On the irregularity of the distribution of the sums of pairs of odd primes

Abstract

Let P 2 (n) denote the number of ways of writing n as a sum of two odd primes. We support a conjecture of Hardy and Littlewood concerning P 2 (n) by showing that it holds in a certain "average" sense. Thereby showing the irregularity of P 2 (n).

The Average Number of Goldbach Representations

“…1 This result has been proved before several times, by Prachar [19] in 1951 and Giordano [5] in 2002.…”

“…From work of Granville [7] [8] the question arose of finding the true size of the error term E 2 (x). This question was almost completely solved by Bhowmik and Schlage-Puchta [1] who proved on RH that E 2 (x) ≪ x(log x) 5 , and also unconditionally that E 2 (x) = Ω(x log log x). This lower bound arises from proving that there exist n for which r 2 (n) > cn log log n so that an indivivual term in the average already makes a contribution of this size.…”

Asymptotics of Goldbach Representations

“…Half a century later Giordano [14] studied the irregularity of g(n) depending on whether or not n is divisible by many small primes by using the Prime Number Theorem for arithmetic progression.…”

Binary Additive Problems: Theorems of Landau and Hardy-Littlwood Type

“…(4) d|n µ 2 (d)/ϕ(d) = n/ϕ(n), while Hardy and Littlewood proved (3) using their tauberian theorem [10]. A modern interesting information about Landau formulas (3),(4) one can find in [8], [12]. In 1998, Dusart [5] obtained the following excellent estimates for the prime counting function π(x) : if x ≥ 355991, then (5) x/ ln x + x/ ln 2 x < π(x) < x/ ln x + x/ ln 2 x + 2.51x/ ln 3 x.…”