Lambert Series
A Lambert series is a series of the form
for |x|<1. Then
where
| [画像: b_N=sum_(n|N)a_n. ] |
(4)
|
The particular case a_n=1 is sometimes denoted
for |beta|<1 (Borwein and Borwein 1987, pp. 91 and 95), where psi_q(z) is a q-polygamma function. Special cases and related sums include
(Borwein and Borwein 1997, pp. 91-92), which arise in the reciprocal Fibonacci and reciprocal Lucas constants.
Some beautiful series of this type include
where mu(n) is the Möbius function, phi(n) is the totient function, d(n)=sigma_0(n) is the number of divisors of n, psi_q(z) is the q-polygamma function, sigma_k(n) is the divisor function, r(n) is the number of representations of n in the form n=A^2+B^2 where A and B are rational integers (Hardy and Wright 1979), theta_3(q) is a Jacobi elliptic function (Bailey et al. 2006), lambda(n) is the Liouville function, and lsb(n) is the least significant bit of n.
See also
Divisor Function, Erdős-Borwein Constant, Lambda Function, Möbius Function, Möbius Transform, Reciprocal Fibonacci Constant, Reciprocal Lucas Constant, Totient FunctionExplore with Wolfram|Alpha
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References
Abramowitz, M. and Stegun, I. A. (Eds.). "Number Theoretic Functions." §24.3.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 826-827, 1972.Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 24-15, 1997.Arndt, J. "On Computing the Generalized Lambert Series." 24 Jun 2012. http://arxiv.org/abs/1202.6525.Bailey, D. H.; Borwein, J. M.; Kapoor, V.; and Weisstein, E. W. "Ten Problems in Experimental Mathematics." Amer. Math. Monthly 113, 481-509, 2006.Borwein, J. M. and Borwein, P. B. "Evaluation of Sums of Reciprocals of Fibonacci Sequences." §3.7 in Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 91-101, 1987.Erdős, P. "On Arithmetical Properties of Lambert Series." J. Indian Math. Soc. 12, 63-66, 1948.Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 257-258, 1979.Referenced on Wolfram|Alpha
Lambert SeriesCite this as:
Weisstein, Eric W. "Lambert Series." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LambertSeries.html