Related papers: Note on the Prime Number Theorem

In this paper, we develop Furstenberg's proof of infinity of primes, and prove several results about prime divisors of sequences of integers, including the celebrated Schur's theorem. In particular, we give a simple proof of a classical…

The goal of this annotated bibliography is to record every publication on the topic of comparative prime number theory (through mid-2024) together with a summary of its results. We use a unified system of notation for the quantities being…

Formal languages are sets of strings of symbols described by a set of rules specific to them. In this note, we discuss a certain class of formal languages, called regular languages, and put forward some elementary results. The properties of…

We will give a simple proof of the ambiguous class number formula.

In this paper we give effective estimates for some classical arithmetic functions defined over prime numbers. First we find the smallest real number $x_0$ so that some inequality involving Chebyshev's $\vartheta$-function holds for every $x…

Prime numbers or primes are man's eternal treasures that have been cherished for several millennia, until today. As their academic ancestors in ancient Mesopotamia, many mathematicians are still trying hard to see primes better. I shall…

This is a pedagogical article cited in the foregoing research note, quant-ph/9911050

In this paper we establish a number of new estimates concerning the prime counting function \pi(x), which improve the estimates proved in the literature. As an application, we deduce a new result concerning the existence of prime numbers in…

In the first part of this expository paper, we present and discuss the interplay of Dirichlet polynomials in some classical problems of number theory, notably the Lindel\"of Hypothesis. We review some typical properties of their means and…

In our work we give the examples using Fermat's Last Theorem for solving some problems from algebra and number theory.

We define and study a transform whose iterates bring to the fore interesting relations between Pisot numbers and primes. Although the relations we describe are general, they take a particular form in the Pisot limit points. We give three…

We pose 100 new conjectures on representations involving primes or related things, which might interest number theorists and stimulate further research. Below are five typical examples: (i) For any positive integer $n$, there exists…

This version of the paper corrects an inaccuracy in the proof of Theorem 2.9 in the published version. The main results remain unchanged.

We introduce and consider a certain probability question involving elementary number theory and the likelihood that a fixed prime will appear in a certain recursively defined factorization of an integer. We derive several convergent…

We survey some different results on the digits of prime numbers, giving a simplified proof of weak forms of a result of Maynard and Mauduit-Rivat.

This note offers an elementary proof of the Siegel-Walfisz theorem for primes in arithmetic progressions.

This work proposes a complete algebraic model for classical information theory. As a precursor the essential probabilistic concepts have been defined and analyzed in the algebraic setting. Examples from probability and information theory…

In this paper we obtain bounds for integer solutions of quadratic polynomials in two variables that represent a natural number. Also we get some results on twin prime numbers. In addition, we use linear functionals to prove some results of…

This is a survey article on the Hardy-Littlewood conjecture about primes in quadratic progressions. We recount the history and quote some results approximating this hitherto unresolved conjecture.

I review the classical theory of likelihood based inference and consider how it is being extended and developed for use in complex models and sampling schemes.