voronin universality theorem - Wolfram|Alpha

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Input interpretation

Voronin universality theorem

Alternate name

one-max property

Definition

Voronin proved the remarkable analytical property of the Riemann zeta function ζ(s) that, roughly speaking, any nonvanishing analytic function can be approximated uniformly by certain purely imaginary shifts of the zeta function in the critical strip.
More precisely, let 0<r<1/4 and suppose that g(s) is a nonvanishing continuous function on the disk left bracketing bar s right bracketing bar <=r that is analytic in the interior. Then for any ϵ>0, there exists a positive real number τ such that
max_( left bracketing bar s right bracketing bar <=r) left bracketing bar ζ(s + 3/4 + i τ) - g(s) right bracketing bar <ϵ.

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Related terms

Riemann hypothesis | Riemann zeta function

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