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Lambert Series

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A Lambert series is a series of the form

F(x)=sum_(n=1)^inftya_n(x^n)/(1-x^n)
(1)

for |x|<1. Then

F(x)=sum_(n=1)^(infty)a_nsum_(m=1)^(infty)x^(mn)
(2)
=sum_(N=1)^(infty)b_Nx^N,
(3)

where

b_N=sum_(n|N)a_n.
(4)

The particular case a_n=1 is sometimes denoted

L(beta)=sum_(n=1)^(infty)(beta^n)/(1-beta^n)
(5)
=sum_(n=1)^(infty)1/(beta^(-n)-1)
(6)
=(psi_beta(1)+ln(1-beta))/(lnbeta)
(7)

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

sum_(n=1)^(infty)(beta^n)/(1+beta^(2n))=1/4[theta_3^2(beta)-1]
(8)
sum_(n=1)^(infty)(beta^(2n+1))/(1+beta^(4n+2))=1/4[theta_3^2(beta)-theta_2^2(beta)]
(9)
=1/4theta_2^2(beta^2)
(10)
sum_(n=1)^(infty)(beta^n)/(1-beta^(2n))=L(beta)-L(beta^2)
(11)
sum_(n=1)^(infty)(beta^(2n+1))/(1-beta^(4n+2))=L(beta)-2L(beta^2)+L(beta^4)
(12)

(Borwein and Borwein 1997, pp. 91-92), which arise in the reciprocal Fibonacci and reciprocal Lucas constants.

Some beautiful series of this type include

sum_(n=1)^(infty)(mu(n)x^n)/(1-x^n)=x
(13)
sum_(n=1)^(infty)(phi(n)x^n)/(1-x^n)=x/((1-x)^2)
(14)
sum_(n=1)^(infty)(x^n)/(1-x^n)=sum_(n=1)^(infty)d(n)x^n
(15)
=(psi_x(1)+ln(1-x))/(lnx)
(16)
sum_(n=1)^(infty)(n^kx^n)/(1-x^n)=sum_(n=1)^(infty)sigma_k(n)x^n
(17)
4sum_(n=1)^(infty)((-1)^(n+1)x^(2n+1))/(1-x^(2n+1))=sum_(n=1)^(infty)r_2(n)x^n
(18)
=theta_3^2(x)-1
(19)
sum_(n=1)^(infty)(lambda(n)x^n)/(1-x^n)=sum_(n=1)^(infty)x^(n^2)
(20)
=1/2[theta_3(x)-1]
(21)
sum_(n=1)^(infty)(lsb(n)x^n)/(1-x^n)=(ln(1-x^2)+psi_(x^2)(1/2))/(ln(x^2)),
(22)

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 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. 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.

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Lambert Series

Cite this as:

Weisstein, Eric W. "Lambert Series." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LambertSeries.html

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