A Lambert series is a series of the form

(1)

for .
Then
where

(4)

The particular case
is sometimes denoted
for
(Borwein and Borwein 1987, pp. 91 and 95), where is a qpolygamma
function. Special cases and related sums include
(Borwein and Borwein 1997, pp. 9192), which arise in the reciprocal
Fibonacci and reciprocal Lucas constants.
Some beautiful series of this type include
where
is the Möbius function, is the totient function,
is the number of divisors
of ,
is the qpolygamma
function,
is the divisor function, is the number of representations of in the form where and are rational integers (Hardy and Wright 1979), is a Jacobi elliptic function (Bailey et al. 2006),
is the Liouville
function, and
is the least significant bit of .
See also
Divisor Function,
ErdősBorwein Constant,
Lambda Function,
Möbius
Function,
Möbius Transform,
Reciprocal
Fibonacci Constant,
Reciprocal Lucas
Constant,
Totient Function
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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. 826827, 1972.Apostol, T. M. Modular
Functions and Dirichlet Series in Number Theory, 2nd ed. New York: SpringerVerlag,
pp. 2415, 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,
481509, 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. 91101, 1987.Erdős, P. "On Arithmetical
Properties of Lambert Series." J. Indian Math. Soc. 12, 6366,
1948.Hardy, G. H. and Wright, E. M. An
Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon
Press, pp. 257258, 1979.Referenced on WolframAlpha
Lambert Series
Cite this as:
Weisstein, Eric W. "Lambert Series." From
MathWorldA Wolfram Web Resource. https://mathworld.wolfram.com/LambertSeries.html
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