# DISCRETE MEASURES WITH DENSE JUMPS INDUCED BY STURMIAN DIRICHLET SERIES

• KWON, DOYONG (Department of Mathematics Chonnam National University)
• Received : 2014.02.06
• Published : 2015.11.30
• 79 7

#### Abstract

Let ($S_{\alpha}(n))_{n{\geq}1}$ be the lexicographically greatest Sturmian word of slope ${\alpha}$ > 0. For a fixed ${\sigma}$ > 1, we consider Dirichlet series of the form ${\nu}_{\sigma}({\alpha})$ := ${\Sigma}_{n=1}^{\infty}s_{\alpha}(n)n^{-{\sigma}}$. This paper studies the singular properties of the real function ${\nu}_{\sigma}$, and the Lebesgue-Stieltjes measure whose distribution is given by ${\nu}_{\sigma}$.

#### Keywords

Dirichlet series;singular function;Sturmian word

#### Acknowledgement

Supported by : National Research Foundation of Korea(NRF)

#### References

1. T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York- Heidelberg, 1976.
2. D. Y. Kwon, Moments of discrete measures with dense jumps induced by ${\beta}$-expansions, J. Math. Anal. Appl. 399 (2013), no. 1, 1-11. https://doi.org/10.1016/j.jmaa.2012.07.014
3. D. Y. Kwon, A one-parameter family of Dirichlet series whose coefficients are Sturmian words, J. Number Theory 147 (2015), 824-835. https://doi.org/10.1016/j.jnt.2014.08.018
4. M. Lothaire, Algebraic Combinatorics on Words, Cambridge University Press, 2002.
5. M. Morse and G. A. Hedlund, Symbolic dynamics II. Sturmian trajectories, Amer. J. Math. 62 (1940), 1-42. https://doi.org/10.2307/2371431