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:

    “Then the justice,
    In fair round belly with good capon lined,
    With eyes severe and beard of formal cut,
    Full of wise saws and modern instances;
    And so he plays his part.”
    —William Shakespeare (1564–1616)

    “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)