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Difference between revisions of "Lambert series"

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The series of functions
 
The series of functions
$$\sum_{n=1}^\infty a_n \frac{x^n}{1-x^n} \ . \label{1}$$
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\begin{equation}\sum_{n=1}^\infty a_n \frac{x^n}{1-x^n} \ . \label{eq1}\end{equation}
 
It was considered by J.H. Lambert (see [[#References|[1]]]) in connection with questions of convergence of [[power series]]. If the series
 
It was considered by J.H. Lambert (see [[#References|[1]]]) in connection with questions of convergence of [[power series]]. If the series
 
$$
 
$$
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\sum_{n=1}^\infty a_n x^n
 
\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:
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converges. The Lambert series is used in certain problems of number theory. Thus, for $|x| < 1$ the sum $\phi(x)$ of the series \eqref{eq1} can be represented as a power series:
 
$$
 
$$
 
\sum_{n=1}^\infty A_n x^n
 
\sum_{n=1}^\infty A_n x^n
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====References====
 
====References====
 
<table>
 
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top"> J.H. Lambert, "Opera Mathematica" , '''1–2''' , O. Füssli  (1946–1948)</TD></TR>
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<TR><TD valign="top">[1]</TD> <TD valign="top"> J.H. Lambert, "Opera Mathematica" , '''1–2''' , O. Füssli  (1946–1948) {{ZBL|0060.01206}}</TD></TR>
 
<TR><TD valign="top">[2]</TD> <TD valign="top"> G.M. Fichtenholz, "Differential und Integralrechnung" , '''2''' , Deutsch. Verlag Wissenschaft.  (1964) {{ZBL|0143.27002}}</TD></TR>
 
<TR><TD valign="top">[2]</TD> <TD valign="top"> G.M. Fichtenholz, "Differential und Integralrechnung" , '''2''' , Deutsch. Verlag Wissenschaft.  (1964) {{ZBL|0143.27002}}</TD></TR>
 
<TR><TD valign="top">[3]</TD> <TD valign="top"> A.G. Postnikov, "Introduction to analytic number theory" , Moscow  (1971)  (In Russian)</TD></TR>
 
<TR><TD valign="top">[3]</TD> <TD valign="top"> A.G. Postnikov, "Introduction to analytic number theory" , Moscow  (1971)  (In Russian)</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> T.M. Apostol, "Modular forms and Dirichlet series in analysis" , Springer  (1976)</TD></TR>
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<TR><TD valign="top">[a1]</TD> <TD valign="top"> T.M. Apostol, "Modular functions and Dirichlet series in analysis" , Springer  (1976) {{ZBL|0332.10017}}</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Rademacher, "Topics in analytic number theory" , Springer  (1973)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Rademacher, "Topics in analytic number theory" , Springer  (1973) {{ZBL|0253.10002}}</TD></TR>
 
<TR><TD valign="top">[a3]</TD> <TD valign="top"> K. Knopp, "Theorie und Anwendung der unendlichen Reihen" , Springer  (1964)  (English translation: Blackie, 1951 &amp; Dover, reprint, 1990)</TD></TR>
 
<TR><TD valign="top">[a3]</TD> <TD valign="top"> K. Knopp, "Theorie und Anwendung der unendlichen Reihen" , Springer  (1964)  (English translation: Blackie, 1951 &amp; Dover, reprint, 1990)</TD></TR>
 
</table>
 
</table>
  
 
{{TEX|done}}
 
{{TEX|done}}

Latest revision as of 07:06, 29 March 2024

The series of functions \begin{equation}\sum_{n=1}^\infty a_n \frac{x^n}{1-x^n} \ . \label{eq1}\end{equation} 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 \eqref{eq1} 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) Zbl 0060.01206
[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 functions and Dirichlet series in analysis" , Springer (1976) Zbl 0332.10017
[a2] H. Rademacher, "Topics in analytic number theory" , Springer (1973) Zbl 0253.10002
[a3] K. Knopp, "Theorie und Anwendung der unendlichen Reihen" , Springer (1964) (English translation: Blackie, 1951 & Dover, reprint, 1990)
How to Cite This Entry:
Lambert series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lambert_series&oldid=52932
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article