arXiv subjects
Publications and source records attributed to Sadegh Nazardonyavi.
Let $n$ be a positive integer. We study the diophantine equation $ab(ab-1)-na=Δ^2$, where $a,b$ are positive integers. We also show that if a system of two congruences is soluble, then an equation which is a translation of Erdős-Straus conjecture is soluble.
Robin's theorem is one of the ingenious reformulation of the Riemann hypothesis (RH). It states that the RH is true if and only if $σ(n) 5040$ where $σ(n)$ is the sum of divisors of $n$ and $γ$ is Euler's constant. In this paper we show that how the RH is delicate in terms of certain subsets of superabundant numbers, namely extremely abundant numbers and some of its specific supersets.
Let σ(n) be the sum of divisors of a positive integer n. Robin's theorem states that the Riemann hypothesis is equivalent to the inequality σ(n) 5040 (γis Euler's constant). It is a natural question in this direction to find a first integer, if exists, which violates this inequality. Following this process, we introduce a new sequence of numbers and call it as extremely abundant numbers. In this paper we show that the Riemann hypothesis is true, if and only if, there are infinitely many of these numbers. Moreover, we investigate some of their properties together with superabundant and colossally abundant numbers.
In this article we present some improved results for Chebyshev's functions $\vartheta$ and $ψ$ using the new zero-free region obtained by H. Kadiri and the calculated the first $10^{13}$ zeros of the Riemann zeta function on the critical line by Xavier Gourdon. The methods in the proofs are similar to those of Rosser-Shoenfeld papers on this subject.
By using new power inequalities we give an elementary proof of an important relation for the Riemann zeta-function |ζ(1-s)| <= |ζ(s)| in the strip 0< Re s<1/2,\ |\Im s| >= 12. Moreover, we establish a sufficient condition of the validity of the Riemann hypothesis in terms of the derivative with respect to Re s of |ζ(s)|^2 and conjecture its necessity.
This article is a collected information from some books and papers, and in most cases the original sentences is reserved about twin prime conjecture.