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Let ${\alpha}$ > 0 be a real number and $(s_{\alpha}(n))_{n{\geq}1}$ be the lexicographically greatest Sturmian word of slope ${\alpha}$. We investigate Dirichlet series of the form ${\sum}^{\infty}_{n=1}s_{\alpha}(n)n^{-s}$. To do this, a generalization of Euler's totient function is required. For a real ${\alpha}$ > 0 and a positive integer n, an arithmetic function ${\varphi}{\alpha}(n)$ is defined to be the number of positive integers m for which gcd(m, n) = 1 and 0 < m/n < ${\alpha}$. Under a condition Re(s) > 1, this paper establishes an identity ${\sum}^{\infty}_{n=1}s_{\alpha}(n)n^{-S}=1+{\sum}^{\infty}_{n=1}{\varphi}_{\alpha}(n)({\zeta}(s)-{\zeta}(s,1+n^{-1}))n^{-s}$.
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참고문헌 (9)
참고문헌 신청
T.M. Apostol / 1976 / Introduction to analytic number theory / Springer-Verlag
J.P. Borel / 1993 / Quelques mots sur la droite projective r eelle / J. Th eor. Nombres Bordeaux 5(1):23~51
L. Euler / Speculationes circa quasdam insignes proprietates numerorum
G.H. Hardy / 2008 / An introduction to the theory of numbers / Oxford University Press
D.Y. Kwon / 2013 / Moments of discrete measures with dense jumps induced by fi- expansions / J. Math. Anal. Appl 399(1):1~11
J.P. Borel / 1993 / Quelques mots sur la droite projective r eelle / J. Th eor. Nombres Bordeaux 5(1):23~51
L. Euler / Speculationes circa quasdam insignes proprietates numerorum
G.H. Hardy / 2008 / An introduction to the theory of numbers / Oxford University Press
D.Y. Kwon / 2013 / Moments of discrete measures with dense jumps induced by fi- expansions / J. Math. Anal. Appl 399(1):1~11
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