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On the value-distribution of iterated integrals of the logarithm of the Riemann zeta-function I: Denseness

  • Kenta Endo and Shōta Inoue EMAIL logo
From the journal Forum Mathematicum

Abstract

We consider iterated integrals of logζ(s) on certain vertical and horizontal lines. Here, the function ζ(s) is the Riemann zeta-function. It is a well-known open problem whether or not the values of the Riemann zeta-function on the critical line are dense in the complex plane. In this paper, we give a result for the denseness of the values of the iterated integrals on the horizontal lines. By using this result, we obtain the denseness of the values of 0tlogζ(12+it)𝑑t under the Riemann Hypothesis. Moreover, we show that, for any m2, the denseness of the values of an m-times iterated integral on the critical line is equivalent to the Riemann Hypothesis.

MSC 2010: 11M06; 11M26

Communicated by Valentin Blomer


Award Identifier / Grant number: 19J11223

Funding statement: The second author is supported by Grant-in-Aid for JSPS Research Fellow (Grant No. 19J11223).

Acknowledgements

The authors would like to deeply thank Professor Kohji Matsumoto for many useful comments and suggestions. They would also like to thank Mr Masahiro Mine for useful discussion.

References

[1] H. Bohr, Zur Theorie der Riemann’schen Zeta-funktion im kritischen Streifen, Acta Math. 40 (1916), 67–100. 10.1007/BF02418541Search in Google Scholar

[2] H. Bohr and R. Courant, Neue Anwendungen der Theorie der Diophantischen Approximationen auf die Riemannsche Zetafunktion, J. Reine Angew. Math. 144 (1914), 249–274. 10.1515/crll.1914.144.249Search in Google Scholar

[3] H. Bohr and B. Jessen, Über die Werteverteilung der Riemannschen Zetafunktion, Erste Mitteilung, Acta Math. 54 (1930), 1–35; Zweite Mitteilung, ibid. 58 (1932), 1–55. 10.1007/BF02547516Search in Google Scholar

[4] E. Bombieri and D. A. Hejhal, On the distribution of zeros of linear combinations of Euler products, Duke Math. J. 80 (1995), no. 3, 821–862. 10.1215/S0012-7094-95-08028-4Search in Google Scholar

[5] T. Chatterjee and S. Gun, On the zeros of generalized Hurwitz zeta functions, J. Number Theory 145 (2014), 352–361. 10.1016/j.jnt.2014.06.004Search in Google Scholar

[6] K. Endo, S. Inoue and M. Mine, On the value distribution of iterated integrals of the logarithm of the Riemann zeta-function II: Probabilistic aspects, in preparation. Search in Google Scholar

[7] R. Garunks̆tis and J. Steuding, On the roots of the equation ζ(s)=a, Abh. Math. Semin. Univ. Hambg. 84 (2014), 1–15. 10.1007/s12188-014-0093-7Search in Google Scholar

[8] T. Hattori and K. Matsumoto, A limit theorem for Bohr-Jessen’s probability measures of the Riemann zeta-function, J. Reine Angew. Math. 507 (1999), 219–232. 10.1515/crll.1999.507.219Search in Google Scholar

[9] S. Inoue, On the logarithm of the Riemann zeta-function and its iterated integrals, preprint (2019), https://arxiv.org/abs/1909.03643. Search in Google Scholar

[10] B. Jessen and A. Wintner, Distribution functions and The Riemann zeta function, Trans. Amer. Math. Soc. 38 (1935), no. 1, 48–88. 10.1090/S0002-9947-1935-1501802-5Search in Google Scholar

[11] A. A. Karatsuba and S. M. Voronin, The Riemann Zeta-Function, De Gruyter Exp. Math. 5, Walter de Gruyter, Berlin, 1992. 10.1515/9783110886146Search in Google Scholar

[12] E. Kowalski and A. Nikeghbali, Mod-Gaussian convergence and the value distribution of ζ(12+it) and related quantities, J. Lond. Math. Soc. (2) 86 (2012), 291–319. 10.1112/jlms/jds003Search in Google Scholar

[13] Y. Lamzouri, Distribution of large values of zeta and L-functions, Int. Math. Res. Not. IMRN 2011 (2011), no. 23, 5449–5503. Search in Google Scholar

[14] Y. Lamzouri, S. Lester and M. Radziwiłł, Discrepancy bounds for the distribution of the Riemann zeta-function and applications, J. Anal. Math. 139 (2019), no. 2, 453–494. 10.1007/s11854-019-0063-1Search in Google Scholar

[15] M. Radziwiłł, Large deviations in Selberg’s limit theorem, preprint (2011), https://arxiv.org/abs/1108.5092. Search in Google Scholar

[16] A. Selberg, Contributions to the theory of the Riemann zeta-function, Collected Papers. Vol. 1, Springer, New York (1989), 214–280. Search in Google Scholar

[17] A. Selberg, Old and new conjectures and results about a class of Dirichlet series, Proceedings of the Amalfi Conference on Analytic Number Theory, Universitá di Salerno, Salerno (1992), 367–385. Search in Google Scholar

[18] S. M. Voronin, Theorem on the “universality” of the Riemann zeta-function (in Russian), Izv. Akad. Nauk SSSR, Ser. Mat. 39 (1975), 475–486; translation in Math. USSR Izv. 9 (1975), 443–445. 10.1070/IM1975v009n03ABEH001485Search in Google Scholar

[19] S. M. Voronin, Ω-theorems in the theory of the Riemann zeta-function (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 2, 424–436; translation in Math. USSR Izv. 32 (1989), no. 2, 429–442. 10.1070/IM1989v032n02ABEH000774Search in Google Scholar

Received: 2020-03-25
Revised: 2020-06-04
Published Online: 2020-10-07
Published in Print: 2021-01-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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