyman-Beurling criterion for the Riemann Hypothesis containing the M\"obius function. The estimate is remarkably sharp in comparison to estimates of other sums containing the M\"obius function. The methods ... More

A criterion related to the Riemann HypothesisMay 28 2017A crucial role in the Nyman-Beurling-B\'aez-Duarte approach to the Riemann Hypothesis is played by the distance \[ d_N^2:=\inf_{A_N}\frac{1}{2\pi}\int_{-\infty}^\infty\left|1-\zeta A_N\left(\frac{1}{2}+it\right)\right|^2\frac{dt}{\frac{1}{4}+t^2}\:, \] ... More

On Müntz-type formulas related to the Riemann zeta functionMay 25 2017The Mellin transform and several Dirichlet series related with the Riemann zeta function are used to deduce some identities similar to the classical M\"untz formula [4]. These formulas are derived in the critical strip and in the half-plane $Re(s)<0$. ... More

Goldbach Representations in Arithmetic Progressions and zeros of Dirichlet L-functionsApr 20 2017Nov 17 2017Assuming a conjecture on distinct zeros of Dirichlet L-functions we get asymptotic results on the average number of representations of an integer as the sum of two primes in arithmetic progression. On the other hand the existence of good error terms gives ... More

On Nontrivial Zeros of Riemann Zeta FunctionApr 17 2017Jul 17 2018Let {\Xi} be a function relating to the Riemann zeta function with . In this paper, we construct a function containing and {\Xi} , and prove that satisfies a nonadjoint boundary value problem to a nonsingular differential equation if is any nontrivial ... More

Reconstruction of the Temporal Component in the Source Term of a (Time-Fractional) Diffusion EquationApr 13 2017In this article, we consider the reconstruction of $\rho(t)$ in the (time-fractional) diffusion equation $(\partial_t^\alpha-\triangle)u(x,t)=\rho(t)g(x)$ ($0<\alpha \le 1$) by the observation at a single point $x_0$. We are mainly concerned with the ... More

Numerical solution of time-dependent problems with fractional power elliptic operatorApr 12 2017An unsteady problem is considered for a space-fractional equation in a bounded domain. A first-order evolutionary equation involves a fractional power of an elliptic operator of second order. Finite element approximation in space is employed. To construct ... More

Zeros of Dirichlet Polynomials via a Density CriterionApr 05 2017We obtain a necessary and sufficient condition in order that a semi-plane of the form $\Re(s)>r$, $r\in \mathbb{R}$, is free of zeros of a given Dirichlet polynomial. The result may be considered a natural generalization of a well-known criterion for ... More

Standard Zero-Free Regions for Rankin--Selberg L-Functions via Sieve TheoryMar 16 2017Dec 10 2017We give a simple proof of a standard zero-free region in the $t$-aspect for the Rankin--Selberg $L$-function $L(s,\pi \times \widetilde{\pi})$ for any unitary cuspidal automorphic representation $\pi$ of $\mathrm{GL}_n(\mathbb{A}_F)$ that is tempered ... More

Discretized Keiper/Li approach to the Riemann HypothesisMar 08 2017Jul 17 2017The Keiper--Li sequence $\{ \lambda _n \}$ is most sensitive to the Riemann Hypothesis asymptotically ($n \to \infty$), but highly elusive both analytically and numerically. We deform it to fully explicit sequences, simpler to analyze and to compute (up ... More

Bounding $S_n(t)$ on the Riemann hypothesisFeb 14 2017Let $S(t) = \tfrac{1}{\pi} \arg \zeta (\frac12 + it)$ be the argument of the Riemann zeta-function at the point $\tfrac12 + it$. For $n \geq 1$ and $t>0$ define its iterates \begin{equation*} S_n(t) = \int_0^t S_{n-1}(\tau) \,{\rm d}\tau\, + \delta_n\,, ... More

Lehmer pairs and derivatives of Hardy's $Z$-functionNov 27 2016Apr 15 2017Occurrences of very close zeros of the Riemann zeta function are strongly connected with Lehmer pairs and with the Riemann Hypothesis. The aim of the present note is to derive a condition for a pair of consecutive simple zeros of the $\zeta$-function ... More

The Geometry of the Mappings by General Dirichlet SeriesOct 26 2016We dealt in a series of previous publications with some geometric aspects of the mappings by functions obtained as analytic continuations to the whole complex plane of general Dirichlet series. Pictures illustrating those aspects contain a lot of other ... More

The Geometry of the Mappings by General Dirichlet SeriesOct 26 2016Nov 15 2016We dealt in a series of previous publications with some geometric aspects of the mappings by functions obtained as analytic continuations to the whole complex plane of general Dirichlet series. Pictures illustrating those aspects contain a lot of other ... More

Zeros of Lattice Sums: 3. Reduction of the Generalised Riemann Hypothesis to Specific GeometriesOct 22 2016Apr 08 2017The location of zeros of the basic double sum over the square lattice is studied. This sum can be represented in terms of the product of the Riemann zeta function and the Dirichlet beta function, so that the assertion that all its non-trivial zeros lie ... More

Zeros of Lattice Sums: 3. Reduction of the Generalised Riemann Hypothesis to Specific GeometriesOct 22 2016The location of zeros of the basic double sum over the square lattice is studied. This sum can be represented in terms of the product of the Riemann zeta function and the Dirichlet beta function, so that the assertion that all its non-trivial zeros lie ... More

Polymeric quantum mechanics and the zeros of the Riemann zeta functionOct 06 2016Oct 12 2017We analize the Berry-Keating model and the Sierra and Rodr\'iguez-Laguna Hamiltonian within the polymeric quantization formalism. By using the polymer representation, we obtain for both models, the associated polymeric quantum Hamiltonians and the corresponding ... More

Polymeric quantum mechanics and the zeros of the Riemann zeta functionOct 06 2016We analize the Berry-Keating model and the Sierra and Rodr\'iguez-Laguna Hamiltonian within the polymeric quantization formalism. By using the polymer representation, we obtain for both models, the associated polymeric quantum Hamiltonians and the corresponding ... More

Polymeric quantum mechanics and the zeros of the Riemann zeta functionOct 06 2016Mar 08 2019We analize the Berry-Keating model and the Sierra and Rodr\'iguez-Laguna Hamiltonian within the polymeric quantization formalism. By using the polymer representation, we obtain for both models, the associated polymeric quantum Hamiltonians and the corresponding ... More

The probability of Riemann's hypothesis being true is equal to 1Sep 24 2016Apr 26 2018Let $P$ be the set of all prime numbers, ${q_1},{q_2}, \cdots ,{q_m} \in P$, $P_k$ be the k-th $(k = 1,2, \cdots m)$ element of $P$ in ascending order of size, ${\alpha _1},{\alpha _2}, \cdots ,{\alpha _m}$ be positive integers, and ${\beta _1},{\beta ... More

Dynamic behavior of the roots of the Taylor polynomials of the Riemann xi function with growing degreeSep 19 2016We establish a uniform approximation result for the Taylor polynomials of the xi function of Riemann which is valid in the entire complex plane as the degree grows. In particular, we identify a domain growing with the degree of the polynomials on which ... More

Further Exploration of Riemann's Functional EquationSep 10 2016A previous exploration of the Riemann functional equation that focussed on the critical line, is extended over the complex plane. Significant results include a simpler derivation of the fundamental equation developed previously, and its generalization ... More

Further Exploration of Riemann's Functional EquationSep 10 2016Nov 09 2016A previous exploration of the Riemann functional equation that focussed on the critical line, is extended over the complex plane. Significant results include a simpler derivation of the fundamental equation developed previously, and its generalization ... More

A quadratic divisor problem and moments of the Riemann zeta-functionSep 08 2016We estimate asymptotically the fourth moment of the Riemann zeta-function twisted by a Dirichlet polynomial of length $T^{\frac14 - \varepsilon}$. Our work relies crucially on Watt's theorem on averages of Kloosterman fractions. In the context of the ... More

Discrete mean square of the Riemann zeta-function over imaginary parts of its zerosAug 30 2016Assume the Riemann hypothesis. In the right-hand side of the critical strip we obtain an asymptotic formula for the discrete mean square of the Riemann zeta-function over imaginary parts of its zeros.

Discrete mean square of the Riemann zeta-function over imaginary parts of its zerosAug 30 2016Dec 07 2017Assume the Riemann hypothesis. On the right-hand side of the critical strip, we obtain an asymptotic formula for the discrete mean square of the Riemann zeta-function over imaginary parts of its zeros.

An extremal problem related to generalizations of the Nyman-Beurling and Báez-Duarte criteriaAug 29 2016We establish generalizations of the Nyman-Beurling and B\'aez-Duarte criteria concerning lack of zeros of Dirichlet $L$-functions in the semi-plane $\Re(s) >1/p$ for $p\in (1,2]$. We pose and solve a natural extremal problem for Dirichlet polynomials ... More

Lower bounds for Weil's Explicit Formula via Selberg's Trace FormulaAug 08 2016We show that the explicit formulae of Weil appear in the contribution of the continuous spectrum to the Arthur-Selberg trace formula for SL(2) in two ways: as a sum over zeroes, and as a sum of distributions. This follows from averaging the Maass-Selberg ... More

$p$-extended Mathieu series from the Schlömilch series point of viewJun 29 2016Motivated by certain current results by Parmar and Pog\'any [9] in which the authors introduced the so-called $p$-extended Mathieu series the main aim of this paper is to present a connection between such series and a various types of Schl\"omilch series. ... More