Related papers: Mertens' theorem and prime number theorem for Selb…
We present a proof given by Euler in his paper {\it ``De serierum determinatione seu nova methodus inveniendi terminos generales serierum"} \cite{E189} (E189:``On the determination of series or a new method of finding the general terms of…
Let $S(t) = \tfrac{1}{\pi} \arg \zeta \big({1/2} + it \big)$ be the argument of the Riemann zeta-function at the point $\tfrac12 + it$. For $n \geq 1$ and $t>0$ define its antiderivatives as \begin{equation*} S_n(t) = \int_0^t S_{n-1}(\tau)…
Assuming the Generalized Riemann Hypothesis we obtain uniform, effective number-field analogues of Mertens' theorems.
We prove mod-Gaussian convergence for a Dirichlet polynomial which approximates $\operatorname{Im}\log\zeta(1/2+it)$. This Dirichlet polynomial is sufficiently long to deduce Selberg's central limit theorem with an explicit error term.…
The present paper is a report on joint work with Alessandro Languasco and Alberto Perelli on our recent investigations on the Selberg integral and its connections to Montgomery's pair-correlation function. We introduce a more general form…
We study the value distribution of the Riemann zeta function near the line $\Re s = 1/2$. We find an asymptotic formula for the number of $a$-values in the rectangle $ 1/2 + h_1 / (\log T)^\theta \leq \Re s \leq 1/2+ h_2 /(\log T)^\theta $,…
This research article provides an unconditional proof of an inequality proposed by Srinivasa Ramanujan involving the Prime Counting Function $\pi(x)$, \begin{align*} (\pi(x))^{2}<\frac{ex}{\log x}\pi\left(\frac{x}{e}\right) \end{align*} for…
In his Classical approximation to the Twin prime problem, Selberg proved that for $x$ sufficiently large, there is an $n \in (x,2x)$ such that $2^{\Omega(n)}+2^{\Omega(n+2)} \leq \lambda$ with $\lambda=14$, where $\Omega(n)$ is the number…
We give a large sieve type inequality for functions supported on primes. As application we prove a conjecture by Elliott, and give bounds for short character sums over primes. The proves uses a combination of the large sieve and the Selberg…
In this note we prove that the Selberg zeta-function associated to a compact Riemann surface is pseudo-prime and right-prime in the sense of a decomposition.
First idea is to compute a quantity like the angular momentum with respect to (0, 0), of an unitary mass of coordinates (<[Xi(s)], =[Xi(s)]) while =[s] is the time, and, <[s] = constant. If we impose that the derivative along <[s], at…
The Riemann Hypothesis has been of central interest to mathematicians for a long time and many unsuccessful attempts have been made to either prove or disprove it. Since the Riemann zeta function is defined as a sum of the infinite number…
The Selberg correlation integrals are averages of the products $\prod_{s=1}^m\prod_{l=1}^n (x_s - z_l)^{\mu_s}$ with respect to the Selberg density. Our interest is in the case $m=1$, $\mu_1 = \mu$, when this corresponds to the $\mu$-th…
This paper has two parts. The first part surveys Euler's work on the constant gamma=0.57721... bearing his name, together with some of his related work on the gamma function, values of the zeta function and divergent series. The second part…
Translation from the Latin of Euler's "Demonstratio theorematis circa ordinem in summis divisorum observatum" (1760). E244 in the Enestroem index. In his previous paper E243, Euler stated the pentagonal number theorem and assuming it proved…
A well-known result, due to Dirichlet and later generalized by de la Vallee-Poussin, expresses a relationship between the sum of fractional parts and the Euler-Mascheroni constant. In this paper, we prove an asymptotic relationship between…
We show how to control the error term in Mertens' formula and related theorems in the context of additive arithmetical semigroups and carry over an old related result of Meissel.
The idea of generating integrals analogous to generating functions is first introduced in this paper. A new proof of the well-known Finite Harmonic Series Theorem in Analysis and Analytical Number Theory is then obtained by the method of…
In 1859, Riemann had announced the following conjecture : the nontrivial roots (zeros) $s=\alpha+i\beta$ of the zeta function, defined by: $$\zeta(s) =\displaystyle \sum_{n=1}^{+\infty}\frac{1}{n^s},\,\mbox{for}\quad \Re(s)>1$$ have real…
In this article, we derive an expression for the complex magnitude of the Dirichlet beta function $\beta(s)$ represented as a Euler prime product and compare with similar results for the Riemann zeta function. We also obtain formulas for…