On a conjecture of R. L. Graham
The number of unsieved integers up to x
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.
Nonvanishing of quadratic Dirichlet -functions at .
Small gaps between zeros of twisted L-functions
Zeros of Fekete polynomials
For an odd prime, we show that the Fekete polynomial has zeros on the unit circle, where . Here is the probability that the function has a zero in , where each is with y . In fact has absolute value at each primitive th root of unity, and we show that if for some then there is a zero of close to this arc.
Maximal sets of integers with distinct divisors.
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