On representing the multiple of a number by a quadratic form
On restricted m-ary partitions.
On restricted sumsets in abelian groups of odd order.
On resultant inequalities
On Riemann Sums and Lebesgue Integrals.
On Riemann's Zeta Function.
On Robin’s criterion for the Riemann hypothesis
Robin’s criterion states that the Riemann Hypothesis (RH) is true if and only if Robin’s inequality is satisfied for , where denotes the Euler(-Mascheroni) constant. We show by elementary methods that if does not satisfy Robin’s criterion it must be even and is neither squarefree nor squarefull. Using a bound of Rosser and Schoenfeld we show, moreover, that must be divisible by a fifth power . As consequence we obtain that RH holds true iff every natural number divisible by a fifth power...
On Rödseth's h-bases A... 0 ...1, a..., 2a..., ..., (k - 2)a..., a...
On Roots of Polynomials with Positive Coefficients
On rough and smooth neighbors.
We study the behavior of the arithmetic functions defined byF(n) = P+(n) / P-(n+1) and G(n) = P+(n+1) / P-(n) (n ≥ 1)where P+(k) and P-(k) denote the largest and the smallest prime factors, respectively, of the positive integer k.
On Rowland's sequence.
On ruled fields
Some results and problems that arise in connection with the foundations of the theory of ruled and rational field extensions are discussed.
On -fibonomials.
On Saito-Kurokawa Descent for congruence subgroups.
On Schertz's class number formula related to elliptic units for some non-Galois extensions
On second 2-descent and non-congruent numbers
We use the so-called second 2-descent method to find several series of non-congruent numbers. We consider three different 2-isogenies of the congruent elliptic curves and their duals, and find a necessary condition to estimate the size of the images of the 2-Selmer groups in the Selmer groups of the isogeny.
On Selberg's Eigenvalue Conjecture.
On self-similar subgroups in the sense of IFS
In this paper, we first give several properties with respect to subgroups of self-similar groups in the sense of iterated function system (IFS). We then prove that some subgroups of -adic numbers are strong self-similar in the sense of IFS.
On semicontinuity of ramification invariants in dimension 2.
On semi-local binomial units in cubic number fields.