Summatory von Mangoldt function

The summatory von Mangoldt function is defined as

$\psi (x)=\sum _{n\leq x}\Lambda (n),\ x\in \mathbb {R} ^{+},\,$

where $\scriptstyle x\,$ is a positive real number and $\scriptstyle \Lambda (n)\,$ is the von Mangoldt function.

Explicit formula

$\scriptstyle \psi (x)\,$ is given by the explicit formula

$\psi (x)=x-\sum _{\rho }{\frac {x^{\rho }}{\rho }}-\log(2\pi )-{\tfrac {1}{2}}\log(1-{\tfrac {1}{x^{2}}})\,$

$=x-\sum _{\rho }{\frac {x^{\rho }}{\rho }}-\log(2\pi )-{\tfrac {1}{2}}{\big [}\log(1-{\tfrac {1}{x}})+\log(1+{\tfrac {1}{x}}){\big ]}\,$

for $\scriptstyle x\,>\,1\,$ and $\scriptstyle x\,$ not a prime or prime power, and the sum is over all nontrivial zeros $\scriptstyle \rho \,$ of the Riemann zeta function $\scriptstyle \zeta (s)\,$, i.e. those in the critical strip, i.e. $\scriptstyle 0\,<\,\Re (\rho )\,<\,1\,$ (which are all on the critical line, i.e. $\scriptstyle \Re (\rho )\,=\,{\frac {1}{2}}\,$ if the Riemann Hypothesis is true,) the sum being interpreted as

$\lim _{t\to \infty }\sum _{|\Im (\rho )|<t}{\frac {x^{\rho }}{\rho }}.\,$

Riemann Hypothesis

The Riemann Hypothesis is equivalent to

$\psi (x)=x+O({\sqrt {x}}\ (\log(x))^{2})\,$