On the distribution of reduced fractions with squarefree denominators
Tom 206 / 2022
Acta Arithmetica 206 (2022), 1-34
MSC: Primary 11M26; Secondary 11B57, 11N56.
DOI: 10.4064/aa210905-20-9
Opublikowany online: 8 November 2022
Streszczenie
We introduce and study a family of functions that are similar to the Riemann zeta function, and we use these functions to explore the distribution in [0,1] of reduced fractions with squarefree denominators. We show that the existence of a zero-free strip of the form $\{\sigma _0\le \,{\rm Re}(s)\le 1\}$ with some $\sigma _0\in (2/3,1)$ for the Riemann zeta function is equivalent to a precise bound on the discrepancy of Farey fractions of squarefree denominators. Our principal tool is an unconditional generalization of a theorem of Blomer that concerns the distribution on average of squarefree integers in arithmetic progressions to large moduli.
