Related papers: On the Goldbach problem with restricted primes

We study an asymptotic formula for average orders of Goldbach representations of an integer as the sum of k primes. We extend the existing result for k=2 to a general k, for which we obtain a better error term. Moreover, we prove an…

Assuming the Riemann Hypothesis, we obtain asymptotic formulas for the average number representations of an even integer as the sum of two primes. We use the method of Bhowmik and Schlage-Puchta and refine their results slightly to obtain a…

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.

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…

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…

Assuming that the Generalized Riemann Hypothesis (GRH) holds, we prove an explicit formula for the number of representations of an integer as a sum of $k\geq 5$ primes. Our error terms in such a formula improve by some logarithmic factors…

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…

Let $\mathcal{P}=\{p_1,p_2,...\}$ be the set of all odd primes arranged in increasing order. A Goldbach partition of the even integer $2k>4$ is a way of writing it as a sum of two primes from $\mathcal{P}$ without regard to order. Let…

We prove that every odd number $N$ greater than 1 can be expressed as the sum of at most five primes, improving the result of Ramar\'e that every even natural number can be expressed as the sum of at most six primes. We follow the circle…

We present a historical account of the asymptotics of classical Goldbach representations with special reference to the equivalence with the Riemann Hypothesis. When the primes are chosen from an arithmetic progression comparable but weaker…

Assuming a conjecture on distinct zeros of Dirichlet L-functions we get asymptotic results on the average number of representations of an integer as the sum of two primes in arithmetic progression. On the other hand the existence of good…

In the present paper we prove that every sufficiently large odd integer $N$ can be represented in the form \begin{equation*} N=p_1+p_2+p_3\,, \end{equation*} where $p_1,p_2,p_3$ are primes, such that $p_1=x^2 + y^2 +1$, $p_2=[n^c]$.

For $H \ge N^{1-\frac{1}{2c}} \ln^2 N$, where $c$ is a fixed non-integer number satisfying $$ \|c\| \ge 3c\left(2^{[c]+1}-1\right)\frac{\ln \ln N}{\ln N}, \qquad c > \frac{4}{3}\left(1 + \frac{52\ln \ln N}{\ln N}\right), $$ we obtain an…

Let $\delta > 1/2$. We prove that if $A$ is a subset of the primes such that the relative density of $A$ in every reduced residue class is at least $\delta$, then almost all even integers can be written as the sum of two primes in $A$. The…

Let $s$, $\ell$ be two integers such that $2\le s\le \ell-1$, $\ell\ge 3$. We prove that a suitable asymptotic formula for the average number of representations of integers $n=\sum_{i=1}^{s} p_{i}^{\ell}$, where $p_i$, $i=1,\dotsc,s$, are…

We investigate the Waring-Goldbach problem of representing a positive integer $n$ as the sum of $s$ $k$th powers of almost equal prime numbers. Define $s_k=2k(k-1)$ when $k\ge 3$, and put $s_2=6$. In addition, put $\theta_2=\frac{19}{24}$,…

For every even integer N, denote by D_{1,2}(N) the number of representations of N as a sum of a prime and an integer having at most two prime factors. In this paper, we give a new lower bound for D_{1,2}(N).

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…

Let $E(X)$ denote the number of even integers below $X$ which are not a sum of two primes. We prove the bound $E(X)=O(X^{\frac{7}{10}})$, where the implicit constant is ineffective. The method applied here also leads to $P(q)=O(q^5)$, where…

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…