{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T00:34:18Z","timestamp":1759970058229,"version":"build-2065373602"},"reference-count":60,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,3]],"date-time":"2025-01-03T00:00:00Z","timestamp":1735862400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, the asymptotic behavior of the modified Mellin transform Z2(s), s=\u03c3+it, 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 Z2(s+i\u03c6(\u03c4)) with a real increasing to +\u221e differentiable functions connected to the growth of the second moment of Z2(s). It is proven that the mass of the limit measure is concentrated at the point expressed as h(s)\u22610. This is used for approximation of h(s) by Z2(s+i\u03c6(\u03c4)).<\/jats:p>","DOI":"10.3390\/axioms14010034","type":"journal-article","created":{"date-parts":[[2025,1,3]],"date-time":"2025-01-03T05:02:02Z","timestamp":1735880522000},"page":"34","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Value Distribution for the Mellin Transform of the Fourth Power of the Riemann Zeta Function"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0530-4874","authenticated-orcid":false,"given":"Virginija","family":"Garbaliauskien\u0117","sequence":"first","affiliation":[{"name":"Institute of Regional Development, \u0160iauliai Academy, Vilnius University, P. Vi\u0161inskio Str. 25, LT-76351 \u0160iauliai, Lithuania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5444-211X","authenticated-orcid":false,"given":"Audron\u0117","family":"Rimkevi\u010dien\u0117","sequence":"additional","affiliation":[{"name":"Faculty of Business and Technologies, \u0160iauli\u0173 Valstybin\u0117 Kolegija, Au\u0161ros Av. 40, LT-76241 \u0160iauliai, Lithuania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8459-6475","authenticated-orcid":false,"given":"Mindaugas","family":"Stoncelis","sequence":"additional","affiliation":[{"name":"Institute of Regional Development, \u0160iauliai Academy, Vilnius University, P. Vi\u0161inskio Str. 25, LT-76351 \u0160iauliai, Lithuania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9248-8917","authenticated-orcid":false,"given":"Darius","family":"\u0160iau\u010di\u016bnas","sequence":"additional","affiliation":[{"name":"Institute of Regional Development, \u0160iauliai Academy, Vilnius University, P. Vi\u0161inskio Str. 25, LT-76351 \u0160iauliai, Lithuania"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,3]]},"reference":[{"key":"ref_1","first-page":"299","article-title":"A relation between the Riemann zeta-function and the hyperbolic Laplacian","volume":"22","author":"Motohashi","year":"1995","journal-title":"Ann. Sc. Norm. Super. Pisa Cl. Sci. IV Ser."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"115","DOI":"10.4064\/aa99-2-2","article-title":"On some conjectures and results for the Riemann zeta-function and Hecke series","volume":"99","year":"2001","journal-title":"Acta Arith."},{"key":"ref_3","first-page":"309","article-title":"The mean square of the error term for the fourth moment of the zeta-function","volume":"69","author":"Motohashi","year":"1994","journal-title":"Proc. Lond. Math. 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