723 research outputs found
Sato--Tate, cyclicity, and divisibility statistics on average for elliptic curves of small height
We obtain asymptotic formulae for the number of primes for which the
reduction modulo of the elliptic curve \E_{a,b} : Y^2 = X^3 + aX + b
satisfies certain ``natural'' properties, on average over integers and
with and , where and are small relative to .
Specifically, we investigate behavior with respect to the Sato--Tate
conjecture, cyclicity, and divisibility of the number of points by a fixed
integer
Fractional parts of Dedekind sums
Using a recent improvement by Bettin and Chandee to a bound of Duke,
Friedlander and Iwaniec~(1997) on double exponential sums with Kloosterman
fractions, we establish a uniformity of distribution result for the fractional
parts of Dedekind sums with and running over rather general
sets. Our result extends earlier work of Myerson (1988) and Vardi (1987). Using
different techniques, we also study the least denominator of the collection of
Dedekind sums on average
for .Comment: Using recent results of S. Bettin and V. Chandee, arXiv 1502.00769,
we have improved some of our result
Integers with a large smooth divisor
We study the function that counts the number of positive
integers which have a divisor with the property that
for every prime dividing . We also indicate some cryptographic
applications of our results
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