Voronin (1975) proved the remarkable analytical property of the Riemann zeta function 
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 
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.,
![lim inf_(T->infty)1/Tmeas{tau in [0,T]:max_(|s|<=r)|zeta(s+3/4+itau)-g(s)|<epsilon}
>0.](/images/equations/VoroninUniversalityTheorem/NumberedEquation2.svg) |
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).
-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.