Explicit bounds for primes in arithmetic progressions

Abstract

We derive explicit upper bounds for various counting functions for primes in arithmetic progressions. By way of example, if q and a are integers with gcd(a, q) = 1 and 3 ≤ q ≤ 10 5 , and θ(x; q, a) denotes the sum of the logarithms of the primes p ≡ a (mod q) with p ≤ x, we show that θ(x; q, a) − x/ϕ(q) < 1 160x log x for all x ≥ 8 • 10 9 , with significantly sharper constants obtained for individual moduli q. We establish inequalities of the same shape for the other standard primecounting functions π(x; q, a) and ψ(x; q, a), as well as inequalities for the nth prime congruent to a (mod q) when q ≤ 1200. For moduli q > 10 5 , we find even stronger explicit inequalities, but only for much larger values of x. Along the way, we also derive an improved explicit lower bound for L(1, χ) for quadratic characters χ, and an improved explicit upper bound for exceptional zeros.

Diophantine equations involving the Euler totient function

“…Then, by Lemma 1, all of the primes in the interval (n/3, n/2] are congruent to 2 (mod 3). Bennett, et al [2] showed that for x ≥ 450, we have…”

An explicit Pólya-Vinogradov inequality via Partial Gaussian sums

“…for some absolute constant c. For long character sums this inequality has remainded the sharpest known and an important problem is to improve on the log q factor in (1). This problem is more or less resolved assuming the Generalized Riemann Hypothesis.…”

“…The second author was supported by Australian Research Council Discovery Project DP160100932. 1 there exist infinitely many integers q and primitive characters χ modulo q, such that S(χ) ≫ √ q log log q, and it was proven by Montgomery and Vaughan [24] that S(χ) has an upper bound of the same order of magnitude assuming the Generalized Riemann Hypothesis.…”

“…Progress on unconditional improvements to (1) has two main themes. The first aims at improving the asymptotic size of the constant c, thus allowing a o(1) term.…”

“…which allows estimation of very short ranges of the parameter N. The bottleneck in the argument is the minor arcs which use the Burgess bound and provide a nontrivial estimate for at best N q 1/4+o (1) . For an explicit variant of Hildebrand's result, the bottleneck switches from the Burgess bound to Montgomery and Vaughan's estimate.…”