On Value Distribution for the Mellin Transform of the Fourth Power of the Riemann Zeta Function

On Value Distribution for the Mellin Transform of the Fourth Power of the Riemann Zeta Function

In this paper, the asymptotic behavior of the modified Mellin transform <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">Z</mi><mn>2</mn&g...

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Bibliographic Details
Main Authors:	Virginija Garbaliauskienė, Audronė Rimkevičienė, Mindaugas Stoncelis, Darius Šiaučiūnas
Format:	Article
Language:	English
Published:	MDPI AG 2025-01-01
Series:	Axioms
Subjects:	limit theorem Mellin transform Riemann zeta function space of analytic functions weak convergence
Online Access:	https://www.mdpi.com/2075-1680/14/1/34
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Summary:	In this paper, the asymptotic behavior of the modified Mellin transform <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">Z</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></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>, of the fourth power of the Riemann zeta function is characterized by weak convergence of probability measures in the space of analytic functions. The main results are devoted to probability measures defined by generalized shifts <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">Z</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>s</mi><mo>+</mo><mi>i</mi><mi>φ</mi><mrow><mo>(</mo><mi>τ</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> with a real increasing to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>+</mo><mo>∞</mo></mrow></semantics></math></inline-formula> differentiable functions connected to the growth of the second moment of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">Z</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. It is proven that the mass of the limit measure is concentrated at the point expressed as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>h</mi><mo>(</mo><mi>s</mi><mo>)</mo><mo>≡</mo><mn>0</mn></mrow></semantics></math></inline-formula>. This is used for approximation of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>h</mi><mo>(</mo><mi>s</mi><mo>)</mo></mrow></semantics></math></inline-formula> by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">Z</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>s</mi><mo>+</mo><mi>i</mi><mi>φ</mi><mrow><mo>(</mo><mi>τ</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula>.
ISSN:	2075-1680