Abstract
We study the mean of the values of the zeta-function on a generalized arithmetic progression on the critical line.
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References
Ingham, A.E.: On two conjectures in the theory of numbers. Am. J. Math. 64, 313–319 (1942)
Lapidus, M.L., van Frankenhuijsen, M.: Fractal Geometry and Number Theory. Complex Dimensions of Fractal Strings and Zeros of Zeta Functions. Birkhäuser Boston, Inc., Boston, MA (2000)
Li, X., Radziwiłł, M.: The Riemann zeta function on vertical arithmetic progressions. Int. Math. Res. Not. IMRN 2015, 325–354 (2015)
Martin, G., Ng, N.: Nonzero values of Dirichlet L-functions in vertical arithmetic progressions. Int. J. Number Theory 9, 813–843 (2013)
Odlyzko, A.M., te Riele, H.J.J.: Disproof of Mertens conjecture. J. Reine Angew. Math. 367, 138–160 (1985)
Özbek, S.S., Steuding, J.: The values of the Riemann zeta-function on arithmetic progressions. In: A. Laurinčikas, et al. (eds.) Analytic and Probabilistic Methods in Number Theory, Proceedings of the Sixth International Conference, Palanga 2016, pp. 149–164, Vilnius University Publishing House, Vilinius (2017)
Putnam, C.R.: On the non-periodicity of the zeros of the Riemann zeta-function. Am. J. Math. 76, 97–99 (1954)
Putnam, C.R.: Remarks on periodic sequences and the Riemann zeta-function. Am. J. Math. 76, 828–830 (1954)
Steuding, J., Wegert, E.: The Riemann zeta function on arithmetic progressions. Exp. Math. 21, 235–240 (2012)
Titchmarsh, E.C.: The Theory of the Riemann Zeta-Function, 2nd edn. Oxford University Press, Oxford (1986). (revised by D.R. Heath-Brown)
van Frankenhuijsen, M.: Arithmetic progressions of zeros of the Riemann zeta-function. J. Number Theory 115, 360–370 (2005)
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The authors would like to express their gratitude to the anonymous referee for her or his careful reading and valuable remarks and corrections.
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Özbek, S.S., Steuding, J. The values of the Riemann zeta-function on generalized arithmetic progressions. Arch. Math. 112, 53–59 (2019). https://doi.org/10.1007/s00013-018-1254-1
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DOI: https://doi.org/10.1007/s00013-018-1254-1