1,010 research outputs found
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 and consider arithmetic progressions , with varying over a given range. We show that for certain specific
values of , the arithmetic progressions 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 -functions, we show that both the average analytic rank and
the average algebraic rank of elliptic curves in families of quadratic twists
are exactly . 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 , and the remaining half . 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 -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 . 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 -functions: A transition in the Ratios Conjecture
We study the -level density of low-lying zeros of quadratic Dirichlet
-functions by applying the -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 . 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 -functions: Lower order terms for extended support
We study the -level density of low-lying zeros of Dirichlet -functions
attached to real primitive characters of conductor at most . Under the
Generalized Riemann Hypothesis, we give an asymptotic expansion of this
quantity in descending powers of , which is valid when the support of
the Fourier transform of the corresponding even test function is
contained in . We uncover a phase transition when the supremum
of the support of reaches , both in the main term and in the
lower order terms. A new lower order term appearing at involves the
quantity , and is analogous to a lower order term which was
isolated by Rudnick in the function field case.Comment: 19 page
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