Sadegh Nazardonyavi

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 < <s < 1/2, |=s | ≥ 12. Moreover, we establish a sufficient condi-tion of the validity of the Riemann hypothesis in terms of the derivative with respect to <s of |ζ(s)|2 and conjecture its necessity.

Let $n$ be a positive integer. We study the diophantine equation $ab(ab-1)-na=\Delta^2$, 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 states that the Riemann hypothesis is equivalent to the inequality σ(n) < en log log n for all n > 5040, where σ(n) is the sum of divisors of n and γ is Euler’s constant. It is natural to seek the first integer, if it exists, that violates this inequality. We introduce the sequence of extremely abundant numbers, a subsequence of superabundant numbers, where one might look for this first violating integer. The Riemann hypothesis is true if and only if there are infinitely many extremely abundant numbers. These numbers have some connection to the colossally abundant numbers. We show the fragility of the Riemann hypothesis with respect to the terms of some supersets of extremely abundant numbers.

We study the equations m∕n=1∕x±1∕y±1∕z, where m=4,5,6,7 and n>1 is a given positive integer. Except for the case m∕n=1∕x+1∕y+1∕z (i.e., all signs are positive), which are known conjectures of Erdős–Straus (m=4), Sierpinski (m=5) and Schinzel (m≥1), we show that the equations (with at least one minus sign) have solutions x,y,z, where z>y>x>0 for sufficiently large n.

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 $\sigma(n) 5040$ where $\sigma(n)$ is the sum of divisors of $n$ and $\gamma$ 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.

By using new power inequalities we give an elementary proof of an important relation for the Riemann zeta-function |\zeta(1-s)| <= |\zeta(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 |\zeta(s)|^2 and conjecture its necessity.

While studying the Riemann zeta-function, observing some graphs related to it and looking for some relation between the Riemann hypothesis and the absolute value of the Riemann zeta function, we noted an interesting relationship between those values, namely, that in the strip 0 5040, where γ is Euler’s constant. Inspired by this inequality, we introduce a sequence of numbers, that we call extremely abundant, and 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 and structure, as well as some properties of superabundant and colossally abundant numbers. Finally we introduce two other sets of numbers, related to extremely abundant numbers, that show how subtle the Riemann hypothesis is.

In this article we present some improved results for Chebyshev&#39;s functions $\vartheta$ and $\psi$ 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.