| Summary: | Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>θ</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> denote the increment of the argument of the product <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>π</mi><mrow><mo>−</mo><mi>s</mi><mo>/</mo><mn>2</mn></mrow></msup><mo>Γ</mo><mrow><mo>(</mo><mi>s</mi><mo>/</mo><mn>2</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula> along the segment connecting the points <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>+</mo><mi>i</mi><mi>t</mi></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>t</mi><mi>n</mi></msub></semantics></math></inline-formula> denote the solution of the equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>θ</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo><mi>π</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo></mrow></semantics></math></inline-formula>. The numbers <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>t</mi><mi>n</mi></msub></semantics></math></inline-formula> are called the Gram points. In this paper, we consider the approximation of a collection of analytic functions by shifts in the Riemann zeta-function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>ζ</mi><mrow><mo>(</mo><mi>s</mi><mo>+</mo><mi>i</mi><msubsup><mi>t</mi><mi>k</mi><msub><mi>α</mi><mn>1</mn></msub></msubsup><mo>)</mo></mrow><mo>,</mo><mo>…</mo><mo>,</mo><mi>ζ</mi><mrow><mo>(</mo><mi>s</mi><mo>+</mo><mi>i</mi><msubsup><mi>t</mi><mi>k</mi><msub><mi>α</mi><mi>r</mi></msub></msubsup><mo>)</mo></mrow><mo>)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>α</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>α</mi><mi>r</mi></msub></mrow></semantics></math></inline-formula> are different positive numbers not exceeding 1. We prove that the set of such shifts approximating a given collection of analytic functions has a positive lower density. For the proof, a discrete limit theorem on weak convergence of probability measures in the space of analytic functions is applied.
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