On products of primes and almost primes in arithmetic progressions

On primes in arithmetic progressions

Let d > 4 and c ∈ (−d, d) be relatively prime integers, and let r(d) be the product of all distinct prime divisors of d. We show that for any sufficiently large integer n (in particular n > 24310 suffices for 4 6 d 6 36) the least positive integer m with 2r(d)k(dk− c) (k = 1, . . . , n) pairwise distinct modulo m is just the first prime p ≡ c (mod d) with p > (2dn − c)/(d − 1). We also conjectu...

Primes in arithmetic progressions

[1] Euler’s proof uses only simple properties of ζ(s), and only of ζ(s) as a function of a real, rather than complex, variable. Given the status of complex number and complex analysis in Euler’s time, this is not surprising. It is slightly more surprising that Dirichlet’s original argument also was a real-variable argument, since by that time, a hundred years later, complex analysis was well-es...

Primes in arithmetic progressions

Strengthening work of Rosser, Schoenfeld, and McCurley, we establish explicit Chebyshev-type estimates in the prime number theorem for arithmetic progressions, for all moduli k ≤ 72 and other small moduli.

1 Arithmetic progressions in primes

Long Arithmetic Progressions of Primes

This is an article for a general mathematical audience on the author’s work, joint with Terence Tao, establishing that there are arbitrarily long arithmetic progressions of primes.