Volume 59, Issue 3

The divisibility of numbers is obtained by iteration of the weighted sum of their integer digits. Then evaluation of the related congruences yields information about the primality of numbers in certain recursive sequences. From the row elements in generalized Delannoy triangles, we can verify the primality of any constellation of numbers. When a number set is not a prime constellation, we can identify factors of their composite numbers. The constellation primality test is proven in all generality, and examples are given for twin primes, prime triplets, and Sophie Germain primes.

Distribution functions of ratio block sequences formed from sequences of positive integers are investigated in the paper. We characterize the case when the set of all distribution functions of a ratio block sequence contains c 0, the greatest possible distribution function. Presented results complete some previously published results.

Using new characteristics of an infinite subset of positive integers we give some estimations of the dispersion of the related block sequence.

In [EGAWA, Y.—SUZUKI, H.: Automorphism groups of Σn+1-invariant trilinear forms, Hokkaido Math. J. 11, (1985), 39–47] are explored Σn+1-invariant symmetric trilinear forms and their automorphisms. In the paper we generalize their results to d-linear symmetric forms for any d ≥ 3.

Given any positive integer n, we prove the existence of infinitely many right triangles with area n and side lengths in certain number fields. This generalizes the famous congruent number problem. The proof allows the explicit construction of these triangles; for this purpose we find for any positive integer n an explicit cubic number field ℚ(λ) (depending on n) and an explicit point P λ of infinite order in the Mordell-Weil group of the elliptic curve Y 2 = X 3 − n 2 X over ℚ(λ).

Using an integral of a hypergeometric function, we give necessary and sufficient conditions for irrationality of Euler’s constant γ. The proof is by reduction to known irrationality criteria for γ involving a Beukers-type double integral. We show that the hypergeometric and double integrals are equal by evaluating them. To do this, we introduce a construction of linear forms in 1, γ, and logarithms from Nesterenko-type series of rational functions. In the Appendix, S. Zlobin gives a change-of-variables proof that the series and the double integral are equal.

The aim of this paper is to find main terms of the star D N* and extremal D N discrepancies of the two dimensional sequence (x n, x n+1), n = 0, 1, 2, ..., N − 1, where x n, n = 0, 1, 2, ..., is the van der Corput sequence. This give a quantitative form of a well-known result that van der Corput sequence is not pseudorandom.

On the basis of the Random Matrix Theory-model several interesting conjectures for the Riemann zeta-function were made during the recent past, in particular, asymptotic formulae for the 2kth continuous and discrete moments of the zeta-function on the critical line, $$ \frac{1} {T}\int\limits_0^T {|\zeta (\tfrac{1} {2} + it)|^{2k} dt} and \frac{1} {{N(T)}}\sum\limits_{0 < \gamma \leqslant {\rm T}} {|\zeta (\tfrac{1} {2} + i(\gamma + \tfrac{\alpha } {L}))|^{2k} } $$, by Conrey, Keating et al. and Hughes, respectively. These conjectures are known to be true only for a few values of k and, even under assumption of the Riemann hypothesis, estimates of the expected order of magnitude are only proved for a limited range of k. We put the discrete moment for k = 1, 2 in relation with the corresponding continuous moment for the derivative of Hardy’s Z-function. This leads to upper bounds for the discrete moments which are off the predicted order by a factor of log T.

In a paper of Thuswaldner and Tichy, a version of Waring’s problem with restrictions on the sum of digits was considered. This paper is devoted to a generalization of their result to arbitrary completely q-additive functions.

Let ℛn(t) denote the set of all reducible polynomials p(X) over ℤ with degree n ≥ 2 and height ≤ t. We determine the true order of magnitude of the cardinality |ℛn(t)| of the set ℛn(t) by showing that, as t → ∞, t 2 log t ≪ |ℛ2(t)| ≪ t 2 log t and t n ≪ |ℛn(t)| ≪ t n for every fixed n ≥ 3. Further, for 1 < n/2 < k < n fixed let ℛk,n(t) ⊂ ℛn(t) such that p(X) ∈ ℛk,n(t) if and only if p(X) has an irreducible factor in ℤ[X] of degree k. Then, as t → ∞, we always have t k+1 ≪ |ℛk,n(t)| ≪ t k+1 and hence |ℛn−1,n (t)| ≫ |ℛn(t)| so that ℛn−1,n (t) is the dominating subclass of ℛn(t) since we can show that |ℛn(t)∖ℛn−1,n (t)| ≪ t n−1(log t)2.On the contrary, if R ns(t) is the total number of all polynomials in ℛn(t) which split completely into linear factors over ℤ, then t 2(log t)n−1 ≪ R ns(t) ≪ t 2 (log t)n−1 (t → ∞) for every fixed n ≥ 2.

For an odd prime number p and an integer n ≥ 0, let h n be the class number of the p n+1st cyclotomic field Q($$ \zeta _{p^{n + 1} } $$). It is known that when p = 3 or 5, h n is odd for all n ≥ 0. We prove that the same holds also when p = 7.

The aim of the paper is the investigation of special infinite series of the form $$ \sum\limits_{n = 0}^\infty {\left( {\frac{{\prod\limits_{s = 0}^{m_1 n} {(s + a)} \prod\limits_{s = 0}^{m_2 n} {(s + b)} }} {{\prod\limits_{s = 0}^{(m_1 + m_2 )n + 1} {(s + a + b)} }}} \right)} ^c \cdot \frac{{P(n)}} {{\theta ^n }} \cdot f_n (n) $$ where (a, b, m 1, m 2, θ, c, P(n)) ∈ ℝ4 × ℂ × {±1} × [n] and $$ \{ f_n (n)\} _{n \in \mathbb{N}_0 } $$ is a sequence of rational functions. A general summation method for the sum above in the case of the special choice of parameters a, b and f n(n) is included. We find the 2m-tuple of rational numbers α i, β j (1 ≤ i ≤ m, 1 ≤ j ≤ m) for which iff and vice versa.

In this paper, we consider Owen’s scrambling of an (m−1, m, d)-net in base b which consists of d copies of a (0, m, 1)-net in base b, and derive an exact formula for the gain coefficients of these nets. This formula leads us to a necessary and sufficient condition for scrambled (m − 1, m, d)-nets to have smaller variance than simple Monte Carlo methods for the class of L 2 functions on [0, 1]d. Secondly, from the viewpoint of the Latin hypercube scrambling, we compare scrambled non-uniform nets with scrambled uniform nets. An important consequence is that in the case of base two, many more gain coefficients are equal to zero in scrambled (m − 1, m, d)-nets than in scrambled Sobol’ points for practical size of samples and dimensions.