Related papers: An Exact Exprsssion of Pi(x)
In this note we prove an inequality involving primes and the product of consecutive primes.
In this work we show that the prime distribution is deterministic. Indeed the set of prime numbers P can be expressed in terms of two subsets of N using three specific selection rules, acting on two sets of prime candidates. The prime…
One is expressed as the sum of the reciprocals of a certain set of integers. We give an elegant proof to the fact applying the polynomial theorem and basic calculus.
We introduce extremely symmetric primes and provide some elementary properties of these.
This paper describes recent advances in the combinatorial method for computing $\pi(x)$, the number of primes $\leq x$. In particular, the memory usage has been reduced by a factor of $\log x$, and modifications for shared- and…
We prove an explicit formula, analogous to the classical explicit formula for $\psi(x)$, for the Ces\`aro-Riesz mean of any order $k>0$ of the number of representations of $n$ as a sum of two primes. Our approach is based on a double Mellin…
A recurrence formula for absolute central moments of Poisson distribution is suggested.
In this work I look at the distribution of primes by calculation of an infinite number of intersections. For this I use the set of all numbers which are not elements of a certain times table in each case. I am able to show that it exists a…
We provide direct elementary proofs of several explicit expressions for Bernoulli numbers and Bernoulli polynomials. As a byproduct of our method of proof, we provide natural definitions for generalized Bernoulli numbers and polynomials of…
This is an expanded account of three lectures on the distribution of prime numbers given at the Montreal NATO school on equidistribution.
The primary purpose of this article is to study the asymptotic and numerical estimates in detail for higher degree polynomials in $\pi(x)$ having a general expression of the form, \begin{align*} P(\pi(x)) - \frac{e x}{\log x} Q(\pi(x/e)) +…
I show a simple expression of the Mill's ratio of the Student's t-Distribution. I use it to prove Conjecture 1 in P. Auer, N. Cesa-Bianchi, and P. Fischer. Finite-time analysis of the multiarmed bandit problem. Mach. Learn.,…
A mathematical interpretation of the usual definition of entropy (for a discrete probability distribution or a trace 1 positive operator) is given. This formulation makes some properties of entropy immediate.
We give an explicit expression of the normalized characters of the symmetric group in terms of the contents of the partition labelling the representation.
We note an interesting and under-expressed fact from Chebyshev's initial bounding for the prime counting function, $\pi(x) := \# \{p \leq x : p \text{ prime}\},$ based upon a selection of fixed coefficients $d\in D$ to show $\psi(x) \asymp…
The pentagonal number theorem is extended to the sequence of the number of integer partitions with all parts equal. The new pentagonal number theorem implies that the distribution of the primes is just a specific detail of the application…
We present a new, elementary, dynamical proof of the prime number theorem.
Quite recently, in [8] the authoor of this paper considered the distribution of primes in the sequence $(S_n)$ whose $n$th term is defined as $S_n=\sum_{k=1}^{2n}p_k$, where $p_k$ is the $k$th prime. Some heuristic arguments and the…
In this paper, we prove a theorem on the distribution of primes in cubic progressions on average.
Numerical study of the distribution of the Riemann zeros differences following the work [1] shows the significance of the function for which the prime sum expression is proposed. Computational results related to this definition explored…