Results 261 to 270 of about 106,424 (299)

On the prime zeta function

BIT, 1968
Riemanns Zeta-Funktion sei \(\displaystyle P(s) = \sum_{(p)} p^{-s}\), \(s = \sigma + i\tau\), über alle Primzahlen \(p\). Die Schwierigkeiten der Berechnung werden außerordentlich groß in der Nähe der imaginären Achse. Das Haselgrove-Miller-Verfahren, das auf halbkonvergierenden, passend zugestutzten Reihen beruht, findet hier Anwendung.
Carl-Erik Fröberg
semanticscholar   +4 more sources

The Riemann Zeta Function and the Prime Number Theorem

, 2020
For a complex number s, we always denote its real part by \(\sigma \) and imaginary part by t. Thus \(s=\sigma +it.\) The Riemann Zeta function is defined as $$ \zeta (s)=\sum ^{\infty }_{n=1} \ \ \frac{1}{n^s} \ \text {in} \ \sigma > 1.$$
T. Shorey
semanticscholar   +3 more sources

An effective description of the roots of bivariates mod pk and the related Igusa’s local zeta function

International Symposium on Symbolic and Algebraic Computation, 2023
Finding roots of a bivariate polynomial f(x1, x2), over a prime field , is a fundamental question with a long history and several practical algorithms are now known.
Sayak Chakrabarti, Nitin Saxena
semanticscholar   +1 more source

Orbit growth of Dyck and Motzkin shifts via Artin–Mazur zeta function

Dynamical systems, 2020
For a discrete dynamical system, the prime orbit and Mertens' orbit counting functions indicate the growth of the closed orbits in the system in a certain way. These functions are analogous to the counting functions for primes in number theory.
Azmeer Nordin, Mohd Salmi Md Noorani, Syahida Che Dzul-Kifli   +2 more
semanticscholar   +1 more source

The Riemann zeta function

The Student Mathematical Library, 2020
The Riemann zeta function has a deep connection with the .distribution of primes. This expository thesis will explain the techniques used in proving the properties of the Riemann zeta function, its analytic continuation to the complex plane, and'the ...
Ernesto Oscar Reyes
semanticscholar   +1 more source

Primes, Arithmetic Functions, and the Zeta Function

2002
In this chapter we will discuss properties of primes and prime decomposition in the ring A = F[T]. Much of this discussion will be facilitated by the use of the zeta function associated to A. This zeta function is an analogue of the classical zeta function which was first introduced by L.
openaire   +2 more sources

The Distribution of Primes and the Riemann Zeta Function

1984
We recall that the problem of the distribution of primes had been raised at least as far back as the Greek antiquity. The proof of our Theorem 3.9, that there are infinitely many primes, appears in Euclid (Book 9, Section 20), and Eratosthenes† devised a systematic method for obtaining all primes up to any given number x.
openaire   +2 more sources

Zeta Function and the Harmonic Ontology of Prime Numbers

This article explores the ontological significance of prime numbers through the lens of the Riemann Zeta Function. By analyzing the harmonic properties and resonant patterns embedded in the distribution of prime numbers, the paper presents a novel interpretation of primes as foundational informational thresholds of reality.
Rezapour, Majid, Rezapour, Ramin
openaire   +1 more source

On the Lefschetz Zeta Function for a Class of Toral Maps

, 2021
P. Berrizbeitia, Marcos J. González, V. Sirvent   +2 more
semanticscholar   +1 more source

Recursive Ecology of Primes: Batesonian Models of the Zeta-Function and Prime Density

This paper explores two foundational results in analytic number theory—the RiemannHypothesis and the Prime Number Theorem—through the lens of recursive ecologicallogic derived from the Bateson Game. We model the Riemann zeta-functionas a frame-sensitive analytic ecology, in which primes serve as coherence attractorsstabilizing recursive structure ...
openaire   +1 more source