Explicit bound for the number of primes in arithmetic progressions assuming the Generalized Riemann Hypothesis

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.

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...

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...

1 Arithmetic progressions in primes

Least primes in arithmetic progressions

For a xed non-zero integer a and increasing function f, we investigate the lower density of the set of integers q for which the least prime in the arithmetic progression a(mod q) is less than qf(q). In particular we conjecture that this lower density is 1 for any f with log x = o(f(x)) and prove this, unconditionally, for f(x) = x=g(x) for any g with log g(x) = o(log x). Under the assumption of...