The Mellin transform of powers of the zeta-function
The moments of infinite series.
The multiplicative and functional independence of Dedekind zeta functions of abelian fields
It is shown that the multiplicative independence of Dedekind zeta functions of abelian fields is equivalent to their functional independence. We also give all the possible multiplicative dependence relations for any set of Dedekind zeta functions of abelian fields.
The n-level densities of low-lying zeros of quadratic Dirichlet L-functions
Previous work by Rubinstein and Gao computed the n-level densities for families of quadratic Dirichlet L-functions for test functions f̂₁, ..., f̂ₙ supported in , and showed agreement with random matrix theory predictions in this range for n ≤ 3 but only in a restricted range for larger n. We extend these results and show agreement for n ≤ 7, and reduce higher n to a Fourier transform identity. The proof involves adopting a new combinatorial perspective to convert all terms to a canonical form,...
The number of primes in a short interval.
The number-theoretic work of Paul Turán
The pair correlation of zeros of Dirichlet L-functions and primes in arithmetic progressions.
The phase transition of the number-theoretical spin chain.
The prime number theorem for Beurling's generalized numbers. New cases
The prime number theorem in short intervals for automorphic L-functions
The probability that a complete intersection is smooth
Given a smooth subscheme of a projective space over a finite field, we compute the probability that its intersection with a fixed number of hypersurface sections of large degree is smooth of the expected dimension. This generalizes the case of a single hypersurface, due to Poonen. We use this result to give a probabilistic model for the number of rational points of such a complete intersection. A somewhat surprising corollary is that the number of rational points on a random smooth intersection...
The product of linear forms in a field of series and the Riemann hypothesis for curves
The product of two Dirichlet series
The quickest proof of the prime number theorem
The Riemann Zeta function and coin tossing.
The second moment of Dirichlet twists of Hecke L-functions
The second moment of quadratic twists of modular L-functions
We study the second moment of the central values of quadratic twists of a modular -function. Unconditionally, we obtain a lower bound which matches the conjectured asymptotic formula, while on GRH we prove the asymptotic formula itself.
The second moment of the symmetric square -functions.
The Selberg zeta function and a local trace formula for Kleinian groups.
The Selberg zeta-function for cocompact discrete subgroups of PSL(2,C)