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A survey on the theory of universality for zeta and $$L$$-functions. (English) Zbl 1381.11070
Kaneko, Masanobu (ed.) et al., Number theory. Plowing and starring through high wave forms. Proceedings of the 7th China-Japan Seminar, Fukuoka, Japan, October 28 – November 1, 2013. Hackensack, NJ: World Scientific (ISBN 978-981-4644-92-1/hbk; 978-981-4644-94-5/ebook). Series on Number Theory and Its Applications 11, 95-144 (2015).
From the text: After the discovery of the universality property of the Riemann zeta-function by S. M. Voronin [Izv. Akad. Nauk SSSR, Ser. Mat. 39, 475–486 (1975; Zbl 0315.10037)], the universality theory of zeta and $$L$$-functions has been studied quite extensively. Various refinements and/or generalizations, such as joint universality, composite universality, hybrid universality, and strong universality were introduced and discussed. In this talk we first survey the history of universality theory, especially explain the above four notions. Then we report a new result, obtained jointly with A. Laurinčikas and L. Steuding [Osaka J. Math. 50, No. 4, 1021–1037 (2013; Zbl 1282.11120); Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Comput. 41, 85–96 (2013; Zbl 1289.11054)], which combines all of these four notions.
For the entire collection see [Zbl 1310.11004].

##### MSC:
 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 11M35 Hurwitz and Lerch zeta functions 11-02 Research exposition (monographs, survey articles) pertaining to number theory
##### Citations:
Zbl 0315.10037; Zbl 1282.11120; Zbl 1289.11054
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