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 $\Vert n\Vert$ of all natural numbers $n\le N$. The first of them is a brute force algorithm, computing all these complexities in time $O(N^2)$ and space $O(N\log^2 N)$. 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 $O(N^{1+\beta})$, where $1+\beta =\log3/\log2\approx1.584963$. 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 $O(N^\alpha)$ and space $O(N\log\log N)$. (Here $\alpha = 1.230175$). 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 $O(N^{1+\beta})$ to $O(N^\alpha)$. 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 $O(N^\alpha)$ and space $O(N^{(1+\beta)/2}\log\log N)$, where $\alpha=1.230175$ and $(1+\beta)/2\approx0.792481$.Comment: 21 pages. v2: We improved the computations to get a better bound for $\alpha

Gaussian variables, polynomials and permanents

AbstractWe prove that given n unit vectors (aj)nj=1 in a complex Hilbert space there exists a unit vector x such that|〈x,a1〉…〈x,an〉|≱n−n2.This answers a question about the best constant in an inequality for polynomials in several variables. The proof depends on a relation between gaussian variables and permanents

On Morrison’s definite integral

Física Teórica, Atómica y Óptic