On a Mean Value Theorem for the Remainder Term in the Prime Number Theorem for Short Arithmetic Progressions

On Elementary Proofs of the Prime Number Theorem for Arithmetic Progressions, without Characters

We consider what one can prove about the distribution of prime numbers in arithmetic progressions, using only Selberg's formula. In particular, for any given positive integer q, we prove that either the Prime Number Theorem for arithmetic progressions, modulo q, does hold, or that there exists a subgroup H of the reduced residue system, modulo q, which contains the squares, such that (x; q; a) ...

Roth’s Theorem on 3-term Arithmetic Progressions

This article is a discussion about the proof of a classical theorem of Roth’s regarding the existence of three term arithmetic progressions in certain subsets of the integers. Before beginning with this task, however, we will take a brief look at the history and motivation behind Roth’s theorem. The questions and ideas surrounding this subject may have begun with a wonderful theorem due to van ...

Faulhaber’s Theorem for Arithmetic Progressions

Abstract. We show that the classical Faulhaber’s theorem on sums of odd powers also holds for an arbitrary arithmetic progression, namely, the odd power sums of any arithmetic progression a + b, a + 2b, . . . , a + nb is a polynomial in na+ n(n+ 1)b/2. The coefficients of these polynomials are given in terms of the Bernoulli polynomials. Following Knuth’s approach by using the central factorial...

A Structure Theorem for Positive Density Sets Having the Minimal Number of 3-Term Arithmetic Progressions

Assuming the well known conjecture that for any γ > 0 and x sufficiently large the interval [x, x+xγ ] always contains a prime number, we prove the following unexpected result: There exist numbers 0 < ρ < 1 arbitrarily close to 0, and arbitrarily large primes q, such that if S is any subset of Z/qZ of density at least ρ, having the least number of 3-term arithmetic progressions among all such s...

Dirichlet’s Theorem on Arithmetic Progressions