Voronin Universality Theorem

Voronin (1975) proved the remarkable analytical property of the Riemann zeta function zeta(s) that, roughly speaking, any nonvanishing analytic function can be approximated uniformly by certain purely imaginary shifts of the zeta function in the critical strip.

More precisely, let 0<r<1/4 and suppose that g(s) is a nonvanishing continuous function on the disk |s|<=r that is analytic in the interior. Then for any epsilon>0, there exists a positive real number tau such that


Moreover, the set of these tau has positive lower density, i.e.,

 lim inf_(T->infty)1/Tmeas{tau in [0,T]:max_(|s|<=r)|zeta(s+3/4+itau)-g(s)|<epsilon} 

Garunkštis (2003) obtained explicit estimates for the first approximating tau and the positive lower density, provided that r is sufficiently small and g(s) sufficiently smooth. The condition that g(s) have no zeros for |s|<=r is necessary.

The Riemann hypothesis is known to be true iff zeta(s) 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).

See also

Riemann Hypothesis, Riemann Zeta Function

This entry contributed by Joern Steuding

Explore with Wolfram|Alpha


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 L-Funktionen." Math. Ann. 85, 115-122, 1922.Garunkštis, R. "The Effective Universality Theorem for the Riemann Zeta Function." Bonner math. Schriften 360, 2003.Karatsuba, A. A. and Voronin, S. M. The Riemann Zeta-Function. Hawthorn, NY: de Gruyter, 1992.Laurinčikas, A. Limit Theorems for the Riemann Zeta-Function. Dordrecht, Netherlands: Kluwer, 1996.Matsumoto, K. "Probabilistic Value-Distribution Theory of Zeta Functions." Sugaku 53, 279-296, 2001. Reprinted in Sugaku Expositions 17, 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.

Referenced on Wolfram|Alpha

Voronin Universality Theorem

Cite this as:

Steuding, Joern. "Voronin Universality Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

Subject classifications