We first consider questions on the distribution of the primes. Using the recent advancement towards the Prime k-tuple Conjecture by Maynard and Tao, we show how to produce infinitely many strings of consecutive primes satisfying specified congruence conditions. We answer an old question of Erdös and Turán by producing strings of consecutive primes whose successive gaps form an increasing (respectively decreasing) sequence. We also show that such strings exist whose successive gaps follow a certain divisibility pattern. Finally, for any coprime integers a and D ≥ 1, we refine a theorem of D. Shiu and find strings of consecutive primes of arbitrary length in the congruence class a mod D. These results were proved jointly with William D. Banks and Tristan Freiberg. We next consider the vertical distribution of the nontrivial zeros of certain Dedekind zeta-functions. In particular, let K be a quadratic number field. Using the mixed second moments of derivatives of the Dedekind zeta-function attached to K on the critical line, we prove the existence of gaps between consecutive zeros of the Dedekind zeta-function attached to K on the critical line which are at least 2.44949... times the average spacing. Finally, assuming the Generalized Riemann Hypothesis and some standard conjectures, we prove upper bounds for moments of arbitrary products of automorphic L-functions and for Dedekind zeta-functions of Galois number fields on the critical line. As an application, we use these bounds to estimate the variance of the coefficients of these zeta- and L-functions in short intervals. We also prove upper bounds for moments of products of central values of automorphic L-functions twisted by quadratic Dirichlet characters and averaged over fundamental discriminants. These results were proved jointly with Micah B. Milinovich.
L-Functions, Moments, Primes In Tuples, Mathematics
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