scholarly journals On Equalities Involving Integrals of the Logarithm of the Riemann ς-Function with Exponential Weight Which Are Equivalent to the Riemann Hypothesis

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Sergey K. Sekatskii ◽  
Stefano Beltraminelli ◽  
Danilo Merlini
Keyword(s):  
Riemann Hypothesis ◽  
Critical Line ◽  
Fourier Transforms ◽  
Riemann Function ◽  
Weight Functions ◽  
Exponential Weight ◽  
Elementary Functions ◽  
Definite Integrals ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Integral equalities involving integrals of the logarithm of the Riemann ς-function with exponential weight functions are introduced, and it is shown that an infinite number of them are equivalent to the Riemann hypothesis. Some of these equalities are tested numerically. The possible contribution of the Riemann function zeroes nonlying on the critical line is rigorously estimated and shown to be extremely small, in particular, smaller than nine milliards of decimals for the maximal possible weight function exp(−2πt). We also show how certain Fourier transforms of the logarithm of the Riemann zeta-function taken along the real (demi)axis are expressible via elementary functions plus logarithm of the gamma-function and definite integrals thereof, as well as certain sums over trivial and nontrivial Riemann function zeroes.

scholarly journals The Values of the Periodic Zeta-Function at the Nontrivial Zeros of Riemann’s Zeta-Function

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2410
Author(s):  
Janyarak Tongsomporn ◽  
Saeree Wananiyakul ◽  
Jörn Steuding
Keyword(s):  
Asymptotic Formula ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Real Line ◽  
Riemann Hypothesis ◽  
Critical Line ◽  
Nontrivial Zeros ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Riemann’S Zeta Function

In this paper, we prove an asymptotic formula for the sum of the values of the periodic zeta-function at the nontrivial zeros of the Riemann zeta-function (up to some height) which are symmetrical on the real line and the critical line. This is an extension of the previous results due to Garunkštis, Kalpokas, and, more recently, Sowa. Whereas Sowa’s approach was assuming the yet unproved Riemann hypothesis, our result holds unconditionally.

scholarly journals The Riemann hypothesis and the zero distribution of angular lattice sums

2011 ◽  
Vol 467 (2133) ◽  
pp. 2462-2478
Author(s):  
Ross C. McPhedran ◽  
Lindsay C. Botten ◽  
Dominic J. Williamson ◽  
Nicolae-Alexandru P. Nicorovici
Keyword(s):  
Riemann Hypothesis ◽  
Critical Line ◽  
Beta Function ◽  
Square Lattice ◽  
Distribution Of Zeros ◽  
Lattice Sums ◽  
Multiplicity One ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Complex Powers

We give analytical results pertaining to the distributions of zeros of a class of sums which involve complex powers of the distance to points in a two-dimensional square lattice and trigonometric functions of their angle. Let denote the product of the Riemann zeta function and the Catalan beta function, and let denote a particular set of angular sums. We then introduce a function that is the quotient of the angular lattice sums with , and use its properties to prove that obeys the Riemann hypothesis for any m if and only if obeys the Riemann hypothesis. We furthermore prove that if the Riemann hypothesis holds, then and have the same distribution of zeros on the critical line (in a sense made precise in the proof). We also show that if obeys the Riemann hypothesis and all its zeros on the critical line have multiplicity one, then all the zeros of every have multiplicity one. We give numerical results illustrating these and other results.

scholarly journals Riemann Hypothesis and Random Walks: The Zeta Case

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2014
Author(s):  
André LeClair
Keyword(s):  
Zeta Function ◽  
Riemann Hypothesis ◽  
Critical Line ◽  
Dirichlet Characters ◽  
The Riemann Zeta Function ◽  
Point Correlation ◽  
Point Correlation Function ◽  
Riemann Zeta ◽  
The Right ◽  
Very High

In previous work, it was shown that if certain series based on sums over primes of non-principal Dirichlet characters have a conjectured random walk behavior, then the Euler product formula for its L-function is valid to the right of the critical line ℜ(s)>12, and the Riemann hypothesis for this class of L-functions follows. Building on this work, here we propose how to extend this line of reasoning to the Riemann zeta function and other principal Dirichlet L-functions. We apply these results to the study of the argument of the zeta function. In another application, we define and study a one-point correlation function of the Riemann zeros, which leads to the construction of a probabilistic model for them. Based on these results we describe a new algorithm for computing very high Riemann zeros, and we calculate the googol-th zero, namely 10100-th zero to over 100 digits, far beyond what is currently known. Of course, use is made of the symmetry of the zeta function about the critical line.

scholarly journals A new approach towards solving the Riemann hypothesis

2021 ◽  
Author(s):  
Almouid Mohammed Hasibul Haque
Keyword(s):  
Fourier Analysis ◽  
Gamma Function ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Complex Analysis ◽  
Critical Line ◽  
New Approach ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Modern Mathematics

In this paper, I attempt to solve one of the most difficult problems in modern mathematics-'The Riemann Hypothesis'. I redefine the gamma function and use that modified form along with some identities from Fourier analysis and concepts from complex analysis to show that all the non-trivial zeros of the Riemann zeta function must lie on the critical line and then by recalling Hardy's theorem I prove the Riemann hypothesis.

scholarly journals Fourier coefficients associated with the Riemann zeta-function

2016 ◽  
Vol 8 (1) ◽  
pp. 16-20
Author(s):  
Yu.V. Basiuk ◽  
S.I. Tarasyuk
Keyword(s):  
Fourier Series ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Fourier Coefficients ◽  
Critical Line ◽  
Series Method ◽  
Fourier Series Method ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

We study the Riemann zeta-function $\zeta(s)$ by a Fourier series method. The summation of $\log|\zeta(s)|$ with the kernel $1/|s|^{6}$ on the critical line $\mathrm{Re}\; s = \frac{1}{2}$ is the main result of our investigation. Also we obtain a new restatement of the Riemann Hypothesis.

On the Size of an Expression in the Nyman–Beurling-Báez–Duarte Criterion for the Riemann Hypothesis

2018 ◽  
Vol 61 (3) ◽  
pp. 622-627
Author(s):  
Helmut Maier ◽  
Michael Th. Rassias
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Crucial Role ◽  
Riemann Hypothesis ◽  
Critical Line ◽  
Nontrivial Zeros ◽  
The Riemann Zeta Function ◽  
Dirichlet Polynomials ◽  
Riemann Zeta

AbstractA crucial role in the Nyman–Beurling–Báez-Duarte approach to the Riemann Hypothesis is played by the distancewhere the infimum is over all Dirichlet polynomialsof length N. In this paper we investigate under the assumption that the Riemann zeta function has four nontrivial zeros off the critical line.

scholarly journals On the value-distribution of iterated integrals of the logarithm of the Riemann zeta-function I: Denseness

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Kenta Endo ◽  
Shōta Inoue
Keyword(s):  
Zeta Function ◽  
Complex Plane ◽  
Riemann Zeta Function ◽  
Open Problem ◽  
Riemann Hypothesis ◽  
Critical Line ◽  
Iterated Integrals ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Iterated Integral

AbstractWe consider iterated integrals of {\log\zeta(s)} on certain vertical and horizontal lines. Here, the function {\zeta(s)} is the Riemann zeta-function. It is a well-known open problem whether or not the values of the Riemann zeta-function on the critical line are dense in the complex plane. In this paper, we give a result for the denseness of the values of the iterated integrals on the horizontal lines. By using this result, we obtain the denseness of the values of {\int_{0}^{t}\log\zeta(\frac{1}{2}+it^{\prime})\,dt^{\prime}} under the Riemann Hypothesis. Moreover, we show that, for any {m\geq 2}, the denseness of the values of an m-times iterated integral on the critical line is equivalent to the Riemann Hypothesis.

scholarly journals The Riemann Hypothesis is most likely true

2021 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Complex Numbers ◽  
Natural Numbers ◽  
Chebyshev Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Sum Of Divisors

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 $\frac{1}{2}$. The Riemann hypothesis belongs to the David Hilbert's list of 23 unsolved problems and it is one of the Clay Mathematics Institute's Millennium Prize Problems. The Robin criterion states that the Riemann hypothesis is true if and only if the inequality $\sigma(n)< e^{\gamma } \times n \times \log \log n$ holds for all natural numbers $n> 5040$, where $\sigma(x)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. The Nicolas criterion states that the Riemann hypothesis is true if and only if the inequality $\prod_{q \leq q_{n}} \frac{q}{q-1} > e^{\gamma} \times \log\theta(q_{n})$ is satisfied for all primes $q_{n}> 2$, where $\theta(x)$ is the Chebyshev function. Using both inequalities, we show that the Riemann hypothesis is most likely true.

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