54 research outputs found
A logarithmic improvement in the Bombieri-Vinogradov theorem
In this paper we improve the best known to date result of
Dress-Iwaniec-Tenenbaum, getting (log x)^2 instead of (log x)^(5/2). We use a
weighted form of Vaughan's identity, allowing a smooth truncation inside the
procedure, and an estimate due to Barban-Vehov and Graham related to Selberg's
sieve. We give effective and non-effective versions of the result.Comment: 17 page
The continuous postage stamp problem
For a real set consider the semigroup , additively generated by
; that is, the set of all real numbers representable as a (finite) sum of
elements of . If is open and non-empty, then is
easily seen to contain all sufficiently large real numbers, and we let . Thus, is the smallest number
with the property that any is representable as indicated above.
We show that if the measure of is large, then is small; more
precisely, writing for brevity \alpha := \mes A we have
G(A) \le
(1-\alpha) \lfloor 1/\alpha \rfloor
\quad &\text{if $0 < \alpha \le 0.1$},
(1-\alpha+\alpha\{1/\alpha\})\lfloor 1/\alpha\rfloor
\quad &\text{if $0.1 \le \alpha \le 0.5$},
2(1-\alpha)
\quad &\text{if $0.5 \le \alpha \le 1$}.
Indeed, the first and the last of these three estimates are the best
possible, attained for and
, respectively; the second is close to
the best possible and can be improved by at most.
The problem studied is a continuous analogue of the linear Diophantine
problem of Frobenius (in its extremal settings due to Erdos and Graham), also
known as the "postage stamp problem" or the "coin exchange problem"
On Taking Square Roots without Quadratic Nonresidues over Finite Fields
We present a novel idea to compute square roots over finite fields, without
being given any quadratic nonresidue, and without assuming any unproven
hypothesis. The algorithm is deterministic and the proof is elementary. In some
cases, the square root algorithm runs in bit operations
over finite fields with elements. As an application, we construct a
deterministic primality proving algorithm, which runs in
for some integers .Comment: 14 page
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