Related papers: On the Distribution of Products of Primes and Powe…
There are two basic number sequences which play a major role in the prime number distribution. The first Number Sequence SQ1 contains all prime numbers of the form 6n+5 and the second Number Sequence SQ2 contains all prime numbers of the…
Asymptotic formulae are established for the number of natural numbers $m$ with largest square-free divisor not exceeding $m^{\vartheta}$, for any fixed positive parameter $\vartheta$. Related counting functions are also considered.
A semiprime is a natural number which can be written as the product of two primes. The asymptotic behaviour of the function $\pi_2(x)$, the number of semiprimes less than or equal to $x$, is studied. Using a combinatorial argument,…
The distribution of prime numbers is here considered. We show a formula for $li^{-1}$ and we study the $\pi(x)$ function and Riemann's hypothesis.
In probability theory, there exist discrete and continuous distributions. Generally speaking, we do not have sufficient kinds and properties of discrete ones compared to the continuous ones. In this paper, we treat the Riemann zeta…
We provide examples of multiplicative functions $f$ supported on the $k$-free integers such that at primes $f(p)=\pm 1$ and such that the partial sums of $f$ up to $x$ are $o(x^{1/k})$. Further, if we assume the Generalized Riemann…
We determine the asymptotic density $\delta_k$ of the set of ordered $k$-tuples $(n_1,...,n_k)\in \N^k, k\ge 2$, such that there exists no prime power $p^a$, $a\ge 1$, appearing in the canonical factorization of each $n_i$, $1\le i\le k$,…
Assuming the Generalized Riemann Hypothesis, we obtain a lower bound within a constant factor of the conjectured asymptotic result for the second moment for primes in an individual arithmetic progression in short intervals. Previous results…
We prove several asymptotic continued fraction expansions of $\pi(x)$, $\Pi(x)$, $\operatorname{li}(x)$, $\operatorname{Ri}(x)$, and related functions, where $\pi(x)$ is the prime counting function, $\Pi(x) = \sum_{k = 1}^\infty…
We show by an inclusion-exclusion argument that the prime $k$-tuple conjecture of Hardy and Littlewood provides an asymptotic formula for the number of consecutive prime numbers which are a specified distance apart. This refines one aspect…
We study some counting questions concerning products of positive integers $u_1, \ldots, u_n$ which form a non-zero perfect square, or more generally, a perfect $k$-th power. We obtain an asymptotic formula for the number of such integers of…
We introduce a new set of prime numbers functions including an exact Generating Function and a Discriminating Function of Prime Numbers neither based on prime number tables nor on algorithms. Instead these functions are defined in terms of…
We provide an asymptotic estimate for certain sums over k-free integers with small prime factors. These sums depend upon a complex parameter \alpha and involve a smooth cut-off f. They are a variation of several classical number-theoretical…
We derive asymptotic estimates for the coefficient of $z^{k}$ in $\left( f\left( z\right) \right) ^{n}$ when $n\rightarrow \infty $ and $k$ is of order $n^{\delta }$, where $0<\delta <1,$ and $f\left( z\right) $ is a power series satisfying…
This research presents the results of a study on the existence and frequency distribution of the shell primes defined herein as prime numbers that result from the calculation of the "half-shell" of an p-dimensional entity of the form…
This paper is concerned with the asymptotic behavior of sums of terms which are a test function f evaluated at successive increments of a discretely sampled semimartingale. Typically the test function is a power function (when the power is…
We study the distribution of primes from a topological viewpoint. Certain conjecture is introduced, and we show that it is equivalent to the Riemann Hypothesis.
A theorem of Meinardus provides asymptotics of the number of weighted partitions under certain assumptions on associated ordinary and Dirichlet generating functions. The ordinary generating functions are closely related to Euler's…
Random integers, sampled uniformly from $[1,x]$, share similarities with random permutations, sampled uniformly from $S_n$. These similarities include the Erd\H{o}s--Kac theorem on the distribution of the number of prime factors of a random…
In this note we prove an inequality involving primes and the product of consecutive primes.