James Maynard: Half-isolated zeros and zero density

Une celebration de la théorie analytique des nombres, une conference en l’honneur d’Andrew Granville/A celebration of analytic number theory, a conference in honor of Andrew Granville September/September 05-09

05 Septembre/September 05:

One well-known barrier to improvements to estimates on primes is if there are arithmetic progressions of zeros of the zeta function on vertical lines. We introduce a new zero-detecting method which is sensitive to the vertical distribution of zeros we show there are few 'half-isolated zeros'. One consequence is if the zeros of the zeta function lie on finitely many vertical lines then we can overcome the standard limitations of zero-density estimates and obtain improved estimates for primes in short intervals. This is joint work with Kyle Pratt and Roger Heath-Brown