Cosmology and zeta (Q2399700)

Let \(Z(t)=e^{i \theta (t)} \zeta \left(\frac{1}{2} +it\right)\) where \(\theta (t)=-\frac{\ln \pi }{2} t+ \Im\left(\ln \Gamma \left(\frac{1}{4} +\frac{it}{2} \right)\right)\). The function \(Z(t)\) takes real values for real \(t\). It is heavily used in the theory of the Riemann zeta function. \textit{J. Moser} in a paper published in 1984, assuming the validity of Riemann's hypothesis, pointed out a similarity between \(|Z(t)|\) and the solution \(R(t)\) of the Einstein-Friedman gravitation field equations. He then suggested a relationship between prime number theory (through the zeta function and \(Z(t)\)) and cosmology [Acta Math. Univ. Comenianae 44--45, 115--125 (1984; Zbl 0581.10016); ibid. 52--53, 49--73 (1987; Zbl 0676.10027)]. The present paper reflects a talk given by the author at the seminar of Analytic Number Theory held at Lomonosov Moscow State University on November 12, 1994. In this talk Karatsuba made a review of Moser's work and added valuable details and comments.