On zero-density estimates and the PNT in short intervals for Beurling generalized numbers
Tom 207 / 2023
Acta Arithmetica 207 (2023), 365-391
MSC: Primary 11M26; Secondary 11M41, 11N05, 11N80.
DOI: 10.4064/aa221223-15-2
Opublikowany online: 11 April 2023
Streszczenie
We study the distribution of zeros of zeta functions associated to Beurling generalized prime number systems whose integers are distributed as . We obtain in particular N(\alpha , T) \ll T^{\frac{c(1{\textstyle-}\alpha )}{1{\textstyle-}\theta }}\log^{9} T for a constant c arbitrarily close to 4, improving significantly the current state of the art. We also investigate the consequences that the zero-density estimates obtained have on the PNT in short intervals. Our proofs crucially rely on an extension of the classical mean value theorem for Dirichlet polynomials to generalized Dirichlet polynomials.
