On the Distribution of Primes

1968 ◽  
Vol 75 (7) ◽  
pp. 764
Author(s):  
Abraham Waksman
Keyword(s):  
Distribution Of Primes

scholarly journals Variants on Andrica’s Conjecture with and without the Riemann Hypothesis

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 289 ◽  
Author(s):  
Matt Visser
Keyword(s):  
Riemann Hypothesis ◽  
Distribution Of Primes

The gap between what we can explicitly prove regarding the distribution of primes and what we suspect regarding the distribution of primes is enormous. It is (reasonably) well-known that the Riemann hypothesis is not sufficient to prove Andrica’s conjecture: ∀n≥1, is p n + 1 - p n ≤ 1 ? However, can one at least get tolerably close? I shall first show that with a logarithmic modification, provided one assumes the Riemann hypothesis, one has p n + 1 /ln p n + 1 - p n /ln p n < 11/25; (n ≥ 1). Then, by considering more general mth roots, again assuming the Riemann hypothesis, I show that p n + 1 m - p n m < 44/(25 e[m < 2]); (n ≥ 3; m > 2). In counterpoint, if we limit ourselves to what we can currently prove unconditionally, then the only explicit Andrica-like results seem to be variants on the relatively weak results below: ln2 pn + 1 - ln2 pn < 9; ln3 pn + 1 - ln3 pn < 52; ln4 pn + 1 - ln4 pn < 991; (n ≥ 1). I shall also update the region on which Andrica’s conjecture is unconditionally verified.

Erratum: Limiting properties of the distribution of primes in an arbitrarily large number of residue classes

2020 ◽  
pp. 1-4
Author(s):  
Lucile Devin
Keyword(s):  
Linear Relation ◽  
Distribution Of Primes ◽  
Residue Classes

Abstract As pointed out by Alexandre Bailleul, the paper mentioned in the title contains a mistake in Theorem 2.2. The hypothesis on the linear relation of the almost periods is not sufficient. In this note, we fix the problem and its minor consequences on other results in the same paper.

scholarly journals Limiting Properties of the Distribution of Primes in an Arbitrarily Large Number of Residue Classes

2020 ◽  
Vol 63 (4) ◽  
pp. 837-849 ◽  
Author(s):  
Lucile Devin
Keyword(s):  
Riemann Hypothesis ◽  
Linear Independence ◽  
Classical Case ◽  
Weak Version ◽  
Distribution Of Primes ◽  
Logarithmic Density ◽  
Fixed Integer ◽  
Residue Classes ◽  
Independence Hypothesis ◽  
And Function

AbstractWe generalize current known distribution results on Shanks–Rényi prime number races to the case where arbitrarily many residue classes are involved. Our method handles both the classical case that goes back to Chebyshev and function field analogues developed in the recent years. More precisely, let $\unicode[STIX]{x1D70B}(x;q,a)$ be the number of primes up to $x$ that are congruent to $a$ modulo $q$. For a fixed integer $q$ and distinct invertible congruence classes $a_{0},a_{1},\ldots ,a_{D}$, assuming the generalized Riemann Hypothesis and a weak version of the linear independence hypothesis, we show that the set of real $x$ for which the inequalities $\unicode[STIX]{x1D70B}(x;q,a_{0})>\unicode[STIX]{x1D70B}(x;q,a_{1})>\cdots >\unicode[STIX]{x1D70B}(x;q,a_{D})$ are simultaneously satisfied admits a logarithmic density.

On the Product of the Primes not Exceeding n

1959 ◽  
Vol 2 (2) ◽  
pp. 119-121 ◽  
Author(s):  
Leo Moser
Keyword(s):  
Simple Proof ◽  
Elementary Theory ◽  
Distribution Of Primes ◽  
Image Position

One of the most elegant results of the elementary theory of the distribution of primes is that1where the product runs over primes. A very simple proof of (1) has recently been given by Erdös and Kalmar [1], [2].

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