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 ON THE RIEMANN HYPOTHESIS AND TACHYONS IN DUAL STRING SCATTERING AMPLITUDES

2006 ◽  
Vol 03 (02) ◽  
pp. 187-199 ◽  
Author(s):  
CARLOS CASTRO
Keyword(s):  
Scattering Amplitudes ◽  
String Theory ◽  
Riemann Hypothesis ◽  
Critical Line ◽  
Open String ◽  
Bosonic String ◽  
Critical Strip ◽  
Veneziano Amplitude ◽  
Riemann Zeta ◽  
Bosonic String Theory

It is the purpose of this work to pursue a novel physical interpretation of the nontrivial Riemann zeta zeros and prove why the location of these zeros zn = 1/2+iλn corresponds physically to tachyonic-resonances/tachyonic-condensates, originating from the scattering of two on-shell tachyons in bosonic string theory. Namely, we prove that if there were nontrivial zeta zeros (violating the Riemann hypothesis) outside the critical line Realz = 1/2 (but inside the critical strip), these putative zeros do not correspond to any poles of the bosonic open string scattering (Veneziano) amplitude A(s, t, u). The physical relevance of tachyonic-resonances/tachyonic-condensates in bosonic string theory, establishes an important connection between string theory and the Riemann Hypothesis. In addition, one has also a geometrical interpretation of the zeta zeros in the critical line in terms of very special (degenerate) triangular configurations in the upper-part of the complex plane.

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 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 Pyruvate Substitutions on Glycoconjugates

2019 ◽  
Vol 20 (19) ◽  
pp. 4929 ◽  
Author(s):  
Hager ◽  
Sützl ◽  
Stefanović ◽  
Blaukopf ◽  
Schäffer
Keyword(s):  
Cell Wall ◽  
Cell Surface ◽  
Early Stage ◽  
Ring Structure ◽  
Genome Sequences ◽  
Cell Wall Component ◽  
Cellular Physiology ◽  
Associated Functions ◽  
Acid Labile ◽  
Insight Into

Glycoconjugates are the most diverse biomolecules of life. Mostly located at the cell surface, they translate into cell-specific “barcodes” and offer a vast repertoire of functions, including support of cellular physiology, lifestyle, and pathogenicity. Functions can be fine-tuned by non-carbohydrate modifications on the constituting monosaccharides. Among these modifications is pyruvylation, which is present either in enol or ketal form. The most commonly best-understood example of pyruvylation is enol-pyruvylation of N-acetylglucosamine, which occurs at an early stage in the biosynthesis of the bacterial cell wall component peptidoglycan. Ketal-pyruvylation, in contrast, is present in diverse classes of glycoconjugates, from bacteria to algae to yeast—but not in humans. Mild purification strategies preventing the loss of the acid-labile ketal-pyruvyl group have led to a collection of elucidated pyruvylated glycan structures. However, knowledge of involved pyruvyltransferases creating a ring structure on various monosaccharides is scarce, mainly due to the lack of knowledge of fingerprint motifs of these enzymes and the unavailability of genome sequences of the organisms undergoing pyruvylation. This review compiles the current information on the widespread but under-investigated ketal-pyruvylation of monosaccharides, starting with different classes of pyruvylated glycoconjugates and associated functions, leading to pyruvyltransferases, their specificity and sequence space, and insight into pyruvate analytics.

scholarly journals Morphogenesis of the Zeta Function in the Critical Strip by Computational Approach

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 285
Author(s):  
Michel Riguidel
Keyword(s):  
Zeta Function ◽  
Numerical Approximation ◽  
Critical Line ◽  
Formal Proof ◽  
Computational Approach ◽  
Partial Sums ◽  
Critical Strip ◽  
Power Functions ◽  
Elementary Functions ◽  
2D And 3D

This article proposes a morphogenesis interpretation of the zeta function by computational approach by relying on numerical approximation formulae between the terms and the partial sums of the series, divergent in the critical strip. The goal is to exhibit structuring properties of the partial sums of the raw series by highlighting their morphogenesis, thanks to the elementary functions constituting the terms of the real and imaginary parts of the series, namely the logarithmic, cosine, sine, and power functions. Two essential indices of these sums appear: the index of no return of the vagrancy and the index of smothering of the function before the resumption of amplification of its divergence when the index tends towards infinity. The method consists of calculating, displaying graphically in 2D and 3D, and correlating, according to the index, the angles, the terms and the partial sums, in three nested domains: the critical strip, the critical line, and the set of non-trivial zeros on this line. Characteristics and approximation formulae are thus identified for the three domains. These formulae make it possible to grasp the morphogenetic foundations of the Riemann hypothesis (RH) and sketch the architecture of a more formal proof.

Partial Euler Products on the Critical Line

2005 ◽  
Vol 57 (2) ◽  
pp. 267-297 ◽  
Author(s):  
Keith Conrad
Keyword(s):  
Asymptotic Behavior ◽  
Elliptic Curve ◽  
Riemann Hypothesis ◽  
Critical Line ◽  
Function Fields ◽  
Number Fields ◽  
Euler Product ◽  
Second Moments ◽  
Precise Sense ◽  
Euler Products

AbstractThe initial version of the Birch and Swinnerton-Dyer conjecture concerned asymptotics for partial Euler products for an elliptic curve L-function at s = 1. Goldfeld later proved that these asymptotics imply the Riemann hypothesis for the L-function and that the constant in the asymptotics has an unexpected factor of. We extend Goldfeld's theorem to an analysis of partial Euler products for a typical L-function along its critical line. The general phenomenon is related to second moments, while the asymptotic behavior (over number fields) is proved to be equivalent to a condition that in a precise sense seemsmuch deeper than the Riemann hypothesis. Over function fields, the Euler product asymptotics can sometimes be proved unconditionally.

scholarly journals Uniqueness of meromorphic functions concerning differential polynomials

2012 ◽  
Vol 43 (1) ◽  
pp. 51-68
Author(s):  
Subhas S. Bhoosnurmath ◽  
Veena L. Pujari ◽  
Anupama J. Patil
Keyword(s):  
Meromorphic Functions ◽  
Simple Technique ◽  
Entire Functions ◽  
Differential Polynomials ◽  
Additional Insight ◽  
Insight Into

In this paper, we present a different and very simple technique to handle various uniqueness problems involving three small entire functions. It also gives a new additional insight into such problems.

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