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

Kenta Endo 2  and Shōta Inoue 1
  • 1 Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, Japan
  • 2 Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, Japan
Kenta Endo and Shōta Inoue
  • Corresponding author
  • Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan
  • Email
  • Search for other articles:
  • degruyter.comGoogle Scholar

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.

  • [1]

    H. Bohr, Zur Theorie der Riemann’schen Zeta-funktion im kritischen Streifen, Acta Math. 40 (1916), 67–100.

    • Crossref
    • Export Citation
  • [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.

  • [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.

    • Crossref
    • Export Citation
  • [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.

  • [5]

    T. Chatterjee and S. Gun, On the zeros of generalized Hurwitz zeta functions, J. Number Theory 145 (2014), 352–361.

    • Crossref
    • Export Citation
  • [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.

  • [7]

    R. Garunks̆tis and J. Steuding, On the roots of the equation ζ ( s ) = a {\zeta(s)=a}, Abh. Math. Semin. Univ. Hambg. 84 (2014), 1–15.

  • [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.

  • [9]

    S. Inoue, On the logarithm of the Riemann zeta-function and its iterated integrals, preprint (2019), https://arxiv.org/abs/1909.03643.

  • [10]

    B. Jessen and A. Wintner, Distribution functions and The Riemann zeta function, Trans. Amer. Math. Soc. 38 (1935), no. 1, 48–88.

    • Crossref
    • Export Citation
  • [11]

    A. A. Karatsuba and S. M. Voronin, The Riemann Zeta-Function, De Gruyter Exp. Math. 5, Walter de Gruyter, Berlin, 1992.

  • [12]

    E. Kowalski and A. Nikeghbali, Mod-Gaussian convergence and the value distribution of ζ ( 1 2 + i t ) {\zeta(\frac{1}{2}+it)} and related quantities, J. Lond. Math. Soc. (2) 86 (2012), 291–319.

    • Crossref
    • Export Citation
  • [13]

    Y. Lamzouri, Distribution of large values of zeta and L-functions, Int. Math. Res. Not. IMRN 2011 (2011), no. 23, 5449–5503.

  • [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.

    • Crossref
    • Export Citation
  • [15]

    M. Radziwiłł, Large deviations in Selberg’s limit theorem, preprint (2011), https://arxiv.org/abs/1108.5092.

  • [16]

    A. Selberg, Contributions to the theory of the Riemann zeta-function, Collected Papers. Vol. 1, Springer, New York (1989), 214–280.

  • [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.

  • [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.

  • [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.

Purchase article
Get instant unlimited access to the article.
$42.00
Log in
Already have access? Please log in.


or
Log in with your institution

Journal + Issues

Forum Mathematicum is a general mathematics journal, which is devoted to the publication of research articles in all fields of pure and applied mathematics, including mathematical physics. Forum Mathematicum belongs to the top 50 journals in pure and applied mathematics, as measured by citation impact.

Search