The distribution of the variance of primes in arithmetic progressions

Hooley conjectured that the variance V(x;q) of the distribution of primes up to x in the arithmetic progressions modulo q is asymptotically x log q, in some unspecified range of q\leq x. On average over 1\leq q \leq Q, this conjecture is known unconditionally in the range x/(log x)^A \leq Q \leq x; this last range can be improved to x^{\frac 12+\epsilon} \leq Q \leq x under the Generalized Riemann Hypothesis (GRH). We argue that Hooley's conjecture should hold down to (loglog x)^{1+o(1)} \leq q \leq x for all values of q, and that this range is best possible. We show under GRH and a linear independence hypothesis on the zeros of Dirichlet L-functions that for moderate values of q, \phi(q)e^{-y}V(e^y;q) has the same distribution as that of a certain random variable of mean asymptotically \phi(q) log q and of variance asymptotically 2\phi(q)(log q)^2. Our estimates on the large deviations of this random variable allow us to predict the range of validity of Hooley's Conjecture.Comment: 26 pages; Modified Definition 2.1, the error term for the variance in Theorem 1.2 and its proo

Residue classes containing an unexpected number of primes

We fix a non-zero integer $a$ and consider arithmetic progressions $a \bmod q$, with $q$ varying over a given range. We show that for certain specific values of $a$, the arithmetic progressions $a \bmod q$ contain, on average, significantly fewer primes than expected.Comment: 18 pages. Added a few remarks, changed the numbering of sections, slightly improved results, and made a few correction

A conditional determination of the average rank of elliptic curves

Under a hypothesis which is slightly stronger than the Riemann Hypothesis for elliptic curve $L$-functions, we show that both the average analytic rank and the average algebraic rank of elliptic curves in families of quadratic twists are exactly $\frac 12$. As a corollary we obtain that under this last hypothesis, the Birch and Swinnerton-Dyer Conjecture holds for almost all curves in our family, and that asymptotically one half of these curves have algebraic rank $0$, and the remaining half $1$. We also prove an analogous result in the family of all elliptic curves. A way to interpret our results is to say that nonreal zeros of elliptic curve $L$-functions in a family have a direct influence on the average rank in this family. Results of Katz-Sarnak and of Young constitute a major ingredient in the proofs.Comment: 27 page

Low-lying zeros of elliptic curve L-functions: Beyond the ratios conjecture

We study the low-lying zeros of L-functions attached to quadratic twists of a given elliptic curve E defined over $\mathbb Q$. We are primarily interested in the family of all twists coprime to the conductor of E and compute a very precise expression for the corresponding 1-level density. In particular, for test functions whose Fourier transforms have sufficiently restricted support, we are able to compute the 1-level density up to an error term that is significantly sharper than the square-root error term predicted by the L-functions Ratios Conjecture.Comment: 33 page

Low-lying zeros of quadratic Dirichlet LL-functions: A transition in the Ratios Conjecture

We study the $1$-level density of low-lying zeros of quadratic Dirichlet $L$-functions by applying the $L$-functions Ratios Conjecture. We observe a transition in the main term as was predicted by the Katz-Sarnak heuristic as well as in the lower order terms when the support of the Fourier transform of the corresponding test function reaches the point $1$. Our results are consistent with those obtained in previous work under GRH and are furthermore analogous to results of Rudnick in the function field case.Comment: 15 page

Low-lying zeros of quadratic Dirichlet LL-functions: Lower order terms for extended support

We study the $1$-level density of low-lying zeros of Dirichlet $L$-functions attached to real primitive characters of conductor at most $X$. Under the Generalized Riemann Hypothesis, we give an asymptotic expansion of this quantity in descending powers of $\log X$, which is valid when the support of the Fourier transform of the corresponding even test function $\phi$ is contained in $(-2,2)$. We uncover a phase transition when the supremum $\sigma$ of the support of $\hat \phi$ reaches $1$, both in the main term and in the lower order terms. A new lower order term appearing at $\sigma=1$ involves the quantity $\hat \phi (1)$, and is analogous to a lower order term which was isolated by Rudnick in the function field case.Comment: 19 page