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.
Read more about this topic: Explicit Formula
Famous quotes containing the words explicit and/or formula:
“... the Ovarian Theory of Literature, or, rather, its complement, the Testicular Theory. A recent camp follower ... of this explicit theory is ... Norman Mailer, who has attributed his own gift, and the literary gift in general, solely and directly to the possession of a specific pair of organs. One writes with these organs, Mailer has said ... and I have always wondered with what shade of ink he manages to do it.”
—Cynthia Ozick (b. 1928)
“Hidden away amongst Aschenbach’s writing was a passage directly asserting that nearly all the great things that exist owe their existence to a defiant despite: it is despite grief and anguish, despite poverty, loneliness, bodily weakness, vice and passion and a thousand inhibitions, that they have come into being at all. But this was more than an observation, it was an experience, it was positively the formula of his life and his fame, the key to his work.”
—Thomas Mann (18751955)