Related papers: A proof of Smale's mean value conjecture
We present a new, elementary, dynamical proof of the prime number theorem.
We formulate several conjectures on mean convex domains in the Euclidean spaces, as well as in more general spaces with lower bonds on their scalar curvatures, and prove a few theorems motivating these conjectures.
We obtain new mean value theorems for exponential sums with very smooth numbers, which provide a power saving against the trivial bound in region where previous bounds do not apply.
We provide a lower bound on the probability that a binomial random variable is exceeding its mean. Our proof employs estimates on the mean absolute deviation and the tail conditional expectation of binomial random variables.
Assuming the Riemann Hypothesis we obtain asymptotic estimates for the mean value of the number of representations of an integer as a sum of two primes. By proving a corresponding Omega-term, we prove that our result is essentially the best…
We report the results of our empirical investigations on the Bateman-Horn conjecture. This conjecture, in its commonly known form, produces rather large deviations when the polynomials involved are not monic. We propose a modified version…
In this paper, using combinatorial and analytic methods, we prove an exact calculating formula on the $2m$-th power mean value of the generalized quadratic Gauss sums for $m\geq 2$. This solves a conjecture of He and Zhang [`On the $2k$-th…
The article provides a counterexample to a conjecture by Blocki-Zwonek.
We provide elementary proof of several congruences involving single sum and multisums of binomial coefficients.
Brown's 1971 paper "Admissible estimators, recurrent diffusions and insoluble boundary value problems" is a landmark in the admissibility literature. It nearly completely settles the issue of admissibility/inadmissibility for estimating the…
It is shown that the formula for the variance of combined series yields surprisingly simple proofs of some well known variance bounds.
Based on the results people have obtained, we try to prove the Jacobian conjecture, but there is a gap in the proof.
A very simple but useful almost sure convergence theorem of probability is given.
We present an astonishingly simple and elegant proof of the celebrated Basel problem.
We show that the naive adaptation of Malle's conjecture to fair counting functions is not true in general.
Two numbers $m$ and $n$ are considered amicable if the sum of their proper divisors, $s(n)$ and $s(m)$, satisfy $s(n) = m$ and $s(m) = n$. In 1981, Pomerance showed that the sum of the reciprocals of all such numbers, $P$, is a constant. We…
In this note, assuming a variant of the Generalized Riemann Hypothesis, which does not exclude the existence of real zeros, we prove an asymptotic formula for the mean value of the representation function for the sum of two primes in…
General considerations on the Equivalence conjectures and a review of few mathematical results.
For most purposes, one can replace the use of Rolle's theorem and the mean value theorem, which are not constructively valid, by the law of bounded change. The proof of two basic results in numerical analysis, the error term for Lagrange…
Several results about the union-closed sets conjecture are presented.