Related papers: New estimates for the $n$th prime number

Let $X$ be a large parameter. We will first give a new estimate for the integral moments of primes in short intervals of the type $(p,p+h]$, where $p\leq X$ is a prime number and $h=\odi{X}$. Then we will apply this to prove that for every…

Let $\mathsf{TH}_k$ denote the $k$-out-of-$n$ threshold function: given $n$ input Boolean variables, the output is $1$ if and only if at least $k$ of the inputs are $1$. We consider the problem of computing the $\mathsf{TH}_k$ function…

This article establishes a new upper bound on the function $\sigma^{*}(n)$, the sum of all coprime divisors of $n$. The article concludes with two questions concerning this function.

In this note, we approximate the average of prime powers in the decomposition of $n!$ into prime numbers.

Assume the Riemann hypothesis throughout. We obtain some new estimates for the size of the set of large values of the error term in the prime number theorem. Our argument is based on an analysis of the behavior of zeros of the Riemann zeta…

In this paper, we are going to prove a famous problem concerning prime numbers. Bertrand postulate states that there is always a prime p with n < p < 2n, if n > 1. Bertrand postulate is not a newer one to be proven, in fact, after his…

We continue to study the distribution of prime numbers $p$, satisfying the condition $\{ p^{\alpha} \} \in I \subset [0; 1)$, in arithmetic progressions. In the paper, we prove an analogue of Bombieri-Vinogradov theorem for $0 < \alpha <…

Under two assumptions, we determine the distribution of the difference between two functions each counting the numbers < x that are in a given arithmetic progression modulo q and the product of two primes. The two assumptions are (i) the…

In this article, we obtain upper bounds on the number of irreducible factors of some classes of polynomials having integer coefficients, which in particular yield some of the well known irreducibility criteria. For devising our results, we…

We give two improved explicit versions of the prime number theorem for primes in arithmetic progression: the first isolating the contribution of the Siegel zero and the second completely explicit, where the improvement is for medium-sized…

We describe a new algorithm that computes the n-th Bernoulli number in n^(4/3 + o(1)) bit operations. This improves on previous algorithms that had complexity n^(2 + o(1)).

Assuming the Riemann Hypothesis, we obtain an upper bound for the 2k-th moment of the derivative of the Riemann zeta-function averaged over the non-trivial zeros of $\zeta(s)$ for every positive integer k. Our bounds are nearly as sharp as…

We establish new inequalities involving classical exponents of Diophantine approximation. This allows for improving on the work of Davenport, Schmidt and Laurent concerning the maximum value of the exponent $\hat{\lambda}_{n}(\zeta)$ among…

In recent years some near-optimal estimates have been established for certain sum-product type estimates. This paper gives some first extremal results which provide information about when these bounds may or may not be tight. The main tool…

New exceptional (i.e. non-repeating) prime number multiplets are given and formulated in terms of arithmetic progressions, along with laws governing them. Accompanying repeating prime number multiplets are pointed out. Prime number…

Using a smoothing function and recent knowledge on the zeros of the Riemann zeta-function, we compute pairs of $(\Delta,x_0)$ such that for all $x \geq x_0$ there exists at least one prime in the interval $(x(1 - \Delta^{-1}), x]$.

We shall given a new effectively computable upper bound of odd perfect numbers whose Euler factors are powers of fixed exponent, improving our old result in T. Yamada, Colloq. Math. 103 (2005), 303--307.

Differentiable real function reproducing primes up to a given number and having a differentiable inverse function is constructed. This inverse function is compared with the Riemann-Von Mangoldt exact expression for the number of primes not…

We investigate lower bounds for the variance in arithmetic progressions of certain multiplicative functions "close" to $1$. Specifically, we consider $\alpha_N$-fold divisor functions, when $\alpha_N$ is a sequence of positive real numbers…

We improve the previuosly known bound for some vertex Folkman numbers.