Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Explicit formula for the average of Goldbach and prime tuples
  representations   Let $\Lambda\left(n\right)$ be the Von Mangoldt function, let \[
r_{G}\left(n\right)=\underset{{\scriptstyle
m_{1}+m_{2}=n}}{\sum_{m_{1},m_{2}\leq
n}}\Lambda\left(m_{1}\right)\Lambda\left(m_{2}\right), \] \[
r_{PT}\left(N,h\right)=\sum_{n=0}^{N}\Lambda\left(n\right)\Lambda\left(n+h\right),\,h\in\mathbb{N}
\] be the counting function of the Goldbach numbers and the counting function
of the prime tuples, respectively. Let $N>2$ be an integer. We will find the
explicit formulae for the averages of $r_{G}\left(n\right)$ and
$r_{PT}\left(N,h\right)$ in terms of elementary functions, the incomplete Beta
function $B_{z}\left(a,b\right)$, series over $\rho$ that, with or without
subscript, runs over the non-trivial zeros of the Riemann Zeta function and the
Dilogarithm function. We will also prove the explicit formulae in an asymptotic
form and a truncated formula for the average of $r_{G}\left(n\right)$. Some
observation about these formulae and the average with Ces\`aro weight \[
\frac{1}{\Gamma\left(k+1\right)}\sum_{n\leq
N}r_{G}\left(n\right)\left(N-n\right)^{k},\,k>0 \] are included.

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The categories of this paper are: nucl-ex
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The categories of this paper are: solv-int
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The categories of this paper are: econ
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The categories of this paper are: chao-dyn
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The categories of this paper are: ao-sci
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The categories of this paper are: supr-con
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The categories of this paper are: q-fin
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The categories of this paper are: quant-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: acc-phys
3
The categories of this paper are: physics
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The categories of this paper are: plasm-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: bayes-an
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The categories of this paper are: math-ph