Zeta Function Universality - Formal Statement

Formal Statement

A mathematically precise statement of universality for the Riemann zeta-function ζ(s) follows.

Let U be a compact subset of the strip

such that the complement of U is connected. Let f : U → C be a continuous function on U which is holomorphic on the interior of U and does not have any zeros in U. Then for any ε > 0 there exists a t ≥ 0 such that

Even more: the lower density of the set of values t which do the job is positive, as is expressed by the following inequality about a limit inferior.

0 < \liminf_{T\to\infty} \frac{1}{T} \,\lambda\!\left( \left\{ t\in \mid \max_{s\in U} |\zeta(s+it)-f(s)| < \varepsilon \right\} \right)

where λ denotes the Lebesgue measure on the real numbers.

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