Related papers: A proof of Smale's mean value conjecture
It is shown that the Mean Value Theorem for arithmetic functions, and simple properties of the zeta function are sufficient to assemble proofs of the Prime Number Theorem, and Dirichlet Theorem. These are among the simplest proofs of the…
We survey the classical results on the prime number theorem
In this paper we give an elementary proof of the local sum conjecture in two dimensions. In a remarkable paper [CMN, arXiv:1810.11340], this conjecture has been established in all dimensions using sophisticated, powerful techniques from a…
We give a counterexample to a recently conjectured variant of the Penrose inequality.
We proved a Bernstein theorem of ancient solutions to mean curvature flow.
Some extensions of an inequality from IMO'2001 are proven by means of the Lagrange multiplier criterion.
In this note, we extend to a composite modulo a recent result of Chan (2016) dealing with mean values of the product of an integer and its multiplicative inverse modulo a prime number.
In this article, we prove an asymptotic formula for mean values of long Dirichlet polynomials with higher order shifted divisor functions, assuming a smoothed additive divisor conjecture for higher order shifted divisor functions. As a…
We find a nontrivial upper bound on the average value of the function M(n) which associates to every positive integer n the minimal Hamming weight of a multiple of n. Some new results about the equation M(n)=M(n') are given.
We show that the sign constancy for the values of certain weighted summatory functions of the von Mangoldt function implies the Riemann hypothesis or the generalized Riemann hypothesis for Dirichlet $L$-functions. While such sign constancy…
We give a one-sentence elementary proof of the combinatorial Fa\`a di Bruno's formula.
A conjecture of Woods from 1972 is disproved.
This paper gives an explicit version of Selberg's 1943 mean-value estimate for the prime number theorem in intervals under the Riemann hypothesis. Two applications are given: for primes in short intervals, and Goldbach numbers (sums of two…
In this note I go through the `proof' of frequentistic confidence intervals and show what it logically implies concerning the value of a physical quantity given an experimental observation (nothing).
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 prove existence of conjugate Galois representations, and we use it to derive a simple method of weight reduction. As a consequence, an alternative proof of the level 1 case of Serre's conjecture follows.
This is a collection of variants of Schanuel's conjecture and the known dependencies between them. It was originally written in 2007, and made available for a time on my webpage. I have been asked by a few people to make it available again…
We put a new conjecture on primes from the point of view of its binary expansions and make a step towards justification.
We answer a question of Slaman and Steel by showing that a version of Martin's conjecture holds for all regressive functions on the hyperarithmetic degrees. A key step in our proof, which may have applications to other cases of Martin's…
We prove a partition identity conjectured by Lassalle (Adv. in Appl. Math. 21 (1998), 457-472).