On Bertrand's problem in an arithmetic progression

On Triples in Arithmetic Progression

On the Second Moment for Primes in an Arithmetic Progression

Abstract. Assuming the Generalized Riemann Hypothesis, we obtain a lower bound within a constant factor of the conjectured asymptotic result for the second moment for primes in an individual arithmetic progression in short intervals. Previous results were averaged over all progression of a given modulus. The method uses a short divisor sum approximation for the von Mangoldt function, together w...

Dirichlet’s Theorem on Primes in an Arithmetic Progression

Our goal is to prove the following theorem: Dirichlet’s Theorem: For any coprime a, b ∈ Z, there are infinitely many primes p such that p ≡ a (mod b). Although the statement of the theorem involves only integers, the simplest proof requires the use of complex numbers and Dirichlet L-series. Most of this paper will therefore be devoted to proving some basic properties of characters and L-series,...

On 4 Squares in Arithmetic Progression

x1 − 2x2 + x3 = 0 x2 − 2x3 + x4 = 0 are given by (x1, x2, x3, x4) = (±1,±1,±1,±1). Now, the above variety is an intersection between 2 quadrics in P. In general – i.e., except for the possibility of the variety being reducible or singular – an intersection between 2 quadrics in P is (isomorphic to) an elliptic curve and there is an algorithm that brings the curve to Weierstraß form by means of ...

Notes on Primes in Arithmetic Progression

The following is a quick set of notes of some properties of Dirichlet characters, in particular, how they are used to prove the infinitude of primes in arithmetic progressions. These notes are from from An Invitation to Modern Number Theory, by myself and Ramin Takloo-Bighash. As this is a modified snippet from the book, references to other parts of the book are displayed as ??. 1. Dirichlet Ch...