Prime Number Theorem - Prime-counting Function in Terms of The Logarithmic Integral

Prime-counting Function in Terms of The Logarithmic Integral

In a handwritten note on a reprint of his 1838 paper "Sur l'usage des séries infinies dans la théorie des nombres", which he mailed to Carl Friedrich Gauss, Johann Peter Gustav Lejeune Dirichlet conjectured (under a slightly different form appealing to a series rather than an integral) that an even better approximation to π(x) is given by the offset logarithmic integral function Li(x), defined by

Indeed, this integral is strongly suggestive of the notion that the 'density' of primes around t should be 1/lnt. This function is related to the logarithm by the asymptotic expansion

So, the prime number theorem can also be written as π(x) ~ Li(x). In fact, it follows from the proof of Hadamard and de la Vallée Poussin that

for some positive constant a, where O(…) is the big O notation. This has been improved to

Because of the connection between the Riemann zeta function and π(x), the Riemann hypothesis has considerable importance in number theory: if established, it would yield a far better estimate of the error involved in the prime number theorem than is available today. More specifically, Helge von Koch showed in 1901 that, if and only if the Riemann hypothesis is true, the error term in the above relation can be improved to

The constant involved in the big O notation was estimated in 1976 by Lowell Schoenfeld: assuming the Riemann hypothesis,

for all x ≥ 2657. He also derived a similar bound for the Chebyshev prime-counting function ψ:

for all x ≥ 73.2.

The logarithmic integral Li(x) is larger than π(x) for "small" values of x. This is because it is (in some sense) counting not primes, but prime powers, where a power pn of a prime p is counted as 1/n of a prime. This suggests that Li(x) should usually be larger than π(x) by roughly Li(x1/2)/2, and in particular should usually be larger than π(x). However, in 1914, J. E. Littlewood proved that this is not always the case. The first value of x where π(x) exceeds Li(x) is probably around x = 10316; see the article on Skewes' number for more details.