Joint approximation of analytic functions by the shifts of Hurwitz zeta-functions in short intervals

Source abstract

Abstract In the article, we obtain that, for algebraically independent over Q {\mathbb{Q}} parameters α 1 , … , α r {\alpha }_{1},\ldots ,{\alpha }_{r} , there are infinitely many shifts ( ζ ( s + i τ , α 1 ) , … , ζ ( s + i τ , α r ) ) \left(\zeta \left(s+i\tau ,{\alpha }_{1}),\ldots ,\zeta \left(s+i\tau ,{\alpha }_{r})) of Hurwitz zeta-functions with τ ∈ [ T , T + H ] \tau \in \left[T,T+H] , T 27 ⁄ 82 ⩽ H ⩽ T 1 ⁄ 2 {T}^{27/82}\leqslant H\leqslant {T}^{1/2} , that approximate any r r -tuple of analytic functions on the strip { s ∈ C : 1 ⁄ 2 < σ < 1 } \left\{s\in {\mathbb{C}}:1/2\lt \sigma \lt 1\right\} . More precisely, the latter set of shifts has a positive density. For the proof, a probabilistic approach is applied.