Universality in Short Intervals of the Riemann Zeta-Function Twisted by Non-Trivial Zeros

Antanas Laurinčikas and Darius Šiaučiūnas
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Antanas Laurinčikas: Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko str. 24, LT-03225 Vilnius, Lithuania
Darius Šiaučiūnas: Regional Development Institute, Šiauliai University, P. Višinskio str. 25, LT-76351 Šiauliai, Lithuania

Mathematics, 2020, vol. 8, issue 11, 1-14

Abstract: Let 0 < γ 1 < γ 2 < ? ? γ k ? ? 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 { γ k } in the intervals [ N , N + M ] with N 1 / 2 + ε ? M ? N , we prove that the set of shifts ζ ( s + i h γ k ) , h > 0 , approximating any non-vanishing analytic function defined in the strip { s ∈ C : 1 / 2 < Re s < 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 ) ) .

Keywords: Montgomery pair correlation conjecture; non-trivial zeros; Riemann zeta-function; universality (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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