Explicit Formula - Riemann's Explicit Formula

Riemann's Explicit Formula

In his 1859 paper On the Number of Primes Less Than a Given Magnitude Riemann found an explicit formula for the normalized prime-counting function π₀(x) which is related to the prime-counting function π(x) by

His formula was given in terms of the related function

which counts primes where a prime power pn counts as 1/n of a prime and which takes the arithmetic mean of the limit from the left and the limit from the right at discontinuities. The normalized prime-counting function can be recovered from this function by

Riemann's formula is then

involving a sum over the non-trivial zeros ρ of the Riemann zeta function. The sum is not absolutely convergent, but may be evaluated by taking the zeros in order of the absolute value of their imaginary part. The function li occurring in the first term is the (unoffset) logarithmic integral function given by the Cauchy principal value of the divergent integral

The terms li(xρ) involving the zeros of the zeta function need some care in their definition as li has branch points at 0 and 1, and are defined by analytic continuation in the complex variable ρ in the region x>1 and Re(ρ)>0. The other terms also correspond to zeros: the dominant term li(x) comes from the pole at s = 1, considered as a zero of multiplicity −1, and the remaining small terms come from the trivial zeros. This formula says that the zeros of the Riemann zeta function control the oscillations of primes around their "expected" positions. (For graphs of the sums of the first few terms of this series see Zagier 1977.)

A simpler variation of Riemann's formula using the normalization of Chebyshev's function ψ rather than π is von-Mangoldt's explicit formula

where for non-integral x, ψ(x) is the sum of log(p) over all prime powers pn less than x. It plays an important role in von Mangoldt's proof of Riemann's explicit formula.