44 research outputs found
Close to Uniform Prime Number Generation With Fewer Random Bits
In this paper, we analyze several variants of a simple method for generating
prime numbers with fewer random bits. To generate a prime less than ,
the basic idea is to fix a constant , pick a
uniformly random coprime to , and choose of the form ,
where only is updated if the primality test fails. We prove that variants
of this approach provide prime generation algorithms requiring few random bits
and whose output distribution is close to uniform, under less and less
expensive assumptions: first a relatively strong conjecture by H.L. Montgomery,
made precise by Friedlander and Granville; then the Extended Riemann
Hypothesis; and finally fully unconditionally using the
Barban-Davenport-Halberstam theorem. We argue that this approach has a number
of desirable properties compared to previous algorithms.Comment: Full version of ICALP 2014 paper. Alternate version of IACR ePrint
Report 2011/48
Sharpenings of Li's criterion for the Riemann Hypothesis
Exact and asymptotic formulae are displayed for the coefficients
used in Li's criterion for the Riemann Hypothesis. For we obtain
that if (and only if) the Hypothesis is true,
(with and explicitly given, also for the case of more general zeta or
-functions); whereas in the opposite case, has a non-tempered
oscillatory form.Comment: 10 pages, Math. Phys. Anal. Geom (2006, at press). V2: minor text
corrections and updated reference
Quantum mechanical potentials related to the prime numbers and Riemann zeros
Prime numbers are the building blocks of our arithmetic, however, their
distribution still poses fundamental questions. Bernhard Riemann showed that
the distribution of primes could be given explicitly if one knew the
distribution of the non-trivial zeros of the Riemann function.
According to the Hilbert-P{\'o}lya conjecture there exists a Hermitean operator
of which the eigenvalues coincide with the real part of the non-trivial zeros
of . This idea encourages physicists to examine the properties of
such possible operators, and they have found interesting connections between
the distribution of zeros and the distribution of energy eigenvalues of quantum
systems. We apply the Mar{\v{c}}henko approach to construct potentials with
energy eigenvalues equal to the prime numbers and to the zeros of the
function. We demonstrate the multifractal nature of these potentials
by measuring the R{\'e}nyi dimension of their graphs. Our results offer hope
for further analytical progress.Comment: 7 pages, 5 figures, 2 table
Physics of the Riemann Hypothesis
Physicists become acquainted with special functions early in their studies.
Consider our perennial model, the harmonic oscillator, for which we need
Hermite functions, or the Laguerre functions in quantum mechanics. Here we
choose a particular number theoretical function, the Riemann zeta function and
examine its influence in the realm of physics and also how physics may be
suggestive for the resolution of one of mathematics' most famous unconfirmed
conjectures, the Riemann Hypothesis. Does physics hold an essential key to the
solution for this more than hundred-year-old problem? In this work we examine
numerous models from different branches of physics, from classical mechanics to
statistical physics, where this function plays an integral role. We also see
how this function is related to quantum chaos and how its pole-structure
encodes when particles can undergo Bose-Einstein condensation at low
temperature. Throughout these examinations we highlight how physics can perhaps
shed light on the Riemann Hypothesis. Naturally, our aim could not be to be
comprehensive, rather we focus on the major models and aim to give an informed
starting point for the interested Reader.Comment: 27 pages, 9 figure