A note on Li’s coefficients for Hecke L-functions

2017 ◽  
Vol 13 (07) ◽  
pp. 1747-1753
Author(s):  
Sami Omar ◽  
Raouf Ouni
Keyword(s):  
Number Theory ◽  
Riemann Hypothesis ◽  
General Class ◽  
Numerical Experiments ◽  
Selberg Class ◽  
Numerical Computations ◽  
Math Phys

Recently, the Li criterion for the Riemann hypothesis has been extended for a general class of [Formula: see text]-functions, so-called the Selberg class [S. Omar and K. Mazhouda, Le critère de Li et l’hypothèse de Riemann pour la classe de Selberg, J. Number Theory 125(1) (2007) 50–58; Corrigendum et addendum à “Le critère de Li et l’hypothèse de Riemann pour la classe de Selberg” [J. Number Theory 125(1) (2007) 50–58], J. Number Theory 130(4) (2010) 1109–1114]. Further numerical computations have been done to verify the positivity of some Li coefficients for the Dirichlet [Formula: see text]-functions and the Hecke [Formula: see text]-functions [S. Omar, R. Ouni and K. Mazhouda, On the zeros of Dirichlet [Formula: see text]-functions, LMS J. Comput. Math. 14 (2011) 140–154; On the Li coefficients for the Hecke [Formula: see text]-functions, Math. Phys. Anal. Geom. 17(1–2) (2014) 67–81]. Based on the latter numerical experiments, it was conjectured that those coefficients are increasing in [Formula: see text]. In this note, we show actually that the Riemann hypothesis holds if and only if the Li coefficients for the Hecke [Formula: see text]-functions are increasing in [Formula: see text].

scholarly journals The Complexity of Mathematics

2020 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Number Theory ◽  
Complexity Theory ◽  
Riemann Hypothesis ◽  
Open Problems ◽  
Mathematical Proofs ◽  
Pure Mathematics ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Goldbach’S Conjecture ◽  
Goldbach's Conjecture

In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in pure mathematics. It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a US 1,000,000 prize for the first correct solution. We prove the Riemann hypothesis using the Complexity Theory. Number theory is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. The Goldbach's conjecture is one of the most important and unsolved problems in number theory. Nowadays, it is one of the open problems of Hilbert and Landau. We show the Goldbach's conjecture is true using the Complexity Theory as well. An important complexity class is 1NSPACE(S(n)) for some S(n). These mathematical proofs are based on if some unary language belongs to 1NSPACE(S(log n)), then the binary version of that language belongs to 1NSPACE(S(n)) and vice versa.

A Proof to the Riemann Hypothesis

2020 ◽  
Author(s):  
Sourangshu Ghosh
Keyword(s):  
Number Theory ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Prime Numbers ◽  
Complex Numbers ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

In this paper, we shall try to prove the Riemann Hypothesis which is a conjecture that the Riemann zeta function hasits zeros only at the negative even integers and complex numbers with real part ½. This conjecture is very importantand of considerable interest in number theory because it tells us about the distribution of prime numbers along thereal line. This problem is one of the clay mathematics institute’s millennium problems and also comprises the 8ththe problem of Hilbert’s famous list of 23 unsolved problems. There have been many unsuccessful attempts in provingthe hypothesis. In this paper, we shall give proof to the Riemann Hypothesis.

scholarly journals The average multiplicative order of a finitely generated subgroup of the rationals modulo primes

2016 ◽  
Vol 12 (08) ◽  
pp. 2147-2158
Author(s):  
Cihan Pehlivan
Keyword(s):  
Prime Number ◽  
Explicit Expression ◽  
Riemann Hypothesis ◽  
Multiplicative Group ◽  
Prime Numbers ◽  
Euler Product ◽  
Finitely Generated ◽  
Numerical Computations ◽  
Multiplicative Order ◽  
Reduction Group

Let [Formula: see text] be a finitely generated subgroup of [Formula: see text]. Let [Formula: see text] be a prime number for which the reduction group [Formula: see text] is a well-defined subgroup of the multiplicative group [Formula: see text], and denote the order of [Formula: see text] by [Formula: see text]. Assuming the Generalized Riemann Hypothesis, we study the average of [Formula: see text] and the average of the powers of [Formula: see text] as [Formula: see text] ranges over prime numbers. The problem was considered in the case of rank 1 by Pomerance and Kurlberg. In the case when [Formula: see text] contains only positive numbers, we give an explicit expression for the involved density in terms of an Euler product. We conclude with some numerical computations.

EXTREME VALUES OF L-FUNCTIONS FROM THE SELBERG CLASS

2013 ◽  
Vol 09 (05) ◽  
pp. 1113-1124 ◽  
Author(s):  
ŁUKASZ PAŃKOWSKI ◽  
JÖRN STEUDING
Keyword(s):  
Diophantine Approximation ◽  
Riemann Hypothesis ◽  
Approximation Theorem ◽  
Extreme Values ◽  
Selberg Class ◽  
Effective Version

We prove estimates for extreme values of L-functions from the Selberg class under assumption of the corresponding analogue of the Riemann hypothesis. The method of proof combines Montgomery's approach with an effective version of Kronecker's diophantine approximation theorem due to Weber.

scholarly journals Fractional calculus and integral transforms of the product of a general class of polynomial and incomplete Fox–Wright functions

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
K. Jangid ◽  
R. K. Parmar ◽  
R. Agarwal ◽  
Sunil D. Purohit
Keyword(s):  
Fractional Calculus ◽  
General Class ◽  
Hypergeometric Functions ◽  
Fractional Integral ◽  
Differential Operators ◽  
Integral Transforms ◽  
Parameter Selection ◽  
Fractional Operators ◽  
Polynomial Class ◽  
Math Phys

Abstract Motivated by a recent study on certain families of the incomplete H-functions (Srivastava et al. in Russ. J. Math. Phys. 25(1):116–138, 2018), we aim to investigate and develop several interesting properties related to product of a more general polynomial class together with incomplete Fox–Wright hypergeometric functions ${}_{p}\Psi _{q}^{(\gamma )}(\mathfrak{t})$ Ψ q ( γ ) p ( t ) and ${}_{p}\Psi _{q}^{(\Gamma )}(\mathfrak{t})$ Ψ q ( Γ ) p ( t ) including Marichev–Saigo–Maeda (M–S–M) fractional integral and differential operators, which contain Saigo hypergeometric, Riemann–Liouville, and Erdélyi–Kober fractional operators as particular cases regarding different parameter selection. Furthermore, we derive several integral transforms such as Jacobi, Gegenbauer (or ultraspherical), Legendre, Laplace, Mellin, Hankel, and Euler’s beta transforms.

The Complexity of Mathematics

2020 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Number Theory ◽  
Complexity Theory ◽  
Riemann Hypothesis ◽  
Open Problems ◽  
Mathematical Proofs ◽  
Pure Mathematics ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Goldbach’S Conjecture ◽  
Goldbach's Conjecture

In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in pure mathematics. It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a US 1,000,000 prize for the first correct solution. We prove the Riemann hypothesis using the Complexity Theory. Number theory is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. The Goldbach's conjecture is one of the most important and unsolved problems in number theory. Nowadays, it is one of the open problems of Hilbert and Landau. We show the Goldbach's conjecture is true using the Complexity Theory as well. An important complexity class is 1NSPACE(S(n)) for some S(n). These mathematical proofs are based on if some unary language belongs to 1NSPACE(S(log n)), then the binary version of that language belongs to 1NSPACE(S(n)) and vice versa.

Short intervals containing numbers without large prime factors

1991 ◽  
Vol 109 (1) ◽  
pp. 1-5 ◽  
Author(s):  
Glyn Harman
Keyword(s):  
Number Theory ◽  
Lower Bound ◽  
Riemann Hypothesis ◽  
Short Intervals ◽  
Prime Factors ◽  
Image Position

We denote, as usual, the number of integers not exceeding x having no prime factors greater than y by Ψ(x, y). We also writeThe function Ψ(x, y) is of great interest in number theory and has been studied by many researchers (see [3], [5] and [6] for example). The function Ψ(x, z, y) has also received some attention (see [2], [4–6]). In this paper we shall try to obtain a positive lower bound for Ψ(x, z, y) with y as small as possible when z is about x½ in magnitude. We note that the approach in [5] and [6] allows y to be much smaller than is permissible here, but requires x/z to be smaller than any power of x in [6] (unless some conjecture like the Riemann Hypothesis is assumed), or needs in [5]. The following result was obtained by Balog[1].

On the Number of Solutions of Some General Types of Equations in a Finite Field

1952 ◽  
Vol 4 ◽  
pp. 343-351 ◽  
Author(s):  
Olin B. Faircloth
Keyword(s):  
Number Theory ◽  
Finite Field ◽  
Riemann Hypothesis ◽  
Quadratic Forms ◽  
Function Fields ◽  
Commutative Rings ◽  
Number Of Solutions ◽  
Fermat's Last Theorem ◽  
Conditional Equation ◽  
And Algebra

The conditional equation f(x1, … , xs) = 0, where f is a polynomial in the x´s with coefficients in a finite field F(pn), is connected with many well-known developments in number theory and algebra, such as: Waring's problem, the arithmetical theory of quadratic forms, the Riemann hypothesis for function fields, Fermat's Last Theorem, cyclotomy, and the theory of congruences in commutative rings.

scholarly journals On the bounded generation of arithmetic SL2

2019 ◽  
Vol 116 (38) ◽  
pp. 18880-18882 ◽  
Author(s):  
Bruce W. Jordan ◽  
Yevgeny Zaytman
Keyword(s):  
Number Theory ◽  
Number Field ◽  
Riemann Hypothesis ◽  
Additional Hypothesis ◽  
Generalized Riemann Hypothesis ◽  
Group Of Units ◽  
Finite Set ◽  
Bounded Generation

Let K be a number field and S be a finite set of primes of K containing the archimedean valuations. Let 𝒪 be the ring of S-integers in K. Morgan, Rapinchuck, and Sury [A. V. Morgan et al., Algebra Number Theory 12, 1949–1974 (2018)] have proved that if the group of units 𝒪× is infinite, then every matrix in SL2(𝒪) is a product of at most 9 elementary matrices. We prove that under the additional hypothesis that K has at least 1 real embedding or S contains a finite place we can get a product of at most 8 elementary matrices. If we assume a suitable generalized Riemann hypothesis, then every matrix in SL2(𝒪) is the product of at most 5 elementary matrices if K has at least 1 real embedding, the product of at most 6 elementary matrices if S contains a finite place, and the product of at most 7 elementary matrices in general.

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