Related papers: Note on the Prime Number Theorem

We survey the classical results of the Dirichlet Approximation Theorem.

We present a new, elementary, dynamical proof of the prime number theorem.

I give some claims on primorial prime numbers for interested readers in number theory.

In the paper, the occurrence of zeros and ones in the binary expansion of the primes is studied. In particular the statement in the title is established. The proof is unconditional.

We prove some theorems which give sufficient conditions for the existence of prime numbers among the terms of a sequence which has pairwise relatively prime terms.

Comparative prime number theory is the study of the {\em{discrepancies}} of distributions when we compare the number of primes in different residue classes. This work presents a list of the problems being investigated in comparative prime…

Building on the concept of pretentious multiplicative functions, we give a new and largely elementary proof of the best result known on the counting function of primes in arithmetic progressions.

Expressions of type $(p^q-1)/(p-1)$ and $a^2+ab+b^2$, where $a, b$ are natural and $p, q$ are prime numbers, are studied.

To determine whether a number is congruent or not is an old and difficult topic and progress is slow. The paper presents a new theorem when a prime number is a congruent number or not. The proof is not necessarily any simpler or shorter…

In this note we generalise a method of Perott to give new proofs that there are infinitely many prime numbers.

We furnish an explicit bound for the prime number theorem in short intervals on the assumption of the Riemann hypothesis.

In this expository article we provide an elegant proof of the one-sided Ingham-Karamata Tauberian theorem. As an application, we present a short deduction of the prime number theorem.

We present a detailed proof of the prime number theorem suitable for a typical undergraduate- or graduate-level complex analysis course. Our presentation is particularly useful for any instructor who seeks to use the prime number theorem…

Dirichlet's theorem on arithmetic progressions called as Dirichlet prime number theorem is a classical result in number theory. Atle Selberg\cite{Selberg} gave an elementary proof of this theorem. In this article we give an alternative…

While the prime numbers have been subject to mathematical inquiry since the ancient Greeks, the accumulated effort of understanding these numbers has - as Marcus du Sautoy recently phrased it - 'not revealed the origins of what makes the…

Definition of the number of prime numbers in the given interval

We announce a number of conjectures associated with and arising from a study of primes and irrationals in $\mathbb{R}$. All are supported by numerical verification to the extent possible.

We survey some results that provide different versions of classical results through different summability methods. Specifically, in order to adapt such classical results, we analyze which properties should satisfy the summability methods.…

In this paper, we will give some estimation for the average error of the prime number theorem.

The theorem presented in this paper allows the creation of large prime numbers (of order up to o(n^2)) given a table of all primes up to n.