A complete Riemann zeta distribution and the Riemann hypothesis

作者：Nakamura Takashi^*

来源：Bernoulli, 2015, 21(1): 604-617.

DOI：10.3150/13-BEJ581

• characteristic function
• Levy-Khintchine representation
• Riemann hypothesis
• zeta distribution

Let a, t is an element of IR, s = sigma + it, Gamma(s) be the Gamma function, zeta(s) be the Riemann zeta function and xi(s) := s(s - 1)pi(-s/2)Gamma(s/2)zeta(s) be the complete Riemann zeta function. We show that Xi(sigma)(t) := xi(sigma -it)/xi(sigma) is a characteristic function for any a E Ilk by giving the probability density function. Next we prove that the Riemann hypothesis is true if and only if each Xi(sigma)(t) is a pretended-infinitely divisible characteristic function, which is defined in this paper, for each 1/2 < sigma <1. Moreover, we show that Xi(sigma)(t) is a pretended-infinitely divisible characteristic function when sigma = 1. Finally we prove that the characteristic function Xi(sigma)(t) is not infinitely divisible but quasi-infinitely divisible for any sigma > 1.

• 出版日期2015-2