Expand all Collapse all | Results 1 - 4 of 4 |
1. CJM 2010 (vol 62 pp. 1155)
Moments of the Critical Values of Families of Elliptic Curves, with Applications
We make conjectures on the moments of the central values of the family
of all elliptic curves and on the moments of the first derivative of
the central values of a large family of positive rank curves. In both
cases the order of magnitude is the same as that of the moments of the
central values of an orthogonal family of $L$-functions. Notably, we
predict that the critical values of all rank $1$ elliptic curves is
logarithmically larger than the rank $1$ curves in the positive rank
family.
Furthermore, as arithmetical applications, we make a conjecture on the
distribution of $a_p$'s amongst all rank $2$ elliptic curves and
show how the Riemann hypothesis can be deduced from sufficient
knowledge of the first moment of the positive rank family (based on an
idea of Iwaniec)
Categories:11M41, 11G40, 11M26 |
2. CJM 2006 (vol 58 pp. 843)
On the One-Level Density Conjecture for Quadratic Dirichlet L-Functions In a previous article, we studied the distribution of ``low-lying"
zeros of the family of quad\-ratic Dirichlet $L$-functions assuming
the Generalized Riemann Hypothesis for all Dirichlet
$L$-functions. Even with this very strong assumption, we were
limited to using weight functions whose Fourier transforms are
supported in the interval $(-2,2)$. However, it is widely believed
that this restriction may be removed, and this leads to what has
become known as the One-Level Density Conjecture for the zeros of
this family of quadratic $L$-functions. In this note, we make use
of Weil's explicit formula as modified by Besenfelder to prove an
analogue of this conjecture.
Category:11M26 |
3. CJM 2001 (vol 53 pp. 866)
Inverse Problems for Partition Functions Let $p_w(n)$ be the weighted partition function defined by the
generating function $\sum^\infty_{n=0}p_w(n)x^n=\prod^\infty_{m=1}
(1-x^m)^{-w(m)}$, where $w(m)$ is a non-negative arithmetic function.
Let $P_w(u)=\sum_{n\le u}p_w(n)$ and $N_w(u)=\sum_{n\le u}w(n)$ be the
summatory functions for $p_w(n)$ and $w(n)$, respectively.
Generalizing results of G.~A.~Freiman and E.~E.~Kohlbecker, we show
that, for a large class of functions $\Phi(u)$ and $\lambda(u)$, an
estimate for $P_w(u)$ of the form
$\log P_w(u)=\Phi(u)\bigl\{1+O(1/\lambda(u)\bigr)\bigr\}$
$(u\to\infty)$ implies an estimate for $N_w(u)$ of the form
$N_w(u)=\Phi^\ast(u)\bigl\{1+O\bigl(1/\log\lambda(u)\bigr)\bigr\}$
$(u\to\infty)$ with a suitable function $\Phi^\ast(u)$ defined in
terms of $\Phi(u)$. We apply this result and related results to
obtain characterizations of the Riemann Hypothesis and the
Generalized Riemann Hypothesis in terms of the asymptotic behavior
of certain weighted partition functions.
Categories:11P82, 11M26, 40E05 |
4. CJM 1998 (vol 50 pp. 563)
Primes in short segments of arithmetic progressions Consider the variance for the number of primes that are both in the
interval $[y,y+h]$ for $y \in [x,2x]$ and in an arithmetic
progression of modulus $q$. We study the total variance
obtained by adding these variances over all the reduced residue
classes modulo $q$. Assuming a strong form of the twin prime
conjecture and the Riemann Hypothesis one can obtain an asymptotic
formula for the total variance in the range when $1 \leq h/q \leq
x^{1/2-\epsilon}$, for any $\epsilon >0$. We show that one can still
obtain some weaker asymptotic results assuming the Generalized Riemann
Hypothesis (GRH) in place of the twin prime conjecture. In their
simplest form, our results are that on GRH the same asymptotic formula
obtained with the twin prime conjecture is true for ``almost all'' $q$
in the range $1 \leq h/q \leq h^{1/4-\epsilon}$, that on averaging
over $q$ one obtains an asymptotic formula in the extended range $1
\leq h/q \leq h^{1/2-\epsilon}$, and that there are lower bounds with
the correct order of magnitude for all $q$ in the range $1 \leq h/q
\leq x^{1/3-\epsilon}$.
Category:11M26 |