Concept

Riemann Xi function

Related publications (12)

On the Sums over Inverse Powers of Zeros of the Hurwitz Zeta Function and Some Related Properties of These Zeros

Recently, we have applied the generalized Littlewood theorem concerning contour integrals of the logarithm of the analytical function to find the sums over inverse powers of zeros for the incomplete gamma and Riemann zeta functions, polygamma functions, an ...

MDPI2024

On the Use of the Generalized Littlewood Theorem Concerning Integrals of the Logarithm of Analytical Functions for the Calculation of Infinite Sums and the Analysis of Zeroes of Analytical Functions

Recently, we have established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of an analytical function to obtain a few new criteria equivalent to the Riemann hypothesis. Here, the same theorem is applied to calcul ...

MDPI2023

Power variations in fractional Sobolev spaces for a class of parabolic stochastic PDEs

Robert Dalang, Carsten Hao Ye Chong

We consider a class of parabolic stochastic PDEs on bounded domains D c Rd that includes the stochastic heat equation but with a fractional power gamma of the Laplacian. Viewing the solution as a process with values in a scale of fractional Sobolev spaces ...

INT STATISTICAL INST2023

On the Density of Coprime Tuples of the Form (n, ⌊f1(n)⌋, …, ⌊fk(n)⌋), Where f_1, …, f_k Are Functions from a Hardy Field

Florian Karl Richter

Let k∈Nk∈Nk \in \mathbb{N} and let f1, …, f k belong to a Hardy field. We prove that under some natural conditions on the k-tuple ( f1, …, f k ) the density of the set {n∈N:gcd(n,⌊f1(n)⌋,…,⌊fk(n)⌋)=1}{n∈N:gcd(n,⌊f1(n)⌋,…,⌊fk(n)⌋)=1}\displaystyle{\big{n \i ...

Springer International Publishing2017

Bilinear forms with Kloosterman sums and applications

Philippe Michel

We prove nontrivial bounds for general bilinear forms in hyper-Kloosterman sums when the sizes of both variables may be below the range controlled by Fourier-analytic methods (Polya-Vinogradov range). We then derive applications to the second moment of cus ...

Annal Mathematics2017

Algebraic twists of modular forms and Hecke orbits

Philippe Michel

We consider the question of the correlation of Fourier coefficients of modular forms with functions of algebraic origin. We establish the absence of correlation in considerable generality (with a power saving of Burgess type) and a corresponding equidistri ...

Springer Verlag2015

Algebraic Trace Functions Over The Primes

Philippe Michel

We study sums over primes of trace functions of l-adic sheaves. Using an extension of our earlier results on algebraic twists of modular forms to the case of Eisenstein series and bounds for Type II sums based on similar applications of the Riemann hypothe ...

Duke Univ Press2014

Generalized Bombieri-Lagarias' Theorem And Generalized Li'S Criterion With Its Arithmetic Interpretation

We show that Li's criterion equivalent to the Riemann hypothesis, i.e., the statement that the sums k(n) = Sigma(rho)(1-(1-1/rho)(n)) over zeros of the Riemann xi-function and the derivatives lambda(n) equivalent to 1/(n-1)! d(n)/dz(n)(z(n-1) ln (xi(z)))ve ...

Springer Verlag2014

Some applications of the Weil-Barner explicit formula

Thorsten Kleinjung

The Weil-Barner explicit formula is applied to the prime counting function and to the problem of numerical partial verification of the Riemann hypothesis. ...

Wiley-V C H Verlag Gmbh2013

On equalities involving integrals of the logarithm of the Riemann zeta-function and equivalent to the Riemann hypothesis

By using the generalized Littlewood theorem about a contour integral involving the logarithm of an analytic function, we show how an infinite number of integral equalities involving integrals of the logarithm of the Riemann zeta-function and equivalent to ...

Springer US2012

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