scholarly journals On the distribution of the order over residue classes

2006 ◽  
Vol 12 (15) ◽  
pp. 121-128 ◽  
Author(s):  
Pieter Moree
Keyword(s):  
Residue Classes

scholarly journals A note on primes in certain residue classes

2018 ◽  
Vol 14 (08) ◽  
pp. 2219-2223
Author(s):  
Paolo Leonetti ◽  
Carlo Sanna
Keyword(s):  
Positive Integer ◽  
Asymptotic Density ◽  
Residue Classes ◽  
Euler Totient Function ◽  
The One ◽  
Positive Integers

Given positive integers [Formula: see text], we prove that the set of primes [Formula: see text] such that [Formula: see text] for [Formula: see text] admits asymptotic density relative to the set of all primes which is at least [Formula: see text], where [Formula: see text] is the Euler totient function. This result is similar to the one of Heilbronn and Rohrbach, which says that the set of positive integer [Formula: see text] such that [Formula: see text] for [Formula: see text] admits asymptotic density which is at least [Formula: see text].

scholarly journals Additive correlation and the inverse problem for the large sieve

2018 ◽  
Vol 168 (2) ◽  
pp. 211-217
Author(s):  
BRANDON HANSON
Keyword(s):  
Inverse Problem ◽  
Energy Estimate ◽  
Large Sieve ◽  
Residue Classes ◽  
Formula Group ◽  
Additive Energy ◽  
Image Position

AbstractLet A ⊆ [1, N] be a set of integers with |A| ≫ $\sqrt N$. We show that if A avoids about p/2 residue classes modulo p for each prime p, then A must correlate additively with the squares S = {n2 : 1 ≤ n ≤ $\sqrt N$}, in the sense that we have the additive energy estimate $$ E(A,S)\gg N\log N. $$ This is, in a sense, optimal.

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