Lambert series

The series of functions $$\sum_{n=1}^\infty a_n \frac{x^n}{1-x^n} \ . \label{1}$$ It was considered by J.H. Lambert (see [1]) in connection with questions of convergence of power series. If the series $$ \sum_{n=1}^\infty a_n $$ converges, then the Lambert series converges for all values of $x$ except $x = \pm 1$; otherwise it converges for those values of $x$ for which the series $$ \sum_{n=1}^\infty a_n x^n $$ converges. The Lambert series is used in certain problems of number theory. Thus, for $|x| < 1$ the sum $\phi(x)$ of the series (1) can be represented as a power series: $$ \sum_{n=1}^\infty A_n x^n $$ where $$ A_n = \sum_{d | n} a_d $$ and the summation is over all divisors $d$ of $n$. In particular, if $a_n = 1$, then $A_n = \tau(n)$, the number of divisors of $n$; if $a_n = n$, then $A_n = \sigma(n)$, the sum of the divisors of $n$. The behaviour of $\phi(x)$ (with suitable $a_n$) as $x \nearrow 1$ is used, for example (see [3]), in the problem of Hardy and Ramanujan on obtaining an asymptotic formula for the number of "unbounded partitions" of a natural number.

Comments

Lambert series also occur in the expansion of Eisenstein series, a particular kind of modular form. See [a1].

References

[1]	J.H. Lambert, "Opera Mathematica" , 1–2 , O. Füssli (1946–1948)
[2]	G.M. Fichtenholz, "Differential und Integralrechnung" , 2 , Deutsch. Verlag Wissenschaft. (1964) Zbl 0143.27002
[3]	A.G. Postnikov, "Introduction to analytic number theory" , Moscow (1971) (In Russian)
[a1]	T.M. Apostol, "Modular forms and Dirichlet series in analysis" , Springer (1976)
[a2]	H. Rademacher, "Topics in analytic number theory" , Springer (1973)
[a3]	K. Knopp, "Theorie und Anwendung der unendlichen Reihen" , Springer (1964) (English translation: Blackie, 1951 & Dover, reprint, 1990)