Related papers: Mertens' Proof of Mertens' Theorem

We make explicit a theorem of Pintz concerning the error term in the prime number theorem. This gives an improved version of the prime number theorem with error term roughly square-root of that which was previously known. We apply this to a…

A work by Nicolas has shown that if it can be proven that a certain inequality holds for all $n$, the Riemann hypothesis is true. This inequality is associated with the Mertens theorem, and hence the Euler totient at $\prod_{k=1}^n p_k$,…

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…

For Hilbert, the consistency of a formal theory T is an infinite series of statements "D is free of contradictions" for each derivation D and a consistency proof is i) an operation that, given D, yields a proof that D is free of…

We survey the classical results on the prime number theorem

A classical theorem of Kempner states that the sum of the reciprocals of positive integers with missing decimal digits converges. This result is extended to much larger families of "missing digits" sets of positive integers with both…

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.

It may seem a funny notion to write about theorems as old and rehashed as Descartes's rule of signs, De Gua's rule or Budan's. Admittedly, these theorems were proved numerous times over the centuries. However, despite the popularity of…

We give a modern account of Agafonov's original proof of his eponymous theorem. The original proof was only reported in Russian in a journal not widely available, and the work most commonly cited in western literature is instead the English…

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

In this article we present the history of auxiliary primes used in proofs of reciprocity laws from the quadratic to Artin's reciprocity law. We also show that the gap in Legendre's proof can be closed with a simple application of Gauss's…

We prove that for a positive integer $k$ the primes in certain kinds of intervals can not distribute too 'uniformly' among the reduced residue classes modulo $k$. Hereby, we prove a generalization of a conjecture of Recaman and establish…

The main purpose of this paper is pedagogical. Despite its importance, all proofs of the correctness of Strassen's famous 1969 algorithm to multiply two 2x2 matrices with only seven multiplications involve some basis-dependent calculations…

In this paper, we derive new properties of the Mertens function and discuss a likely upper bound of the absolute value of the Mertens function $\sqrt{\log{x!}}>|M(x)|$ when $x>1$. Using this likely bound we show that we have a sufficient…

This paper has been withdrawn by the author due to a crucial error in the proof of Theorem 1.

We give a proof of the inconsistency of PM arithmetic, classical set theory and related systems, incidentally exposing an error in Goedel's own proof of Goedel's Theorems. The inconsistency proof, that formulae of the form R and ~R occur as…

I present a simple, elementary proof of Morley's theorem, highlighting the naturalness of this theorem.

In this paper, we discuss an alternative approach to determine an asymptotic equivalent of the partial sum of the reciprocals of prime numbers. This well-known result, related to Merten's second theorem, is usually derived through methods…

The purpose of this article is to present my new proof of the the construction and the convergence theorem of spectral sequences of filtered complexes, which is much shorter and cleaner than the "standard" proof.

The proofs first generated by automated theorem provers are far from optimal by any measure of simplicity. In this paper I describe a technique for simplifying automated proofs. Hopefully this discussion will stimulate interest in the…