Related papers: An Exact Formula for the Prime Counting Function

The aim of this article is to investigate how various Riemann Hypotheses would follow only from properties of the prime numbers. To this end, we consider two classes of $L$-functions, namely, non-principal Dirichlet and those based on cusp…

In this article, with a new approach, which is not discussed in the literature yet, the limit of the Riemann zeta function or Euler-Riemann zeta function is approximately explored by applying Dirichlet's rearrangement theorem for absolutely…

The Riemann hypothesis is equivalent to the $\varpi$-form of the prime number theorem as $\varpi(x) =O(x\sp{1/2} \log\sp{2} x)$, where $\varpi(x) =\sum\sb{n\le x}\ \bigl(\Lambda(n) -1\big)$ with the sum running through the set of all…

Let the summatory function of the M\"{o}bius function be denoted $M(x)$. We deduce in this article conditional results concerning $M(x)$ assuming the Riemann Hypothesis and a conjecture of Gonek and Hejhal on the negative moments of the…

As long as people have studied mathematics, they have wanted to know how many primes there are. Getting precise answers is a notoriously difficult problem, and the first suitable technique, due to Riemann, inspired an enormous amount of…

In this paper, by introducing a new operation in the vector space of Laurent series, the author derived explicit series for the values of $\zeta$-funtion at positive integers, where $\zeta$ denotes the Riemann zeta function. The values of…

We study Birkhoff sums over rotations (series of the form $\sum_{r=1}^{N}\phi(r\alpha)$), in which the summed function $\phi$ may be unbounded at the origin. Estimates of these sums have been of significant interest and application in pure…

We exploit transformations relating generalized $q$-series, infinite products, sums over integer partitions, and continued fractions, to find partition-theoretic formulas to compute the values of constants such as $\pi$, and to connect sums…

In this paper we extend the Zeta function regularization technique, which gives a meaningful solution to divergent power series, in order to assign finite values to divergent integral of certain transcendental functions $f(x)$. The…

We will derive a function that eliminates any sequence of equidistant numbers from the integer numbers, then we will derive its inverse. Then we will use the Sequence elimination function to eliminate the multiples of the prime numbers from…

Factorial series played a major role in Stirling's classic book "Methodus Differentialis" (1730), but now only a few specialists still use them. This article wants to show that this neglect is unjustified, and that factorial series are…

It is known that there are infinitely-many prime numbers which take the form of a polynomial of degree one with integer coefficients, this is Dirichlet's theorem. We use an elementary sieving argument together with bounds on the prime…

We conjecture average counting functions for prime $k$-tuples based on a gamma distribution hypothesis for prime powers. The conjecture is closely related to the Hardy-Littlewood conjecture for $k$-tuples but yields better estimates.…

Let [t] be the integral part of the real number t and let 1 P be the characteristic function of the primes. Denote by $\pi$ G (x) the number of primes in the floor function set G(x) := {[ x n ] : 1 n x} and by S 1 P (x) the number of primes…

The paper provides the proof of the Rimann's conjecture. The results of the works of A. M. Odlyzko and H. te Riile "Disproof of the Conjecture", which gives a disproof of the Mertens hypothesis, using to prove the Riemann's hypothesis. This…

We are interested in Moebius function and related topics!

Let phi denote Euler's phi function. For a fixed odd prime we give an asymptotic series expansion in the sense of Poincare for the number E_q(x) of n<=x such that q does not divide phi(n). Thereby we improve on a recent theorem of B.K.…

We study the Dirichlet series associated with the integers whose radix-$b$ representation misses certain (fixed) digits. The existence of a meromorphic continuation to the entire complex plane, which was already well-known as a general fact…

For a real number $\alpha$ the Hilbert spaces $\mathscr{D}_\alpha$ consists of those Dirichlet series $\sum_{n=1}^\infty a_n/n^s$ for which $\sum_{n=1}^\infty |a_n|^2/[d(n)]^\alpha < \infty$, where $d(n)$ denotes the number of divisors of…

We prove a prime number theorem first for the classical Rankin-Selberg L-function $L(s,\pi\times\pi')$ over any Galois extension with $\pi$ and $\pi'$ unitary automorphic cuspidal representations of $GL_n$ and $GL_m$ respectively with at…