The Number of Goldbach Representations of an Integer
نویسندگان
چکیده
Let Λ be the von Mangoldt function andR(n)= ∑ h+k=nΛ(h)Λ(k) be the counting function for the Goldbach numbers. Let N ≥ 2 and assume that the Riemann Hypothesis holds. We prove that N ∑ n=1 R(n) = N2 2 − 2 ∑ ρ Nρ+1 ρ(ρ+ 1) +O(N log N), where ρ = 1/2+iγ runs over the non-trivial zeros of the Riemann zeta-function ζ(s). This improves a recent result by Bhowmik and Schlage-Puchta.
منابع مشابه
Adelic Singular Series and the Goldbach Conjecture
The purpose of this paper is to show how adelic ideas might be used to make progress on the Goldbach Conjecture. In particular, we present a new Schwartz function which is able to keep track of the number of prime factors of an integer. We then use this, along with the Ono/Igusa adelic methods for Diophantine equations, to present an infinite sum whose evaluation would prove or disprove the ver...
متن کاملConvolutions of the Von Mangoldt Function over Residue Classes
We consider a Dirichlet series associated to the number of representations of an integer as a sum of primes from certain residue classes. Using techniques due to Matsumoto and Egami [4], we prove similar results about the analytic behaviour of this series. This is related to the Goldbach conjecture with residue class conditions.
متن کاملOn Randomness of Goldbach Sequences
We consider the use of Goldbach numbers as random sequences. The randomness is analyzed in terms of the autocorrelation function of the sequence of number of partitions. The distinct representations of an even number n as the sum of two primes is a local maximum for multiples of the product of the consecutive smallest primes less than the number. Specific partitions, which we call Goldbach elli...
متن کاملRAPPORT New experimental results concerning the Goldbach conjecture
The Goldbach conjecture states that every even integer 4 can be written as a sum of two prime numbers. It is known to be true up to 4 10. In this paper, new experiments on a Cray C916 supercomputer and on an SGI compute server with 18 R8000 CPUs are described, which extend this bound to 10. Two consequences are that (1) under the assumption of the Generalized Riemann hypothesis, every odd numbe...
متن کاملUpper bounds on the solutions to n = p+m^2
ardy and Littlewood conjectured that every large integer $n$ that is not a square is the sum of a prime and a square. They believed that the number $mathcal{R}(n)$ of such representations for $n = p+m^2$ is asymptotically given by begin{equation*} mathcal{R}(n) sim frac{sqrt{n}}{log n}prod_{p=3}^{infty}left(1-frac{1}{p-1}left(frac{n}{p}right)right), end{equation*} where $p$ is a prime, $m$ is a...
متن کامل