On Artin's conjecture on average and short character sums

2024/12/17 by Klurman, Oleksiy, Shparlinski, Igor E., Teräväinen, Joni
#11A07 #11L40 #11N25 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2412.13355

Abstract

Let N_a(x) denote the number of primes up to x for which the integer a is a primitive root. We show that N_a(x) satisfies the asymptotic predicted by Artin's conjecture for almost all 1≤ a≤ exp((log log x)²). This improves on a result of Stephens (1969). A key ingredient in the proof is a new short character sum estimate over the integers, improving on the range of a result of Garaev (2006).

Related

• Counting primes with a given primitive root, uniformly
• Generalized Artin Primitive Root Conjecture
• Uniform bounds for the density in Artin's conjecture on primitive roots
• Refinements of Artin's primitive root conjecture
• Uniformly counting primes with a given primitive root and in an\n arithmetic progression