Explicit bounds for primes in arithmetic progressions

By Michael A. Bennett, Greg Martin, Kevin O'Bryant, A. Rechnitzer. 2018

99 citations recorded by the source corpus.

We derive explicit upper bounds for various functions counting primes in arithmetic progressions. By way of example, if $q$ and $a$ are integers with $\gcd(a,q)=1$ and $3 \leq q \leq 10^5$, and $\theta(x;q,a)$ denotes the sum of the logarithms of the primes $p \equiv a \pmod{q}$ with $p \leq x$, we show that $$ \bigg| \theta (x; q, a) - \frac{x}{\phi (q)} \bigg| 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 $L(1,\chi)$ for quadratic characters $\chi$, and an improved explicit upper bound for exceptional zeros.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/57089ac731473bdc004a8760b7cbd63a87a349e2

Citations in the corpus

• The Riemann hypothesis is true up to 3·1012 (2020, unknown): “;x≤ 2.169·1025), |θ− x| ≤ √ x 8π log2x, (599 <x≤ 2.169·1025), |π(x)− li(x)| ≤ √ x 8π log2x, (2657 <x≤ 2.169·1025). We also mention that some results from the expansive article by Bennett et al. [2] can be improved, as can bounds on weighted sums of Λ(n) as given by Ramar´e [25]. 3.2 Zero-free ”