Visions in Arithmetic and Beyond: Celebrating Peter Sarnak's Work and Impact

Bounds for Dirichlet polynomials play an important role in several questions connected to the distribution of primes. For example, they can be used to bound the number of zeroes of the Riemann zeta function in vertical strips, which is relevant to the distribution of primes in short intervals.  A Dirichlet polynomial is a trigonometric polynomial of the form

$D(t) = \sum_{n = N}^{2N} b_n n^{it}$

The main question is about the size of the superlevel sets of $D(t)$.  We normalize so that the coefficients have norm at most 1, and then we study the superlevel set $|D(t)| > N^\sigma$ for some exponent sigma between 1/2 and 1.  

For large values of sigma, Montgomery proved very strong bounds for the superlevel sets.  But for sigma $\le$ 3/4, the best known bounds follow from a very simple orthogonality argument (and they don't appear to be sharp).  We improve the known bounds for sigma close to 3/4.   Work in progress.  Joint with James Maynard.