| Summary: | Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="fraktur">a</mi><mo>=</mo><mo>{</mo><msub><mi>a</mi><mi>m</mi></msub><mo>}</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="fraktur">b</mi><mo>=</mo><mo>{</mo><msub><mi>b</mi><mi>m</mi></msub><mo>}</mo></mrow></semantics></math></inline-formula> be two periodic sequences of complex numbers, and, additionally, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">a</mi></semantics></math></inline-formula> is multiplicative. In this paper, the joint approximation of a pair of analytic functions by shifts <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>ζ</mi><msub><mi>n</mi><mi>T</mi></msub></msub><mrow><mo>(</mo><mi>s</mi><mo>+</mo><mi>i</mi><mi>τ</mi><mo>;</mo><mi mathvariant="fraktur">a</mi><mo>)</mo></mrow><mo>,</mo><msub><mi>ζ</mi><msub><mi>n</mi><mi>T</mi></msub></msub><mrow><mo>(</mo><mi>s</mi><mo>+</mo><mi>i</mi><mi>τ</mi><mo>,</mo><mi>α</mi><mo>;</mo><mi mathvariant="fraktur">b</mi><mo>)</mo></mrow><mo>)</mo></mrow></semantics></math></inline-formula> of absolutely convergent Dirichlet series <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ζ</mi><msub><mi>n</mi><mi>T</mi></msub></msub><mrow><mo>(</mo><mi>s</mi><mo>;</mo><mi mathvariant="fraktur">a</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ζ</mi><msub><mi>n</mi><mi>T</mi></msub></msub><mrow><mo>(</mo><mi>s</mi><mo>,</mo><mi>α</mi><mo>;</mo><mi mathvariant="fraktur">b</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> involving the sequences <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">a</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">b</mi></semantics></math></inline-formula> is considered. Here, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>n</mi><mi>T</mi></msub><mo>→</mo><mo>∞</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>n</mi><mi>T</mi></msub><mo>≪</mo><msup><mi>T</mi><mn>2</mn></msup></mrow></semantics></math></inline-formula> as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mo>→</mo><mo>∞</mo></mrow></semantics></math></inline-formula>. The coefficients of these series tend to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>a</mi><mi>m</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>b</mi><mi>m</mi></msub></semantics></math></inline-formula>, respectively. It is proved that the set of the above shifts in the interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>]</mo></mrow></semantics></math></inline-formula> has a positive density. This generalizes and extends the Mishou joint universality theorem for the Riemann and Hurwitz zeta-functions.
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