scholarly journals Bounding the log-derivative of the zeta-function

2021 ◽  
Author(s):  
Andrés Chirre ◽  
Felipe Gonçalves
Keyword(s):  
Zeta Function ◽  
Riemann Hypothesis ◽  
Critical Strip ◽  
Explicit Bounds ◽  
Riemann’S Zeta Function

AbstractAssuming the Riemann hypothesis we establish explicit bounds for the modulus of the log-derivative of Riemann’s zeta-function in the critical strip.

scholarly journals An Integral Equation for Riemann’s Zeta Function and Its Approximate Solution

2020 ◽  
Vol 2020 ◽  
pp. 1-29
Author(s):  
Michael Milgram
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Zeta Function ◽  
Complex Plane ◽  
Strong Correlation ◽  
Riemann Hypothesis ◽  
Critical Strip ◽  
The Real ◽  
Analytic Expression ◽  
Riemann’S Zeta Function

Two identities extracted from the literature are coupled to obtain an integral equation for Riemann’s ξs function and thus ζs indirectly. The equation has a number of simple properties from which useful derivations flow, the most notable of which relates ζs anywhere in the critical strip to its values on a line anywhere else in the complex plane. From this, both an analytic expression for ζσ+it, everywhere inside the asymptotic t⟶∞ critical strip, as well as an approximate solution can be obtained, within the confines of which the Riemann Hypothesis is shown to be true. The approximate solution predicts a simple, but strong correlation between the real and imaginary components of ζσ+it for different values of σ and equal values of t; this is illustrated in a number of figures.

scholarly journals An Analytical and Numerical Detour for the Riemann Hypothesis

Information ◽  
2021 ◽  
Vol 12 (11) ◽  
pp. 483
Author(s):  
Michel Riguidel
Keyword(s):  
Riemann Hypothesis ◽  
Meromorphic Functions ◽  
Critical Line ◽  
Numerical Approach ◽  
Product Formula ◽  
Critical Strip ◽  
Associated Functions ◽  
Poles And Zeros ◽  
Riemann’S Zeta Function ◽  
Insight Into

From the functional equation of Riemann’s zeta function, this article gives new insight into Hadamard’s product formula. The function and its family of associated functions, expressed as a sum of rational fractions, are interpreted as meromorphic functions whose poles are the poles and zeros of the function. This family is a mathematical and numerical tool which makes it possible to estimate the value of the function at a point in the critical strip from a point on the critical line .Generating estimates of at a given point requires a large number of adjacent zeros, due to the slow convergence of the series. The process allows a numerical approach of the Riemann hypothesis (RH). The method can be extended to other meromorphic functions, in the neighborhood of isolated zeros, inspired by the Weierstraß canonical form. A final and brief comparison is made with the and functions over finite fields.

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 A HIDDEN SYMMETRY RELATED TO THE RIEMANN HYPOTHESIS WITH THE PRIMES INTO THE CRITICAL STRIP

2010 ◽  
Vol 03 (04) ◽  
pp. 555-563 ◽  
Author(s):  
Stefano Beltraminelli ◽  
Danilo Merlini ◽  
Sergey Sekatskii
Keyword(s):  
Numerical Experiment ◽  
Zeta Function ◽  
Riemann Hypothesis ◽  
Global Symmetry ◽  
Special Choice ◽  
Critical Strip ◽  
Hidden Symmetry ◽  
The Riemann Zeta Function ◽  
The Subject ◽  
Riemann Zeta

In this note concerning integrals involving the logarithm of the Riemann Zeta function, we extend some treatments given in previous pioneering works on the subject and introduce a more general set of Lorentz measures. We first obtain two new equivalent formulations of the Riemann Hypothesis (RH). Then with a special choice of the measure we formulate the RH as a "hidden symmetry", a global symmetry which connects the region outside the critical strip with that inside the critical strip. The Zeta function with all the primes appears as argument of the Zeta function in the critical strip. We then illustrate the treatment by means of a simple numerical experiment. The representation we obtain goes a little more in the direction to believe that RH may eventually be true.

scholarly journals A CENTRAL LIMIT THEOREM FOR THE ZEROES OF THE ZETA FUNCTION

2014 ◽  
Vol 10 (02) ◽  
pp. 483-511 ◽  
Author(s):  
BRAD RODGERS
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Zeta Function ◽  
Central Limit ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Riemann Hypothesis ◽  
Matrix Theory ◽  
Riemann’S Zeta Function

On the assumption of the Riemann hypothesis, we generalize a central limit theorem of Fujii regarding the number of zeroes of Riemann's zeta function that lie in a mesoscopic interval. The result mirrors results of Spohn and Soshnikov and others in random matrix theory. In an appendix we put forward some general theorems regarding our knowledge of the zeta zeroes in the mesoscopic regime.

On the explicit formula related to Riemann's zeta-function

2015 ◽  
Vol 11 (08) ◽  
pp. 2451-2486
Author(s):  
Xian-Jin Li
Keyword(s):  
Hilbert Space ◽  
Zeta Function ◽  
Explicit Formula ◽  
Finite Fields ◽  
Riemann Hypothesis ◽  
Algebraic Curves ◽  
Nauk Sssr ◽  
Akad Nauk Sssr ◽  
Curves Over Finite Fields ◽  
Riemann’S Zeta Function

In 1940, Weil [Sur les fonctions algébriques à corps de constantes finis, C. R. Acad. Sci. Paris210 (1940) 592–594] proved the Riemann hypothesis for curves over finite fields. It follows from the Castelnuovo–Severi defect inequality concerning correspondences between algebraic curves (see [A. Mattuck and J. Tate, On the inequality of Castelnuovo–Severi, Abh. Math. Sem. Univ. Hamburg22 (1958) 295–299]). An important step in the proof of Castelnuovo–Severi's defect inequality is the invariance of the Castelnuovo–Severi defect under trivial correspondences, so that the degree of divisors can be modified by adding multiples of trivial correspondences. In the number field case, the Weil distribution Δ(h) (see [A. Weil, Sur les formules explicites de la théorie des nombres, Izv. Akad. Nauk SSSR Ser. Mat.36 (1972) 3–18]) corresponds to the Castelnuovo–Severi defect. Functions of the form ∑ξ∈K* f(ξx) with f in the Schwartz–Bruhat space S(𝔸)0 correspond to trivial correspondences. In this paper, we show that the two terms [Formula: see text] and [Formula: see text] in the Weil distribution can be chosen to be zero by adding "trivial correspondences" to h while keeping the Weil distribution essentially unchanged. As an application of this result, the Weil distribution is expressed as the spectral trace of an operator on a Hilbert space.

scholarly journals On the integer part of the reciprocal of the Riemann zeta function tail at certain rational numbers in the critical strip

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
WonTae Hwang ◽  
Kyunghwan Song
Keyword(s):  
Zeta Function ◽  
Rational Number ◽  
Riemann Zeta Function ◽  
Rational Numbers ◽  
Critical Strip ◽  
Integer Part ◽  
Integral Points ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

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

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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