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.

Read more about this topic:  Zeta Function Universality

Famous quotes containing the words formal and/or statement:

    “On every formal visit a child ought to be of the party, by way of provision for discourse.”
    —Jane Austen (1775–1817)

    “Most personal correspondence of today consists of letters the first half of which are given over to an indexed statement of why the writer hasn’t written before, followed by one paragraph of small talk, with the remainder devoted to reasons why it is imperative that the letter be brought to a close.”
    —Robert Benchley (1889–1945)