| Summary: | Let <inline-formula><math display="inline"><semantics><mrow><mn>0</mn><mo><</mo><msub><mi>γ</mi><mn>1</mn></msub><mo><</mo><msub><mi>γ</mi><mn>2</mn></msub><mo><</mo><mo>⋯</mo><mo>⩽</mo><msub><mi>γ</mi><mi>k</mi></msub><mo>⩽</mo><mo>⋯</mo></mrow></semantics></math></inline-formula> be the sequence of imaginary parts of non-trivial zeros of the Riemann zeta-function <inline-formula><math display="inline"><semantics><mrow><mi>ζ</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>. Using a certain estimate on the pair correlation of the sequence <inline-formula><math display="inline"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>γ</mi><mi>k</mi></msub><mo stretchy="false">}</mo></mrow></semantics></math></inline-formula> in the intervals <inline-formula><math display="inline"><semantics><mrow><mo>[</mo><mi>N</mi><mo>,</mo><mi>N</mi><mo>+</mo><mi>M</mi><mo>]</mo></mrow></semantics></math></inline-formula> with <inline-formula><math display="inline"><semantics><mrow><msup><mi>N</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn><mo>+</mo><mi>ε</mi></mrow></msup><mo>⩽</mo><mi>M</mi><mo>⩽</mo><mi>N</mi></mrow></semantics></math></inline-formula>, we prove that the set of shifts <inline-formula><math display="inline"><semantics><mrow><mi>ζ</mi><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mi>i</mi><mi>h</mi><msub><mi>γ</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><mrow><mi>h</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, approximating any non-vanishing analytic function defined in the strip <inline-formula><math display="inline"><semantics><mrow><mo stretchy="false">{</mo><mi>s</mi><mo>∈</mo><mi mathvariant="double-struck">C</mi><mo>:</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo><</mo><mi>Re</mi><mi>s</mi><mo><</mo><mn>1</mn><mo stretchy="false">}</mo></mrow></semantics></math></inline-formula> with accuracy <inline-formula><math display="inline"><semantics><mrow><mi>ε</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> has a positive lower density in <inline-formula><math display="inline"><semantics><mrow><mo>[</mo><mi>N</mi><mo>,</mo><mi>N</mi><mo>+</mo><mi>M</mi><mo>]</mo></mrow></semantics></math></inline-formula> as <inline-formula><math display="inline"><semantics><mrow><mi>N</mi><mo>→</mo><mo>∞</mo></mrow></semantics></math></inline-formula>. Moreover, this set has a positive density for all but at most countably <inline-formula><math display="inline"><semantics><mrow><mi>ε</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. The above approximation property remains valid for certain compositions <inline-formula><math display="inline"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>.
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