= Selberg symmetry formula
{c}
The elementary Selberg symmetry formula can be written
$$
\psi(x)+\frac1{\log x}\sum_{n\leq x}\Lambda(n)\psi(x/n)=2x+O\left(\frac{x}{\log x}\right).
$$
Use the weights $\lambda_d=\mu(d)\log(x/d)/\log x$. Their <divisor sum> is $1_{n=1}+\Lambda(n)/\log x$. Apply them to the <integrated Chebyshev sum>, then use the <Möbius harmonic logarithmic moments>. The <Chebyshev estimate> shows this is equivalent, with an $O(x)$ error, to $\sum_{n\leq x}\Lambda(n)\log n+\sum_{mn\leq x}\Lambda(m)\Lambda(n)=2x\log x+O(x)$.
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