Zeta function universality

A sketch of the proof presented in (Voronin and Karatsuba, 1992)[3] follows. We consider only the case where U is a disk centered at 3/4:

${\displaystyle U=\{s\in \mathbb {C}$ :|s-3/4|<r\}\quad {\mbox{with}}\quad 0<r<1/4}

and we will argue that every non-zero holomorphic function defined on U can be approximated by the ζ-function on a vertical translation of this set.

Passing to the logarithm, it is enough to show that for every holomorphic function g : U → C and every ε > 0 there exists a real number t such that

${\displaystyle \left|\ln \zeta (s+it)-g(s)\right|<\varepsilon \quad {\text{for all}}\quad s\in U.}$

We will first approximate g(s) with the logarithm of certain finite products reminiscent of the Euler product for the ζ-function:

${\displaystyle \zeta (s)=\prod _{p\in \mathbb {P} }\left(1-{\frac {1}{p^{s}}}\right)^{-1},}$

where P denotes the set of all primes.

If ${\displaystyle \theta =(\theta _{p})_{p\in \mathbb {P} }}$ is a sequence of real numbers, one for each prime p, and M is a finite set of primes, we set

${\displaystyle \zeta _{M}(s,\theta )=\prod _{p\in M}\left(1-{\frac {e^{-2\pi i\theta _{p}}}{p^{s}}}\right)^{-1}.}$

We consider the specific sequence

${\displaystyle {\hat {\theta }}=\left({\frac {1}{4}},{\frac {2}{4}},{\frac {3}{4}},{\frac {4}{4}},{\frac {5}{4}},\ldots \right)}$

and claim that g(s) can be approximated by a function of the form ${\displaystyle \ln(\zeta _{M}(s,{\hat {\theta }}))}$ for a suitable set M of primes. The proof of this claim utilizes the Bergman space, falsely named Hardy space in (Voronin and Karatsuba, 1992),[3] in H of holomorphic functions defined on U, a Hilbert space. We set

${\displaystyle u_{k}(s)=\ln \left(1-{\frac {e^{-\pi ik/2}}{p_{k}^{s}}}\right)}$

where p_k denotes the k-th prime number. It can then be shown that the series

${\displaystyle \sum _{k=1}^{\infty }u_{k}}$

is conditionally convergent in H, i.e. for every element v of H there exists a rearrangement of the series which converges in H to v. This argument uses a theorem that generalizes the Riemann series theorem to a Hilbert space setting. Because of a relationship between the norm in H and the maximum absolute value of a function, we can then approximate our given function g(s) with an initial segment of this rearranged series, as required.

By a version of the Kronecker theorem, applied to the real numbers ${\displaystyle {\frac {\ln 2}{2\pi }},{\frac {\ln 3}{2\pi }},{\frac {\ln 5}{2\pi }},\ldots ,{\frac {\ln p_{N}}{2\pi }}}$ (which are linearly independent over the rationals) we can find real values of t so that ${\displaystyle \ln(\zeta _{M}(s,{\hat {\theta }}))}$ is approximated by ${\displaystyle \ln(\zeta _{M}(s+it,0))}$. Further, for some of these values t, ${\displaystyle \ln(\zeta _{M}(s+it,0))}$ approximates ${\displaystyle \ln(\zeta (s+it))}$, finishing the proof.

The theorem is stated without proof in § 11.11 of (Titchmarsh and Heath-Brown, 1986),[4] the second edition of a 1951 monograph by Titchmarsh; and a weaker result is given in Thm. 11.9. Although Voronin's theorem is not proved there, two corollaries are derived from it:

1) Let   ${\displaystyle {\tfrac {1}{2}}<\sigma <1}$   be fixed. Then the curve

${\displaystyle \gamma (t)=(\zeta (\sigma +it),\zeta '(\sigma +it),\dots ,\zeta ^{(n-1)}(\sigma +it))}$

is dense in ${\displaystyle \mathbb {C} ^{n}.}$

2) Let   ${\displaystyle \Phi }$   be any continuous function, and let   ${\displaystyle h_{1},h_{2},\dots ,h_{n}}$   be real constants.

Then ${\displaystyle \zeta (s)}$ cannot satisfy the differential-difference equation

${\displaystyle \Phi \{\zeta (s+h_{1}),\zeta '(s+h_{1}),\dots ,\zeta ^{(n_{1})}(s+h_{1}),\zeta (s+h_{2}),\zeta '(s+h_{2}),\dots ,\zeta ^{(n_{2})}(s+h_{2}),\dots \}=0}$

unless   ${\displaystyle \Phi }$   vanishes identically.