Universality in Short Intervals of the Riemann Zeta-Function Twisted by Non-Trivial Zeros
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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)).
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2000 ◽
Vol 80
(1)
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pp. 31-49
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1975 ◽
Vol 81
(1)
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pp. 139-143
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2009 ◽
Vol 85
(99)
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pp. 1-17
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