On the Rank of Elliptic Curves in Elementary Cubic Extensions

We give a method for explicitly constructing an elementary cubic extension L over which an elliptic curve ED:y2+Dy=x3 (D∈Q∗) has Mordell-Weil rank of at least a given positive integer by finding a close connection between a 3-isogeny of ED and a generic polynomial for cyclic cubic extensions. In our method, the extension degree [L:Q] often becomes small.

Hybrid Moments of the Riemann Zeta-Function

The “hybrid” moments ∫T2Tζ1/2+itk∫t-Gt+Gζ1/2+ixldxmdt Tε≪G=GT≪T of the Riemann zeta-function ζs on the critical line Res=1/2 are studied. The expected upper bound for the above expression is Oε(T1+εGm). This is shown to be true for certain specific values of k,l,m∈N, and the explicitly determined range of G=G(T;k,l,m). The application to a mean square bound for the Mellin transform function of ζ1/2+ix4 is given.

New General Theorems and Explicit Values of the Level 13 Analogue of Rogers-Ramanujan Continued Fraction

We prove general theorems for the explicit evaluations of the level 13 analogue of Rogers-Ramanujan continued fraction and find some new explicit values. This work is a sequel to some recent works of S. Cooper and D. Ye.

Using Continued Fractions to Compute Iwasawa Lambda Invariants of Imaginary Quadratic Number Fields

Let l>3 be a prime such that l≡3 (mod 4) and Q(l) has class number 1. Then Hirzebruch and Zagier noticed that the class number of Q(-l) can be expressed as h(-l)=(1/3)(b1+b2+⋯+bm)-m where the bi are partial quotients in the “minus” continued fraction expansion l=[[b0;b1,b2,…,bm¯]]. For an odd prime p≠l, we prove an analogous formula using these bi which computes the sum of Iwasawa lambda invariants λp(-l)+λp(-4) of Q(-l) and Q(-1). In the case that p is inert in Q(-l), the formula pleasantly simplifies under some additional technical assumptions.

Representing and Counting the Subgroups of the Group Zm×Zn

We deduce a simple representation and the invariant factor decompositions of the subgroups of the group Zm×Zn, where m and n are arbitrary positive integers. We obtain formulas for the total number of subgroups and the number of subgroups of a given order.

On Second Order Gap Balancing Numbers

We consider the Diophantine equation 1k+2k+⋯+x-1k=x+2k+x+3k+⋯+x+rk for some natural numbers x, k, and r, and we call 2x+1 as kth order 2-gap balancing number. It was also proved that there are infinitely many first order 2-gap balancing numbers. In this paper, we show that the only second order 2-gap balancing number is 1.

A Mean Value Formula for Elliptic Curves

It is proved in this paper that, for any point on an elliptic curve, the mean value of x-coordinates of its n-division points is the same as its x-coordinate and that of y-coordinates of its n-division points is n times that of its y-coordinate.

Algebraic Numbers Satisfying Polynomials with Positive Rational Coefficients

A theorem of Dubickas, affirming a conjecture of Kuba, states that a nonzero algebraic number is a root of a polynomial f 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 f 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.

On the Equation y2=x3-pqx

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.

Generations of Correlation Averages

We give a general link between weighted Selberg integrals of any arithmetic function f and averages of f 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 dk(n), gauged by the Cesaro weight in the short interval n∈x-H,x+H and improved by these some recent results by Ivić. The same link provides, also, an unconditional improvement. Then, some remarkable conditional implications on the 2kth moments of Riemann zeta function on the critical line are derived. We also give general requirements on f that allow our treatment for f weighted Selberg integrals.