J. Newman observed that the full strength of Ikehara's theorem is not needed for the prime number theorem, and one can get away with a special case that is much easier to prove.In a handwritten note on a reprint of his 1838 paper "", which he mailed to Gauss, Dirichlet conjectured (under a slightly different form appealing to a series rather than an integral) that an even better approximation to is given by the offset logarithmic integral function , defined by

Indeed, this integral is strongly suggestive of the notion that the "density" of primes around should be .