On Artin's conjecture on average and short character sums

: Klurman, Oleksiy; Shparlinski, Igor E.; Teräväinen, Joni

Publisher: WILEY

: HOBOKEN

: 2025

: Bulletin of the London Mathematical Society

: BULLETIN OF THE LONDON MATHEMATICAL SOCIETY

: B LOND MATH SOC

: 15

: 0024-6093

: 1469-2120

DOI: https://doi.org/10.1112/blms.70103

: https://doi.org/10.1112/blms.70103

: https://research.utu.fi/converis/portal/detail/Publication/498668440

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).

Bulletin of London Math Soc - 2025 - Klurman - On Artin s conjecture on average and short character sums.pdf

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Australian Research Council, Grant/Award Numbers: DP230100530, DP230100534; Knut and AliceWallenberg
Fellowship; European Union’s Horizon Europe Research and Innovation Programme under Marie Skłodowska-Curie, Grant/Award Number: 101058904; Academy of Finland, Grant/Award Number: 362303