Vishal Gupta

Large values of Dirichlet polynomials with characters

Dirichlet polynomials are useful in the study of the Riemann zeta function & Dirichlet L functions, serving as approximations to them via the approximate functional equation. If there are many zeroes of a Dirichlet L function off the critical line then there is a corresponding Dirichlet polynomial taking unusually large values. Thus understanding how often Dirichlet polynomials can be large gives bounds on the number of zeroes of a Dirichlet L function in vertical strips, known as zero density estimates, which help us understand the distribution of primes in short intervals. An adaptation of Guth-Maynard's argument for the Riemann zeta function yields an improvement to large values estimates for Dirichlet polynomials with characters. This yields a zero density estimate of the form $\sum_{\chi \bmod q} N(\sigma,T,\chi) \ll (qT)^{7/3(1-\sigma)+o(1)}$, improving the exponent of 12/5 due to Huxley, with further improvement possible if q is sufficiently smooth. This is joint work with Yung Chi Li.