The Minkowski question mark function, and topics related to continued fractions, modular world

  1. Transfer operator for the Gauss' continued fraction map. I. Structure of the eigenvalues and trace formulas. The newest version. The talk on this topic given in February, 2015.
  2. Transfer operator for the Gauss' continued fraction map. II. Fine arithmetic of the decomposition formulas (planned in 2018).
  3. Transfer operator for the Gauss' continued fraction map. III. Selberg zeta function and structure of the eigenvalues of the Mayer-Ruelle operator (planned in 2018).
  4. The modular group and words in its two generators, Lith. Math. J. 57 (1) (2017), 1-12.
  5. The Minkowski ?(x), a class of singular measures, quasi-modular and mean-modular forms. I. The newest version. "The Minkowski question mark function, quasi-modular forms and the Dedekind eta-function", the talk on this topic given in March, 2017.
  6. The map from the set of nontrivial zeros of the Riemann zeta function to the set of eigenvalues of the Gauss-Kuzmin-Wirsing operator.
  7. Fourier-Stieltjes coefficients of the Minkowski question mark function, Anal. Probab. Methods Number Theory, Proceedings of the Fifth Intern. Conf. in Honour of J. Kubilius, Palanga, Lithuania, 4-10 September 2011 (2012), 19-33.
  8. The Minkowski ?(x) function and Salem's problem, Comptes Rendus Mathématique 350 (3-4) (2012), 137-140.
  9. Semi-regular continued fractions and an exact formula for the moments of the Minkowski question mark function, Ramanujan J. 25 (3) (2011), 359-367.
  10. The Minkowski question mark function: explicit series for the dyadic period function and moments, Math. Comp. 79 (269) (2010), 383-418; Addenda and corrigenda, Math. Comp. 80 (276) (2011), 2445-2454.
  11. The moments of Minkowski question mark function: the dyadic period function, Glasg. Math. J. 52 (1) (2010), 41-64.
  12. Generating and zeta functions, structure, spectral and analytic properties of the moments of the Minkowski question mark function, Involve 2 (2) (2009), 121-159.
  13. An asymptotic formula for the moments of Minkowski question mark function in the interval [0,1], Lith. Math. J. 48 (4) (2008), 357-367.

Projective and general flows: relations to topology, algebraic geometry, differential geometry, differential equations and mathematical physics

  1. Projective and polynomial superflows. I. Latest version is here (110 pages).
  2. Projective and polynomial superflows. II. O(3) and the icosahedral group.
  3. Projective and polynomial superflows. III. Finite subgroups of U(2).
  4. Projective and polynomial superflows. IV. Arithmetic of the orbits.
  5. Planar 2-homogeneous commutative rational vector fields.
  6. General theory of unramified projective flows.
  7. Beltrami vector fields with an icosahedral symmetry.
  8. Beltrami vector fields with polyhedral symmetries.
  9. The projective translation equation and rational plane flows. II. Corrections and additions, Aequationes Math. 91 (5) (2017), 871-907
  10. Algebraic and abelian solutions to the projective translation equation, Aequationes Math. 90 (4) (2016), 727-763
  11. The projective translation equation and unramified 2-dimensional flows with rational vector fields, Aequationes Math. 89 (3) (2015), 873-913
  12. The projective translation equation and rational plane flows. I, Aequationes Math. 85 (3) (2013), 273-328.
  13. Multi-variable translation equation which arises from homothety, Aequationes Math. 80 (3) (2010), 335-350.

Other: number theory, partitions

  1. Algebraic functions with Fermat property, eigenvalues of transfer operator and Riemann zeros, and other open problems.
  2. Congruence properties of the function that counts compositions into powers of 2, J. Integer Seq. 13 (2010), Article: 10.5.3, 9 p.
  3. A curious proof of Fermat's little theorem, Amer. Math. Monthly 116 (4) (2009), 362-364.
  4. Functional equation related to quadratic and norm forms, Lith. Math. J. 45 (2) (2005), 153-172.
  5. Prime and composite numbers as integer parts of powers (with A. Dubickas), Acta Math. Hungar. 105 (3) (2004), 249-256.
  6. Dirichlet series associated with strongly q-multiplicative functions, Ramanujan J. 8 (1) (2004), 13-21.
  7. Generalization of the Rödseth-Gupta theorem on binary partitions, Lith. Math. J. 43 (2) (2003), 123-132.

Ph.D. thesis: "Integral transforms of the Minkowski question mark function" (2008).

Popular papers, notes
  1. Author of IMO 6, 1997, "American Mathematical Monthly" problems 11484, 11661.
  2. The moments of the Minkowski question mark function, Minkowski-Alkauskas constants by Steven R. Finch.
  3. Author of the sequences A265434 and A220420 in "The On-Line Encyclopedia of Integer Sequences".
  4. Friendly paths (solution of the problem 11484) Amer. Math. Monthly 119 (2) (2012), 167-168. (The official solution contains few typos: b-a-1 and b-a+2 should be replaced by b-a and b-a+1 in the displayed inequality).
  5. m-nariniai skaidiniai, (1999). Course-work. This paper, written in Lithuanian, contains new proof of a known result, some new results, one of which was proved by other authors 16 years later.
  6. Recursive construction of a series converging to the eigenvalues of the Gauss-Kuzmin-Wirsing operator, (2010). This paper is not for publication since it was superseded by this paper.
  7. Dirichlet series associated with Thue-Morse sequence; Vilnius University, Department of Mathematics, preprint (2001). This work is badly written, maybe some results are incorrect; but it contains the explicit construction of the function F(x) which satisfies F'(x)=4F(2x). This very function was independently rediscovered by Japanese mathematician in this paper few years later. Some day in the near future I will rewrite this preprint, since certain results are really interesting.
  8. Exact lower bound for the house of algebraic integer with abelian Galois group (2003). The result of the note is a rediscovery of the old result of A. Schinzel.
  9. Splitting of primes in Q(5v2), tutorial (2004); inspired by the excellent introduction to algebraic number theory by H. Stark.
  10. A question from complex dynamics (2008).
  11. Solutions of IMO 2005.
  12. Prime numbers in arithmetic progressions, Alfa+Omega 2 (4), 4 pages, (1997).
  13. The germs of important mathematical results in pupil olympiads, Alfa+Omega 2 (4), 5 pages, (1997).
  14. Inequalities, Alfa+Omega 1 (5), 6 pages, 1998.
  15. Functional equations in pupil olympiads, Alfa+Omega 1 (7), 14 pages, (1999).
  16. Commutative polynomials, Alfa+Omega 1 (7), 4 pages, (2001).
  17. Several brochures (with co-authors) on Lithuanian Mathematical Olympiads, 1996-2000.