Abstract
Let 0<γ1<γ2<⋯⩽γk⩽⋯ be the sequence of imaginary parts of non-trivial zeros of the Riemann zeta-function ζ(s). Using a certain estimate on the pair correlation of the sequence {γk} in the intervals [N,N+M] with N1/2+ε⩽M⩽N, we prove that the set of shifts ζ(s+ihγk), h>0, approximating any non-vanishing analytic function defined in the strip {s∈C:1/2<Res<1} with accuracy ε>0 has a positive lower density in [N,N+M] as N→∞. Moreover, this set has a positive density for all but at most countably ε>0. The above approximation property remains valid for certain compositions F(ζ(s)).
Highlights
The Riemann zeta-function ζ (s), s = σ + it, is defined, for σ > 1, by ∞ ζ (s) = ∑ ms = m =1 ∏ p 1− ps −1, where the infinite product is taken over all prime numbers, and has analytic continuation over the whole complex plane, except for the point s = 1 which is a simple pole with residue 1
Where the infinite product is taken over all prime numbers, and has analytic continuation over the whole complex plane, except for the point s = 1 which is a simple pole with residue 1
Voronin obtained [2] the infinite-dimensional version of the Bohr–Courant theorem, proving the so-called universality of ζ (s)
Summary
Voronin obtained [2] the infinite-dimensional version of the Bohr–Courant theorem, proving the so-called universality of ζ (s) This means that every non-vanishing analytic function in the strip D = {s ∈ C : 1/2 < σ < 1} can be approximated by shifts ζ (s + iτ ). All above theorems are non-effective in the sense that any concrete shift approximating a given analytic function is not known. This shortcoming leads to the idea of universality in intervals as short as possible containing τ with approximating property.
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