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

Abstract In the article, we obtain that, for algebraically independent over <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="double-struck">Q</m:mi> </m:math> {\mathbb{Q}} parameters <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>α</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo form="prefix">,</m:mo> <m:mrow> <m:mo>…</m:mo> </m:mrow> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>α</m:mi> </m:mrow> <m:mrow> <m:mi>r</m:mi> </m:mrow> </m:msub> </m:math> {\alpha }_{1},\ldots ,{\alpha }_{r} , there are infinitely many shifts <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>ζ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>s</m:mi> <m:mo>+</m:mo> <m:mi>i</m:mi> <m:mi>τ</m:mi> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>α</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mrow> <m:mo>…</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mi>ζ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>s</m:mi> <m:mo>+</m:mo> <m:mi>i</m:mi> <m:mi>τ</m:mi> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>α</m:mi> </m:mrow> <m:mrow> <m:mi>r</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \left(\zeta \left(s+i\tau ,{\alpha }_{1}),\ldots ,\zeta \left(s+i\tau ,{\alpha }_{r})) of Hurwitz zeta-functions with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>τ</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mi>T</m:mi> <m:mo>,</m:mo> <m:mi>T</m:mi> <m:mo>+</m:mo> <m:mi>H</m:mi> </m:mrow> <m:mo>]</m:mo> </m:mrow> </m:math> \tau \in \left[T,T+H] , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>T</m:mi> </m:mrow> <m:mrow> <m:mn>27</m:mn> <m:mo>⁄</m:mo> <m:mn>82</m:mn> </m:mrow> </m:msup> <m:mo>⩽</m:mo> <m:mi>H</m:mi> <m:mo>⩽</m:mo> <m:msup> <m:mrow> <m:mi>T</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> <m:mo>⁄</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:math> {T}^{27/82}\leqslant H\leqslant {T}^{1/2} , that approximate any <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>r</m:mi> </m:math> r -tuple of analytic functions on the strip <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:mi>s</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="double-struck">C</m:mi> <m:mo>:</m:mo> <m:mn>1</m:mn> <m:mo>⁄</m:mo> <m:mn>2</m:mn> <m:mo>&lt;</m:mo> <m:mi>σ</m:mi> <m:mo>&lt;</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>}</m:mo> </m:mrow> </m:math> \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.