Zeta function universality

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In mathematics, the universality of zeta-functions is the remarkable property of the Riemann zeta-function and other, similar, functions, such as the Dirichlet L-functions, to approximate an arbitrary holomorphic function arbitrarily well.

[edit] Definition

A mathematically precise statement of universality for the Riemann zeta-function follows.

Let U be a compact subset of the strip

\{ s\in \mathbb{C} | 1/2 < \mbox{Re } s < 1 \}

such that U has a connected complement. Let f(s) be a non-vanishing continuous function on U which is holomorphic on the interior of U. Then, for any ε > 0, one has

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

where λ denotes the Lebesgue measure on the real numbers.

[edit] Universality on other zeta functions

A similar universality property has been shown for the Lerch zeta-function. The Dirichlet L-functions show not only universality, but a certain kind of joint universality that allow any set of functions to be approximated by the same value(s) of t in different L-functions, where each function to be approximated is paired with a different L-function. Sections of the Lerch zeta-function have also been shown to have a form of joint universality.

[edit] References

  • B. Bagchi (1982). "A Universality theorem for Dirichlet L-functions". Mat. Z. 181: pp.319-334.
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