More precisely, let and suppose that is a nonvanishing continuous
function on the disk that is analytic in the interior. Then for any ,
there exists a positive real number such that
Moreover, the set of these has positive lower density, i.e.,
Garunkštis (2003) obtained explicit estimates for the first approximating
and the positive lower density, provided that is sufficiently small and sufficiently smooth. The condition that have no zeros for is necessary.
The Riemann hypothesis is known to be true iff can approximate itself uniformly in the sense of Voronin's
theorem (Bohr 1922, Bagchi 1987). It is also known that there exists a rich zoo of
Dirichlet series having this or some similar
universality property (Karatsuba 1992, Laurinčikas 1996, Matsumoto 2001).
Bagchi, B. "Recurrence in Topological Dynamics and the Riemann Hypothesis." Acta Math. Hungar.50, 227-240, 1987.Bohr,
H. "Über eine Quasi-Periodische Eigenschaft Dirichletscher Reihen mit Anwendung
auf die Dirichletschen -Funktionen." Math. Ann.85, 115-122, 1922.Garunkštis,
R. "The Effective Universality Theorem for the Riemann Zeta Function."
Bonner math. Schriften360, 2003.Karatsuba, A. A.
and Voronin, S. M. The
Riemann Zeta-Function. Hawthorn, NY: de Gruyter, 1992.Laurinčikas,
Theorems for the Riemann Zeta-Function. Dordrecht, Netherlands: Kluwer, 1996.Matsumoto,
K. "Probabilistic Value-Distribution Theory of Zeta Functions." Sugaku53,
279-296, 2001. Reprinted in Sugaku Expositions17, 51-71, 2004.Voronin,
S. M. "Theorem on the Universality of the Riemann Zeta Function."
Izv. Akad. Nauk SSSR, Ser. Matem.39, 475-486, 1975. Reprinted in Math.
USSR Izv.9, 443-445, 1975.