The so-called explicit formula ψ(x) = x - sum_ρ x^ρ/ρ - ln(2π) - 1/2 ln(1 - x^(-2)) gives an explicit relation between prime numbers and Riemann zeta function zeros for x>1 and x not a prime or prime power. Here, ψ(x) is the summatory Mangoldt function (also known as the second Chebyshev function), and the second sum is over all nontrivial zeros ρ of the Riemann zeta function ζ(s), i.e., those in the critical strip so 0<ℜ[ρ]<1.

Chebyshev functions | Mangoldt function | prime number theorem | Riemann-von Mangoldt formula | Riemann zeta function