1935

Distribution Functions and the Riemann Zeta Function

Borge Jessen; Aurel Wintner

Transactions of the American Mathematical Society, Vol. 38, No. 1. (Jul., 1935), pp. 48-88.

It is one of the first papers that applied the probability theory to the theory of the Riemann zeta function. Consider the factors in the Euler product. The logarithm of each of the factors maps the vertical line to a curve. The uniform distribution on a large interval on the vertical line induces a distribution on the curve and therefore a (singular) distribution on the plane. This is a distribution corresponding to 1 factor of the vector product.

Next we argue that linear independence of leads to an almost independence of the distributions (here Jessen and Winther use the theory of almost periodic functions) and consequently to the existence of the limit of the partial sums of the distributions by the law of large numbers.

The key to making this idea more precise is the combination of the following theorems. They provide a substitute for the concept of independence:

Theorem 23: The vector function x(t) possesses an asymptotic distribution function if and only if the mean value exists and

holds uniformly in every sphere in for any admissible sequence of sets . Then the characteristic function of is .

Theorem 28: If , …, are almost periodic functions with independent moduli then the asymptotic distribution function of is , where , …, are the asymptotic distribution functions of , …, respectively.