Twisted Von Mangoldt estimate implying the Riemann hypothesis
= Twisted Von Mangoldt estimate implying the Riemann hypothesis
{c}
If, for some real $u$,
$$
\sum_{n\leq x}\Lambda(n)n^{iu}
=\frac{x^{1+iu}}{1+iu}+O\left(x^{1/2}(\log x)^2\right),
$$
then partial summation continues $-\zeta'(s-iu)/\zeta(s-iu)$ meromorphically to $\Re s>1/2$ with no pole except at $1+iu$. The zeta function has no zero to the right of the critical line, and its functional equation then implies the Riemann hypothesis.