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 manifestation of poetry in external life is formal perfection. True sentiment grows within, and art must represent internal phenomena externally.”
    —Franz Grillparzer (1791–1872)

    “One is apt to be discouraged by the frequency with which Mr. Hardy has persuaded himself that a macabre subject is a poem in itself; that, if there be enough of death and the tomb in one’s theme, it needs no translation into art, the bold statement of it being sufficient.”
    —Rebecca West (1892–1983)