2,377 research outputs found
Dynamical zeta functions and Kummer congruences
We establish a connection between the coefficients of Artin-Mazur
zeta-functions and Kummer congruences. This allows to settle positively the
question of the existence of a map T such that the number of fixed points of
the n-th iterate of T is equal to the absolute value of the 2n-th Euler number.
Also we solve a problem of Gabcke related to the coefficients of Riemann-Siegel
formula.Comment: 12 pages, AMS-LaTe
Algorithms for determining integer complexity
We present three algorithms to compute the complexity of all
natural numbers . The first of them is a brute force algorithm,
computing all these complexities in time and space . The
main problem of this algorithm is the time needed for the computation. In 2008
there appeared three independent solutions to this problem: V. V. Srinivas and
B. R. Shankar [11], M. N. Fuller [7], and J. Arias de Reyna and J. van de Lune
[3]. All three are very similar. Only [11] gives an estimation of the
performance of its algorithm, proving that the algorithm computes the
complexities in time , where . The other two algorithms, presented in [7] and
[3], were very similar but both superior to the one in [11]. In Section 2 we
present a version of these algorithms and in Section 4 it is shown that they
run in time and space . (Here ).
In Section 2 we present the algorithm of [7] and [3]. The main advantage of
this algorithm with respect to that in [11] is the definition of kMax in
Section 2.7. This explains the difference in performance from
to .
In Section 3 we present a detailed description a space-improved algorithm of
Fuller and in Section 5 we prove that it runs in time and space
, where and
.Comment: 21 pages. v2: We improved the computations to get a better bound for
$\alpha
The n-th prime asymptotically
A new derivation of the classic asymptotic expansion of the n-th prime is
presented. A fast algorithm for the computation of its terms is also given,
which will be an improvement of that by Salvy (1994).
Realistic bounds for the error with \li^{-1}(n), after having retained the
first m terms, for , are given. Finally, assuming the Riemann
Hypothesis, we give estimations of the best possible such that, for , we have where is the sum of the first four terms
of the asymptotic expansion
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