| Summary: | By the Bagchi theorem, the Riemann hypothesis (all non-trivial zeros lie on the critical line) is equivalent to the self-approximation of 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></mrow></semantics></math></inline-formula> by 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></mrow></semantics></math></inline-formula>. In this paper, it is determined that the Riemann hypothesis is equivalent to the positivity of density of the set of the above shifts approximating <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> with all but at most countably many accuracies <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ε</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. Also, the analogue of an equivalent in terms of positive density in short intervals is discussed.
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