Explicit Formulas

Abstract

In this chapter our goal is to derive the explicit formula

$$\psi (x) = x - \sum\limits_p {\frac{{{x^\rho }}}{\rho } - \frac{{\zeta '(0)}}{{\zeta (0)}} - \frac{1}{2}\log (1 - {x^{ - 2}})} $$

where the sum is over the nontrivial zeros ρ of ζ(s). The method will then be used to derive the result

$$\psi (x) = x + O({x^{1/2}}{\log ^2}x) $$

assuming the Riemann hypothesis. A similar result can be obtained for primes in arithmetic progressions.