| Summary: | The Hurwitz zeta-function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mo>(</mo><mi>s</mi><mo>,</mo><mi>α</mi><mo>)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>=</mo><mi>σ</mi><mo>+</mo><mi>i</mi><mi>t</mi></mrow></semantics></math></inline-formula>, with parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo>⩽</mo><mn>1</mn></mrow></semantics></math></inline-formula> is a generalization of the Riemann zeta-function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mo>(</mo><mi>s</mi><mo>)</mo></mrow></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mo>(</mo><mi>s</mi><mo>,</mo><mn>1</mn><mo>)</mo><mo>=</mo><mi>ζ</mi><mo>(</mo><mi>s</mi><mo>)</mo></mrow></semantics></math></inline-formula>) and was introduced at the end of the 19th century. The function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mo>(</mo><mi>s</mi><mo>,</mo><mi>α</mi><mo>)</mo></mrow></semantics></math></inline-formula> plays an important role in investigations of the distribution of prime numbers in arithmetic progression and has applications in special function theory, algebraic number theory, dynamical system theory, other fields of mathematics, and even physics. The function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mo>(</mo><mi>s</mi><mo>,</mo><mi>α</mi><mo>)</mo></mrow></semantics></math></inline-formula> is the main example of zeta-functions without Euler’s product (except for the cases <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula>), and its value distribution is governed by arithmetical properties of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>. For the majority of zeta-functions, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mo>(</mo><mi>s</mi><mo>,</mo><mi>α</mi><mo>)</mo></mrow></semantics></math></inline-formula> for some <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> is universal, i.e., its shifts <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mo>(</mo><mi>s</mi><mo>+</mo><mi>i</mi><mi>τ</mi><mo>,</mo><mi>α</mi><mo>)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>τ</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>, approximate every analytic function defined in the strip <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><mi>s</mi><mo>:</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo><</mo><mi>σ</mi><mo><</mo><mn>1</mn><mo>}</mo></mrow></semantics></math></inline-formula>. For needs of effectivization of the universality property for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mo>(</mo><mi>s</mi><mo>,</mo><mi>α</mi><mo>)</mo></mrow></semantics></math></inline-formula>, the interval for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula> must be as short as possible, and this can be achieved by using the mean square estimate for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mo>(</mo><mi>σ</mi><mo>+</mo><mi>i</mi><mi>t</mi><mo>,</mo><mi>α</mi><mo>)</mo></mrow></semantics></math></inline-formula> in short intervals. In this paper, we obtain the bound <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo>(</mo><mi>H</mi><mo>)</mo></mrow></semantics></math></inline-formula> for that mean square over the interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mi>T</mi><mo>−</mo><mi>H</mi><mo>,</mo><mi>T</mi><mo>+</mo><mi>H</mi><mo>]</mo></mrow></semantics></math></inline-formula>, with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>T</mi><mrow><mn>27</mn><mo>/</mo><mn>82</mn></mrow></msup><mo>⩽</mo><mi>H</mi><mo>⩽</mo><msup><mi>T</mi><mi>σ</mi></msup></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><mn>2</mn><mo><</mo><mi>σ</mi><mo>⩽</mo><mn>7</mn><mo>/</mo><mn>12</mn></mrow></semantics></math></inline-formula>. This is the first result on the mean square for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mo>(</mo><mi>s</mi><mo>,</mo><mi>α</mi><mo>)</mo></mrow></semantics></math></inline-formula> in short intervals. In forthcoming papers, this estimate will be applied for proof of universality for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mo>(</mo><mi>s</mi><mo>,</mo><mi>α</mi><mo>)</mo></mrow></semantics></math></inline-formula> and other zeta-functions in short intervals.
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