Related papers: A proof of Smale's mean value conjecture

The conjecture by Steklov was solved negatively by Rakhmanov in 1979. His original proof was based on the formula for orthogonal polynomial obtained by adding point masses to the measure of orthogonality. In this note, we show how this…

We formulate and discuss a conjecture which would extend a classical inequality of Bernstein.

The paper deals with some elementary problems about various mean value properties and their connections to harmonic functions and random walks.

Assuming the Riemann hypothesis, we obtain a formula for the mean value of the $k$-derivative of $\zeta'/\zeta$, depending on the pair correlation of zeros of the Riemann zeta-function. This formula allows us to obtain new equivalences to…

We prove an elementary lemma concerning primitive amalgams and use it to greatly simplify the proof of the Sims conjecture in the case of almost simple groups.

Over one year ago, a very long preprint posted on arXiv [arXiv:1709.03771] and HAL announced a proof of Lehmer's Conjecture (and of other related results). Unfortunately, as was remarked by several specialists, this proof contains a (at…

We continue investigations on the average number of representations of a large positive integer as a sum of given powers of prime numbers. The average is taken over a short interval, whose admissible length depends on whether or not we…

More than a century ago, G. Kowalewski stated that for each n continuous functions on a compact interval [a,b], there exists an n-point quadrature rule (with respect to Lebesgue measure on [a,b]), which is exact for given functions. Here we…

We prove Dejean's conjecture. Specifically, we show that Dejean's conjecture holds for the last remaining open values of n, namely 15 <= n <= 26.

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

A conjecture is given that, if true, could lead to an algorithm for computing definite sums of rational functions.

We generalise our still-wide-open $q$-rious positivity conjecture from 2011 to a $q$-rious unimodality conjecture.

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

We give some explicit calculations for stable distributions and convergence to them, mainly based on less explicit results in Feller (1971). The main purpose is to provide ourselves with easy reference to explicit formulas and examples.…

A short, fairly self-contained proof is given of the Poincar\'e Conjecture. In the previous version there was an error on Page 8. This gap has now been filled.

We prove the following conjecture, due to Tomaszewski (1986): Let $X= \sum_{i=1}^{n} a_{i} x_{i}$, where $\sum_i a_i^2=1$ and each $x_i$ is a uniformly random sign. Then $\Pr[|X|\leq 1] \geq 1/2$. Our main novel tools are local…

The paper gives a unified and simple proof of both theorems and Cousin's theorem.

We prove the numerical Hodge standard conjecture for the square of a simple abelian variety of prime dimension, and also in some related cases.

Let $d_i(m)$ denote the coefficients of the Boros-Moll polynomials. Moll's minimum conjecture states that the sequence $\{i(i+1)(d_i^2(m)-d_{i-1}(m)d_{i+1}(m))\}_{1\leq i \leq m}$ attains its minimum with $i=m$. This conjecture is a…

The purpose of this short note is to present a simplified proof of Serre's modularity conjecture using the strong modularity lifting results currently available. This second version includes extra details on definitions and proofs than the…