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:

    “The conviction that the best way to prepare children for a harsh, rapidly changing world is to introduce formal instruction at an early age is wrong. There is simply no evidence to support it, and considerable evidence against it. Starting children early academically has not worked in the past and is not working now.”
    —David Elkind (20th century)

    “The most distinct and beautiful statement of any truth must take at last the mathematical form.”
    —Henry David Thoreau (1817–1862)