Glasnik Matematicki, Vol. 56, No. 1 (2021), 79-94.

EXTREMAL BEHAVIOUR OF ± 1-VALUED COMPLETELY MULTIPLICATIVE FUNCTIONS IN FUNCTION FIELDS

Nikola Lelas

Faculty of Mathematics, University of Belgrade, 11000 Belgrade, Serbia
e-mail: dzoni@matf.bg.ac.rs

Abstract.   We investigate the classical Pólya and Turán conjectures in the context of rational function fields over finite fields 𝔽q. Related to these two conjectures we investigate the sign of truncations of Dirichlet L-functions at point s=1 corresponding to quadratic characters over 𝔽q[t], prove a variant of a theorem of Landau for arbitrary sets of monic, irreducible polynomials over 𝔽q[t] and calculate the mean value of certain variants of the Liouville function over 𝔽q[t].

2010 Mathematics Subject Classification.   11T06, 11N80, 11N37

Key words and phrases.   Liouville function, quadratic characters, Möbius function, function fields

Full text (PDF) (access from subscribing institutions only)

https://doi.org/10.3336/gm.56.1.06

References:

  1. P. Borwein, S. Choi and M. Coons, Completely multiplicative functions taking values in {-1,1 }, Trans. Amer. Math. Soc. 362 (2010), 6279-6291.
    MathSciNet     CrossRef

  2. A. Granville and K. Soundararajan, Negative values of truncations to L(1,χ), Proceedings of the Gauss-Dirichlet Conference, Göttingen, 2005, Clay Math. Proc. 7, Amer. Math. Soc., Providence, 2007.
    MathSciNet    

  3. C. B. Haselgrove, A disproof of a conjecture of Pólya, Mathematika, 5 (1958), 141-145.
    MathSciNet     CrossRef

  4. P. Humphries, The Mertens and Pólya conjectures in function fields, Master of Philosophy thesis, Australian National University, Australia, 2012.

  5. O. Klurman, Mean values and correlations of multiplicative functions: The ''pretentious'' approach, Doctoral dissertation, University of Montreal, Canada, 2017.

  6. H. von Mangoldt, Beweis der Gleichung ∑k=1∞ μ(k)/k=0, Sitzungsber. Königl. Preuss. Akad. Wiss. Berlin, 1897, 849-851.

  7. G. Pólya, Verschiedene bemerkungen zur zahlentheorie, Jahresber. Deutsch. Math.-Verein, 28 (1919), 31-40.

  8. M. Rosen, Number theory in function fields, Springer, New York, 2000.
    MathSciNet     CrossRef

  9. M. Skałba, On Euler-von Mangoldt's equation, Colloq. Math. 69 (1995), 143-145.
    MathSciNet     CrossRef

  10. T. Tao, A remark on partial sums involving the Möbius function, Bull. Aust. Math. Soc. 81 (2010), 343-349.
    MathSciNet     CrossRef

  11. P. Turán, On some approximate Dirichlet-polynomials in the theory of zeta-function of Riemann, Munksgaard, Copenhagen, 1948.
    MathSciNet    
Glasnik Matematicki Home Page