Let $\Sing(n) = 2 \prod\limits_{p>2}\left(1-\frac{1}{(p-1)^2}\right) \prod\limits_{\substack {p\mid n\\ p>2}} \left(\frac{p-1}{p-2}\right)$ if $n$ is even and $\Sing(n) =0$ if $n$ is odd, be the singular series of the Goldbach problem. Let $\nu\geq 1$ be a fixed real number. We prove that $$\sum_{n\leq X} \Sing(n)^\nu = c_1X +c_2(\log X)^\nu + O((\log X)^{\nu-1/3}),$$ where $c_1,c_2$ and the implicit constant depend on $\nu$. As a consequence, we improve the known results on the positive proportion of Goldbach numbers in short intervals.
A singular series average and Goldbach numbers in short intervals
LANGUASCO, ALESSANDRO
1998
Abstract
Let $\Sing(n) = 2 \prod\limits_{p>2}\left(1-\frac{1}{(p-1)^2}\right) \prod\limits_{\substack {p\mid n\\ p>2}} \left(\frac{p-1}{p-2}\right)$ if $n$ is even and $\Sing(n) =0$ if $n$ is odd, be the singular series of the Goldbach problem. Let $\nu\geq 1$ be a fixed real number. We prove that $$\sum_{n\leq X} \Sing(n)^\nu = c_1X +c_2(\log X)^\nu + O((\log X)^{\nu-1/3}),$$ where $c_1,c_2$ and the implicit constant depend on $\nu$. As a consequence, we improve the known results on the positive proportion of Goldbach numbers in short intervals.File in questo prodotto:
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