We consider zeta functions ζ(s; \( \mathfrak{a} \)) given by Dirichlet series with multiplicative periodic coefficients and prove that, for some classes of functions F , the functions F(ζ(s; \( \mathfrak{a} \))) have infinitely many zeros in the critical strip. For example, this is true for sin(ζ(s; \( \mathfrak{a} \))).
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References
G. H. Hardy, “Sur les zéros de la function ζ(s) de Riemann,” C. R. Acad. Sci. Paris, 158, 1012–1014 (1914).
H. M. Bui, B. Conrey, and M. P. Young, “More than 41 % of the zeros of the zeta function are on the critical line,” Acta Arith., 150, 35–64 (2011).
X. Gourdon, “The 1013 first zeros of the Riemann zeta function and zeros computation at very large height,” http://numbers.computation.free.fr (2004).
W. Schnee, “Die Funktionalgleichung der Zetafunction und der Dirichletschen Reichen mit Periodishen Koeffizienten,” Math. Z., 31, 378–390 (1930).
J. Steuding, “On Dirichlet series with periodic coefficients,” Ramanujan J., 6, 295–306 (2002).
J. Steuding, “Value Distribution of L-Functions,” Lect. Notes Math., 1877 (2007).
A. Laurinčikas, “Universality of composite functions of periodic zeta functions,” Mat. Sb., 203, No. 11, 105–120 (2012); English transl.: Sb. Math., 203, No. 11, 1631–1646 (2012).
S. M. Voronin, “Theorem on the ‘universality’ of the Riemann zeta function,” Izv. Akad. Nauk SSSR, Ser. Mat., 39, 475–486 (1975); English transl.: Math. USSR Izv., 9, 443–453 (1975).
A. Laurinčikas, Limit Theorems for the Riemann Zeta Function, Kluwer Acad. Publ., Dordrecht, etc. (1996).
B. Bagchi, The Statistical Behavior and Universality Properties of the Riemann Zeta Function and Other Allied Dirichlet Series, Ph. D. Thesis, Calcutta (1981).
B. Bagchi, “A joint universality theorem for Dirichlet L-functions,” Math. Z., 181, 319–334 (1982).
J. Sander and J. Steuding, “Joint universality for sums and products of Dirichlet L-functions,” Analysis, 26, 285–312 (2006).
J. Kaczorowski, “Some remarks on the universality of periodic L-functions,” New Directions in Value Distribution Theory of Zeta and L-Functions, R. Steuding and J. Steuding, Eds, Shaker Verlag, Aachen (2009), pp. 113–120.
A. Laurinčikas and D. Šiaučiūnas, “Remarks on the universality of periodic zeta functions,” Mat. Zametki, 80, No. 4, 561–568 (2006); English transl.: Math. Notes, 80, No. 3-4, 532–538 (2006).
S. N. Mergelyan, “Uniform approximations to functions of complex variable,” Usp. Mat. Nauk, 7, 31–122 (1954); English transl.: Amer. Math. Soc. Trans., 101, 99 (1954).
J. L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain [in Russian], Izd. Inostr. Lit., Moscow (1961).
I. I. Privalov, Introduction to the Theory of Functions of Complex Variable [in Russian], Nauka, Moscow (1967).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 6, pp. 857–862, June, 2013.
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Laurinčikas, A., Šiaučiūnas, D. On Zeros of Periodic Zeta Functions. Ukr Math J 65, 953–958 (2013). https://doi.org/10.1007/s11253-013-0832-4
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DOI: https://doi.org/10.1007/s11253-013-0832-4