Publications (3)0 Total impact
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ABSTRACT: Stephen D. Miller showed that, assuming the generalized Riemann Hypothesis,
every entire $L$-function of real archimedian type has a zero in the interval
$\frac12+i t$ with $-t_0 < t < t_0$, where $t_0\approx 14.13$ corresponds to
the first zero of the Riemann zeta function. We give an example of a self-dual
degree-4 $L$-function whose first positive imaginary zero is at $t_1\approx
14.496$. In particular, Miller's result does not hold for general
$L$-functions. We show that all $L$-functions satisfying some additional
(conjecturally true) conditions have a zero in the interval $(-t_2,t_2)$ with
$t_2\approx 22.661$.
11/2012;
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ABSTRACT: We describe a tabulation of (conjecturally) modular elliptic curves over the
field Q(sqrt(5)) up to the first curve of rank 2. Using an efficient
implementation of an algorithm of Lassina Dembele, we computed tables of
Hilbert modular forms of weight (2,2) over Q(sqrt(5)), and via a variety of
methods we constructed corresponding elliptic curves, including (again,
conjecturally) all elliptic curves over Q(sqrt(5)) that have conductor with
norm less than or equal to 1831.
02/2012;
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ABSTRACT: We obtain explicit bounds on the moments of character sums, refining
estimates of Montgomery and Vaughan. As an application we obtain results on the
distribution of the maximal magnitude of character sums normalized by the
square root of the modulus, finding almost double exponential decay in the tail
of this distribution.
10/2011;
