{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:23:56Z","timestamp":1760059436582,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,6,17]],"date-time":"2025-06-17T00:00:00Z","timestamp":1750118400000},"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, we obtain approximation theorems of classes of analytic functions by shifts L(\u03bb,\u03b1,s+i\u03c4) of the Lerch zeta-function for \u03c4\u2208[T,T+H] where H\u2208[T27\/82,T1\/2]. The cases of all parameters, \u03bb,\u03b1\u2208(0,1], are considered. If the set {log(m+\u03b1):m\u2208N0} is linearly independent over Q, then every analytic function in the strip {s=\u03c3+it\u2208C:\u03c3\u2208(1\/2,1)} is approximated by the above shifts.<\/jats:p>","DOI":"10.3390\/axioms14060472","type":"journal-article","created":{"date-parts":[[2025,6,17]],"date-time":"2025-06-17T03:37:07Z","timestamp":1750131427000},"page":"472","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Approximation of Analytic Functions Using Shifts of the Lerch Zeta-Function in Short Intervals"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7671-0282","authenticated-orcid":false,"given":"Antanas","family":"Laurin\u010dikas","sequence":"first","affiliation":[{"name":"Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko Str. 24, LT-03225 Vilnius, Lithuania"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1007\/BF02612318","article-title":"Note sur la fonction K(w,x,s) = \u2211n\u2a7e0 exp{2\u03c0inx}(n + w)\u2212s","volume":"11","author":"Lerch","year":"1887","journal-title":"Acta Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1515\/crll.1889.105.127","article-title":"Untersuchung einer aus vier Elementen gebildeten Reihe","volume":"105","author":"Lipschitz","year":"1889","journal-title":"J. Reine Angew. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"161","DOI":"10.2140\/pjm.1951.1.161","article-title":"On the Lerch zeta function","volume":"1","author":"Apostol","year":"1951","journal-title":"Pac. J. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"117","DOI":"10.2140\/pjm.1956.6.117","article-title":"Note on the Lerch zeta function","volume":"6","author":"Oberhettinger","year":"1956","journal-title":"Pac. J. Math."},{"key":"ref_5","first-page":"111","article-title":"New proof and extension of the functional equality of Lerch\u2019s zeta-function","volume":"14","year":"1971","journal-title":"Ann. Univ. Sci. Budap. Rolando E\u00f6tv\u00f6s Sect. Math."},{"key":"ref_6","first-page":"403","article-title":"Two new proofs of Lerch\u2019s functional equation","volume":"32","author":"Berndt","year":"1972","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_7","unstructured":"Erd\u00e9lyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F.G. (1953). Higher Transcendental Functions, McGraw-Hill."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/PL00000117","article-title":"On the Hurwitz-Lerch zeta-function","volume":"59","author":"Kanemitsu","year":"2000","journal-title":"Aequ. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1515\/form.2011.047","article-title":"The Lerch zeta-function I. Zeta integrals","volume":"24","author":"Lagarias","year":"2012","journal-title":"Forum Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1515\/form.2011.048","article-title":"The Lerch zeta-function II. Analytic continuation","volume":"24","author":"Lagarias","year":"2012","journal-title":"Forum Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1186\/s40687-015-0049-2","article-title":"The Lerch zeta-function III. Polylogarithms and special values","volume":"3","author":"Lagarias","year":"2016","journal-title":"Res. Math. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1186\/s40687-016-0082-9","article-title":"The Lerch zeta-function IV. Hecke operators","volume":"3","author":"Lagarias","year":"2016","journal-title":"Res. Math. Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"443","DOI":"10.1070\/IM1975v009n03ABEH001485","article-title":"Theorem on the \u201cuniversality\u201d of the Riemann zeta-function","volume":"9","author":"Voronin","year":"1975","journal-title":"Math. USSR Izv."},{"key":"ref_14","unstructured":"Kaneko, M., Kanemitsu, S., and Liu, J. (2015). A survey on the theory of universality for zeta and L-functions. Number Theory: Plowing and Starring Through High Wave Forms, Proceedings of the 7th China-Japan Seminar (Fukuoka 2013), Fukuoka, Japan, 28 October\u20131 November 2013, World Scientific Publishing Co."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Laurin\u010dikas, A., and Garunk\u0161tis, R. (2002). The Lerch Zeta-Function, Kluwer Academic Publishers.","DOI":"10.1007\/978-94-017-6401-8"},{"key":"ref_16","unstructured":"Bagchi, B. (1981). The Statistical Behaviour and Universality Properties of the Riemann Zeta-Function and Other Allied Dirichlet Series. [Ph.D. Thesis, Indian Statistical Institute]."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"894","DOI":"10.1134\/S0037446618050130","article-title":"Universality of the periodic Hurwitz zeta function with rational parameter","volume":"59","author":"Mochov","year":"2018","journal-title":"Sib. Math. J."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"829","DOI":"10.1007\/s00365-021-09561-2","article-title":"On the value distribution of Hurwitz zeta-function with algebraic irrational parameter","volume":"55","author":"Sourmelidis","year":"2022","journal-title":"Constr. Approx."},{"key":"ref_19","first-page":"107","article-title":"\u201cAlmost\u201d universality of the Lerch zeta-function","volume":"24","year":"2019","journal-title":"Math. Commun."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1016\/j.jnt.2019.04.006","article-title":"Universality of the Riemann zeta-function in short intervals","volume":"204","year":"2019","journal-title":"J. Number Theory"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1007\/s10986-024-09631-5","article-title":"Notes on universality in short intervals and exponential shifts","volume":"64","author":"Andersson","year":"2024","journal-title":"Lith. Math. J."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Billingsley, P. (1999). Convergence of Probability Measures, John Wiley & Sons. [2nd ed.].","DOI":"10.1002\/9780470316962"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Heyer, H. (1977). Probability Measures on Locally Compact Groups, Springer.","DOI":"10.1007\/978-3-642-66706-0"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Gutauskien\u0117, B., Laurin\u010dikas, A., and \u0160iau\u010di\u016bnas, D. (2025). On the mean square estimate for the Lerch zeta-function in short intervals. Lith. Math. J., submitted.","DOI":"10.3390\/axioms13080510"},{"key":"ref_25","unstructured":"Ivi\u010d, A. (1985). The Riemann Zeta-Function, John Wiley & Sons."},{"key":"ref_26","unstructured":"Prachar, K. (1967). Distribution of Prime Numbers, Mir. (In Russian)."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Conway, J.B. (1973). Functions of One Complex Variable, Springer.","DOI":"10.1007\/978-1-4615-9972-2"},{"key":"ref_28","unstructured":"Mergelyan, S.N. (1954). Uniform approximations to functions of a complex variable. American Mathematical Society Translations, American Mathematical Society. No. 101."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/6\/472\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:53:16Z","timestamp":1760032396000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/6\/472"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,17]]},"references-count":28,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2025,6]]}},"alternative-id":["axioms14060472"],"URL":"https:\/\/doi.org\/10.3390\/axioms14060472","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,6,17]]}}}