Involve: A Journal of Mathematics

  • Involve
  • Volume 2, Number 2 (2009), 121-159.

Generating and zeta functions, structure, spectral and analytic properties of the moments of the Minkowski question mark function

Giedrius Alkauskas

Full-text: Open access


In this paper we are interested in moments of the Minkowski question mark function ?(x). It appears that, to some extent, the results are analogous to results obtained for objects associated with Maass wave forms: period functions, L-series, distributions. These objects can be naturally defined for ?(x) as well. Various previous investigations of ?(x) are mainly motivated from the perspective of metric number theory, Hausdorff dimension, singularity and generalizations. In this work it is shown that analytic and spectral properties of various integral transforms of ?(x) do reveal significant information about the question mark function. We prove asymptotic and structural results about the moments, calculate certain integrals which involve ?(x), define an associated zeta function, generating functions, Fourier series, and establish intrinsic relations among these objects.

Article information

Involve, Volume 2, Number 2 (2009), 121-159.

Received: 29 January 2008
Accepted: 29 December 2008
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11A55: Continued fractions {For approximation results, see 11J70} [See also 11K50, 30B70, 40A15] 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72} 26A30: Singular functions, Cantor functions, functions with other special properties
Secondary: 11F99: None of the above, but in this section

Minkowski question mark function Farey tree period functions distribution moments


Alkauskas, Giedrius. Generating and zeta functions, structure, spectral and analytic properties of the moments of the Minkowski question mark function. Involve 2 (2009), no. 2, 121--159. doi:10.2140/involve.2009.2.121.

Export citation


  • G. Alkauskas, “An asymptotic formula for the moments of Minkowski question mark function in the interval $[0,1]$”, Lithuanian Math. J. 48:4 (2008), 357–367.
  • G. Alkauskas, “The moments of Minkowski question mark function: the dyadic period function”, preprint. submitted.
  • K. I. Babenko, “On a problem of Gauss”, Dokl. Akad. Nauk SSSR 238:5 (1978), 1021–1024.
  • O. R. Beaver and T. Garrity, “A two-dimensional Minkowski $?(x)$ function”, J. Number Theory 107:1 (2004), 105–134.
  • C. Bonanno, S. Graffi, and S. Isola, “Spectral analysis of transfer operators associated to Farey fractions”, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 19:1 (2008), 1–23.
  • N. Calkin and H. S. Wilf, “Recounting the rationals”, Amer. Math. Monthly 107:4 (2000), 360–363.
  • P. Contucci and A. Knauf, “The phase transition of the number-theoretical spin chain”, Forum Math. 9:4 (1997), 547–567.
  • P. Cvitanović, K. Hansen, J. Rolf, and G. Vattay, “Beyond the periodic orbit theory”, Nonlinearity 11:5 (1998), 1209–1232.
  • A. Denjoy, “Sur une fonction réelle de Minkowski”, J. Math. Pures Appl. 17 (1938), 105–151.
  • A. Denjoy, “La fonction minkowskienne complexe uniformisée détermine les intervalles de validité des transformations de la fonction reélle”, C. R. Acad. Sci. Paris 242 (1956), 1924–1930.
  • A. Denjoy, “La fonction minkowskienne complexe uniformisée éclaire la genèse des fractions continues canoniques réelles”, C. R. Acad. Sci. Paris 242 (1956), 1817–1823.
  • A. Denjoy, “Propriétés différentielles de la fonction minkowskienne réelle. Statistique des fractions continues”, C. R. Acad. Sci. Paris 242 (1956), 2075–2079.
  • K. Dilcher and K. B. Stolarsky, “A polynomial analogue to the Stern sequence”, Int. J. Number Theory 3:1 (2007), 85–103.
  • A. Dushistova and N. G. Moshchevitin, “On the derivative of the Minkowski question mark function $?(x)$”, preprint.
  • M. D. Esposti, S. Isola, and A. Knauf, “Generalized Farey trees, transfer operators and phase transitions”, preprint.
  • S. R. Finch, Mathematical constants, vol. 94, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2003.
  • P. Flajolet and B. Vallée, “Continued fraction algorithms, functional operators, and structure constants”, Theoret. Comput. Sci. 194:1-2 (1998), 1–34.
  • R. Girgensohn, “Constructing singular functions via Farey fractions”, J. Math. Anal. Appl. 203:1 (1996), 127–141.
  • P. J. Grabner, P. Kirschenhofer, and R. F. Tichy, “Combinatorial and arithmetical properties of linear numeration systems”, Combinatorica 22:2 (2002), 245–267. Special issue: Paul Erdős and his mathematics.
  • M. J. S. Haran, The mysteries of the real prime, vol. 25, London Mathematical Society Monographs. New Series, The Clarendon Press Oxford University Press, New York, 2001.
  • S. Isola, “On the spectrum of Farey and Gauss maps”, Nonlinearity 15:5 (2002), 1521–1539.
  • M. Kesseb öhmer and B. O. Stratmann, “A multifractal analysis for Stern-Brocot intervals, continued fractions and Diophantine growth rates”, J. Reine Angew. Math. 605 (2007), 133–163.
  • M. Kesseb öhmer and B. O. Stratmann, “Fractal analysis for sets of non-differentiability of Minkowski's question mark function”, J. Number Theory 128:9 (2008), 2663–2686.
  • A. Y. Khinchin, Continued fractions, The University of Chicago Press, Chicago, Ill.-London, 1964.
  • J. R. Kinney, “Note on a singular function of Minkowski”, Proc. Amer. Math. Soc. 11 (1960), 788–794.
  • A. N. Kolmogorov and S. V. Fomin, Elementy teorii funktsii i funktsionalnogo analiza, Sixth ed., “Nauka”, Moscow, 1989. With a supplement, “Banach algebras”, by V. M. Tikhomirov.
  • M. Kontsevich and D. Zagier, “Periods”, pp. 771–808 in Mathematics unlimited–-2001 and beyond, Springer, Berlin, 2001.
  • J. C. Lagarias, “The Farey shift and the Minkowski $?$-function”, unpublished manuscript, 1991.
  • J. C. Lagarias and C. P. Tresser, “A walk along the branches of the extended Farey tree”, IBM J. Res. Develop. 39:3 (1995), 283–294.
  • M. Lamberger, “On a family of singular measures related to Minkowski's $?(x)$ function”, Indag. Math. (N.S.) 17:1 (2006), 45–63.
  • M. A. Lavrentjev and B. V. Shabat, Methods in the theory of functions of complex variable, Nauka, Moscow, 1987.
  • J. B. Lewis, “Spaces of holomorphic functions equivalent to the even Maass cusp forms”, Invent. Math. 127:2 (1997), 271–306.
  • J. Lewis and D. Zagier, “Period functions for Maass wave forms. I”, Ann. of Math. (2) 153:1 (2001), 191–258.
  • Y. I. Manin and M. Marcolli, “Continued fractions, modular symbols, and noncommutative geometry”, Selecta Math. (N.S.) 8:3 (2002), 475–521.
  • N. Moshchevitin and M. Vielhaber, “Moments for generalized Farey-Brocot partitions”, preprint.
  • M. Newman, “Recounting the rationals, continued”, Amer. Math. Monthly 110 (2003), 642–643.
  • H. Okamoto and M. Wunsch, “A geometric construction of continuous, strictly increasing singular functions”, Proc. Japan Acad. Ser. A Math. Sci. 83:7 (2007), 114–118.
  • G. Panti, “Multidimensional continued fractions and a Minkowski function”, Monatsh. Math. 154:3 (2008), 247–264.
  • J. Paradís, P. Viader, and L. Bibiloni, “The derivative of Minkowski's $?(x)$ function”, J. Math. Anal. Appl. 253:1 (2001), 107–125.
  • G. Ramharter, “On Minkowski's singular function”, Proc. Amer. Math. Soc. 99:3 (1987), 596–597.
  • S. Reese, “Some Fourier-Stieltjes coefficients revisited”, Proc. Amer. Math. Soc. 105:2 (1989), 384–386.
  • B. Reznick, “Regularity property of the Stern enumeration of the rationals”, preprint.
  • F. Ryde, “Arithmetical continued fractions”, Lunds universitets arsskrift N. F. Avd. 2. 22:2 (1922), 1–182.
  • F. Ryde, “On the relation between two Minkowski functions”, J. Number Theory 17:1 (1983), 47–51.
  • R. Salem, “On some singular monotonic functions which are strictly increasing”, Trans. Amer. Math. Soc. 53 (1943), 427–439.
  • J.-P. Serre, A course in arithmetic, Springer-Verlag, New York, 1973. Translated from the French, Graduate Texts in Mathematics, No. 7.
  • M. A. Stern, “Über eine zahlentheoretische Funktion”, J. Reine Angew. Math. 55 (1858), 193–220.
  • J. Steuding, 2006. personal communication.
  • R. F. Tichy and J. Uitz, “An extension of Minkowski's singular function”, Appl. Math. Lett. 8:5 (1995), 39–46.
  • L. Vepštas, “The Minkowski question mark, ${\sf GL}(2,\mathbb{Z})$ and the modular monoid”, preprint, 2004,
  • P. Viader, J. Paradís, and L. Bibiloni, “A new light on Minkowski's $?(x)$ function”, J. Number Theory 73:2 (1998), 212–227.
  • E. Wirsing, “On the theorem of Gauss-Kusmin-Lévy and a Frobenius-type theorem for function spaces”, Acta Arith. 24 (1973/74), 507–528. Collection of articles dedicated to Carl Ludwig Siegel on the occasion of his seventy-fifth birthday, V.
  • E. Wirsing, “J. Steuding's Problem”, unpublished manuscript, 2006.
  • D. Zagier, “New points of view on the Selberg zeta function”, unpublished manuscript, 2001.