The Density of the Zeros of the Zeta Function and the Problem of the Distribution of Prime Numbers in Short Intervals

1993 ◽  
pp. 94-101
Author(s):  
Anatolij A. Karatsuba ◽  
Melvyn B. Nathanson
Keyword(s):  
Zeta Function ◽  
Prime Numbers ◽  
Short Intervals

On prime numbers of special kind on short intervals

2006 ◽  
Vol 79 (5-6) ◽  
pp. 848-853
Author(s):  
N. N. Mot’kina
Keyword(s):  
Special Kind ◽  
Prime Numbers ◽  
Short Intervals

Almost-primes in arithmetic progressions and short intervals

1978 ◽  
Vol 83 (3) ◽  
pp. 357-375 ◽  
Author(s):  
D. R. Heath-Brown
Keyword(s):  
Prime Numbers ◽  
Arithmetic Progressions ◽  
Short Intervals ◽  
Almost Primes ◽  
Primes In Arithmetic Progressions ◽  
Image Position

In this paper we shall investigate the occurrence of almost-primes in arithmetic progressions and in short intervals. These problems correspond to two well-known conjectures concerning prime numbers. The first conjecture is that, if (l, k) = 1, there exists a prime p satisfying

scholarly journals Modified L-functions

2008 ◽  
Vol 48 ◽  
Author(s):  
Eugenijus Stankus
Keyword(s):  
Zeta Function ◽  
Prime Numbers ◽  
Simple Pole

The sequence of generalized prime numbers q0 = 1, qn = pkn+1 -1, n ∈ N, and the corresponding zeta-function Zk(s) = \prodp>2( 1 - (pk - 1)-s)-1 , s = σ + it, are analyzed. The analyticity of Zk(s) in the domain σ > 0, except for a simple pole s = 1/k , is proved.

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.

Sums of Two Squares in Short Intervals

2000 ◽  
Vol 52 (4) ◽  
pp. 673-694 ◽  
Author(s):  
Antal Balog ◽  
Trevor D. Wooley
Keyword(s):  
Prime Numbers ◽  
Positive Dimension ◽  
Short Intervals ◽  
Sums Of Two Squares ◽  
Sieving Process

AbstractLet denote the set of integers representable as a sum of two squares. Since can be described as the unsifted elements of a sieving process of positive dimension, it is to be expected that hasmany properties in common with the set of prime numbers. In this paper we exhibit “unexpected irregularities” in the distribution of sums of two squares in short intervals, a phenomenon analogous to that discovered by Maier, over a decade ago, in the distribution of prime numbers. To be precise, we show that there are infinitely many short intervals containing considerably more elements of than expected, and infinitely many intervals containing considerably fewer than expected.

scholarly journals SIGN CHANGES IN SHORT INTERVALS OF COEFFICIENTS OF SPINOR ZETA FUNCTION OF A SIEGEL CUSP FORM OF GENUS 2

2014 ◽  
Vol 10 (02) ◽  
pp. 327-339 ◽  
Author(s):  
EMMANUEL ROYER ◽  
JYOTI SENGUPTA ◽  
JIE WU
Keyword(s):  
Zeta Function ◽  
Cusp Form ◽  
Short Intervals ◽  
Sign Changes ◽  
Genus 2 ◽  
Quantitative Results ◽  
Voronoi Formula ◽  
Siegel Cusp Form

In this paper, we establish a Voronoi formula for the spinor zeta function of a Siegel cusp form of genus 2. We deduce from this formula quantitative results on the number of its positive (respectively, negative) coefficients in some short intervals.

Sign in / Sign up

Export Citation Format

Share Document