Related papers: The Average Number of Goldbach Representations

We prove that a suitable explicit formula for the Cesaro-averaged number of representations of an integer as a sum of two primes holds in short intervals.

In the present paper we prove that under the assumption of the GRH (Generalized Riemann Hypothesis) each sufficiently large odd integer can be expressed as the sum of a prime and two isolated primes.

It is shown that every sufficiently large even integer is a sum of two primes and exactly 13 powers of 2. Under the Generalized Rieman Hypothesis one can replace 13 by 7. Unlike previous work on this problem, the proof avoids numerical…

In this paper, we exhibit an asymptotic formula for the number of representations of a large integer as a sum of a fixed power of Piatetski-Shapiro primes, thereby establishing a variant of Waring-Goldbach problem with primes from a sparse…

In order to study the analytic properties of the Goldbach generating function we consider a smooth version, similar to the Chebyshev function for the Prime Number Theorem. In this paper, we obtain explicit numerical estimates for the…

We investigate the average number of representations of a positive integer as the sum of $k + 1$ perfect $k$-th powers of primes. We extend recent results of Languasco and the last Author, which dealt with the case $k = 2$ [6] and $k = 3$…

Let $\Lambda\left(n\right)$ be the Von Mangoldt function, let \[ r_{G}\left(n\right)=\underset{{\scriptstyle m_{1}+m_{2}=n}}{\sum_{m_{1},m_{2}\leq n}}\Lambda\left(m_{1}\right)\Lambda\left(m_{2}\right), \] \[…

We prove that suitable asymptotic formulae in short intervals hold for the problems of representing an integer as a sum of a prime square and a square, or a prime square. Such results are obtained both assuming the Riemann Hypothesis and in…

We consider the Linnik--Goldbach problem of writing all large even integers as the sum of two primes and a fixed number of powers of 2. We show that, under the generalised Riemann hypothesis, one can use 6 powers of two. In addition, we…

Given a finitely generated multiplicative subgroup of rational numbers $\Gamma$, assuming the Generalized Riemann Hypothesis, we determine an asymptotic formula for average over prime numbers, powers of the order of the reduction group…

We prove that a good average order on the Goldbach generating function implies that the real parts of the non-trivial zeros of the Riemann zeta function are strictly less than 1. This together with existing results establishes an…

We prove that assuming the Generalized Riemann Hypothesis every even integer larger than $\exp(\exp(15.85))$ can be written as the sum of a prime number and a number that has at most two prime factors.

We prove that a suitable asymptotic formula for the average number of representations of integers $n=p_{1}^{3}+p_{2}^{3}+p_{3}^{3}+p_{4}^{3}$, where $p_1,p_2,p_3,p_4$ are prime numbers, holds in intervals shorter than the the ones…

We prove that suitable asymptotic formulae in short intervals hold for the problems of representing an integer as a sum of a prime and a square, or a prime square. Such results are obtained both assuming the Riemann Hypothesis and in the…

The Goldbach conjecture states that every even integer greater than 2 can be expressed as the sum of two prime numbers. This conjecture was first proposed by German mathematician Christian Goldbach in 1742 and, despite being obviously true,…

We prove the following result: Let $N \geq 2$ and assume the Riemann Hypothesis (RH) holds. Then \[ \sum_{n=1}^{N} R(n) =\frac{N^{2}}{2} -2 \sum_{\rho} \frac{N^{\rho + 1}}{\rho (\rho + 1)} + O(N \log^{3}N), \] where $\rho=1/2+i\gamma$ runs…

The famous strongly binary Goldbach's conjecture asserts that every even number $2n \geq 8$ can always be expressible as the sum of two distinct odd prime numbers. We use a new approach to dealing with this conjecture. Specifically, we…

The binary Goldbach conjecture asserts that every even integer greater than $4$ is the sum of two primes. In a preceding paper we have proved that there exists a positive integer $K_\alpha$ such that every even integer $x > p_k^2$ can be…

We give an asymptotic formula for the mean value of the number of representations of an integer as sum of two squares known as the Gauss circle problem.

For g,n coprime integers, let l_g(n) denote the multiplicative order of g modulo n. Motivated by a conjecture of Arnold, we study the average of l_g(n) as n <= x ranges over integers coprime to g, and x tending to infinity. Assuming the…