Journal of Numbers http://www.hindawi.com The latest articles from Hindawi Publishing Corporation © 2014 , Hindawi Publishing Corporation . All rights reserved. On the Equation Wed, 16 Jul 2014 08:01:19 +0000 http://www.hindawi.com/journals/jn/2014/825634/ We consider certain quartic twists of an elliptic curve. We establish the rank of these curves under the Birch and Swinnerton-Dyer conjecture and obtain bounds on the size of Shafarevich-Tate group of these curves. We also establish a reduction between the problem of factoring integers of a certain form and the problem of computing rational points on these twists. Iftikhar A. Burhanuddin and Ming-Deh A. Huang Copyright © 2014 Iftikhar A. Burhanuddin and Ming-Deh A. Huang. All rights reserved. Algebraic Numbers Satisfying Polynomials with Positive Rational Coefficients Wed, 16 Jul 2014 07:55:12 +0000 http://www.hindawi.com/journals/jn/2014/296828/ A theorem of Dubickas, affirming a conjecture of Kuba, states that a nonzero algebraic number is a root of a polynomial with positive rational coefficients if and only if none of its conjugates is a positive real number. A certain quantitative version of this result, yielding a growth factor for the coefficients of similar to the condition of the classical Eneström-Kakeya theorem of such polynomial, is derived. The bound for the growth factor so obtained is shown to be sharp for some particular classes of algebraic numbers. Vichian Laohakosol and Suton Tadee Copyright © 2014 Vichian Laohakosol and Suton Tadee. All rights reserved. Generations of Correlation Averages Wed, 18 Jun 2014 06:02:20 +0000 http://www.hindawi.com/journals/jn/2014/140840/ We give a general link between weighted Selberg integrals of any arithmetic function and averages of correlations in short intervals, proved by the elementary dispersion method (our version of Linnik’s method). We formulate conjectural bounds for the so-called modified Selberg integral of the divisor functions , gauged by the Cesaro weight in the short interval and improved by these some recent results by Ivić. The same link provides, also, an unconditional improvement. Then, some remarkable conditional implications on the 2th moments of Riemann zeta function on the critical line are derived. We also give general requirements on that allow our treatment for weighted Selberg integrals. Giovanni Coppola and Maurizio Laporta Copyright © 2014 Giovanni Coppola and Maurizio Laporta. All rights reserved. Complex Roots of Unity and Normal Numbers Thu, 12 Jun 2014 10:55:29 +0000 http://www.hindawi.com/journals/jn/2014/437814/ Given an arbitrary prime number , set . We use a clever selection of the values of , in order to create normal numbers. We also use a famous result of André Weil concerning Dirichlet characters to construct a family of normal numbers. Jean Marie De Koninck and Imre Kátai Copyright © 2014 Jean Marie De Koninck and Imre Kátai. All rights reserved. Continued Fractions of Order Six and New Eisenstein Series Identities Tue, 27 May 2014 06:29:32 +0000 http://www.hindawi.com/journals/jn/2014/643241/ We prove two identities for Ramanujan’s cubic continued fraction and a continued fraction of Ramanujan, which are analogues of Ramanujan’s identities for the Rogers-Ramanujan continued fraction. We further derive Eisenstein series identities associated with Ramanujan’s cubic continued fraction and Ramanujan’s continued fraction of order six. Chandrashekar Adiga, A. Vanitha, and M. S. Surekha Copyright © 2014 Chandrashekar Adiga et al. All rights reserved. On a Rankin-Selberg -Function over Different Fields Sun, 27 Apr 2014 11:42:06 +0000 http://www.hindawi.com/journals/jn/2014/314173/ Given two unitary automorphic cuspidal representations and defined on and , respectively, with and being Galois extensions of , we consider two generalized Rankin-Selberg -functions obtained by forcefully factoring   and  . We prove the absolute convergence of these -functions for . The main difficulty in our case is that the two extension fields may be completely unrelated, so we are forced to work either “downstairs” in some intermediate extension between and , or “upstairs” in some extension field containing the composite extension . We close by investigating some special cases when analytic continuation is possible and show that when the degrees of the extension fields and are relatively prime, the two different definitions give the same generating function. Tim Gillespie Copyright © 2014 Tim Gillespie. All rights reserved. On the Distribution of -Tuples of -Free Numbers Sun, 13 Apr 2014 00:00:00 +0000 http://www.hindawi.com/journals/jn/2014/537606/ For k, being a fixed integer ≥2, a positive integer n is called k-free number if n is not divisible by the kth power of any integer >1. In this paper, we studied the distribution of r-tuples of k-free numbers and derived an asymptotic formula. Ting Zhang and Huaning Liu Copyright © 2014 Ting Zhang and Huaning Liu. All rights reserved. Sums of Products Involving Power Sums of Integers Thu, 27 Feb 2014 16:34:23 +0000 http://www.hindawi.com/journals/jn/2014/158351/ A sequence of rational numbers as a generalization of the sequence of Bernoulli numbers is introduced. Sums of products involving the terms of this generalized sequence are then obtained using an application of Faà di Bruno's formula. These sums of products are analogous to the higher order Bernoulli numbers and are used to develop the closed form expressions for the sums of products involving the power sums which are defined via the Möbius function μ and the usual power sum of a real or complex variable . The power sum is expressible in terms of the well-known Bernoulli polynomials by . Jitender Singh Copyright © 2014 Jitender Singh. All rights reserved.