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The series of functions
 
The series of functions
 
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$$\sum_{n=1}^\infty a_n \frac{x^n}{1-x^n} \ . \label{1}$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057340/l0573401.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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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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$$
It was considered by J.H. Lambert (see [[#References|[1]]]) in connection with questions of convergence of [[Power series|power series]]. If the series
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\sum_{n=1}^\infty a_n
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057340/l0573402.png" /></td> </tr></table>
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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
 
+
$$
converges, then the Lambert series converges for all values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057340/l0573403.png" /> except <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057340/l0573404.png" />; otherwise it converges for those values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057340/l0573405.png" /> for which the series
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\sum_{n=1}^\infty a_n x^n
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057340/l0573406.png" /></td> </tr></table>
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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 (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057340/l0573407.png" /> the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057340/l0573408.png" /> of the series (1) can be represented as a power series:
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\sum_{n=1}^\infty A_n x^n
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057340/l0573409.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
 
 
 
 
where
 
where
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$$
 +
A_n = \sum_{d | n} a_d
 +
$$
 +
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 divisors|sum of the divisors]] of $n$. The behaviour of $\phi(x)$ (with suitable $a_n$) as $x \nearrow 1$ is used, for example (see [[#References|[3]]]), in the problem of Hardy and Ramanujan on obtaining an asymptotic formula for the number of  "unbounded partitions"  of a natural number.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057340/l05734010.png" /></td> </tr></table>
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====Comments====
 
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Lambert series also occur in the expansion of Eisenstein series, a particular kind of [[modular form]]. See [[#References|[a1]]].
and the summation is over all divisors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057340/l05734011.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057340/l05734012.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057340/l05734013.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057340/l05734014.png" />, the number of divisors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057340/l05734015.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057340/l05734016.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057340/l05734017.png" />, the sum of the divisors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057340/l05734018.png" />. The behaviour of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057340/l05734019.png" /> (with suitable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057340/l05734020.png" />) as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057340/l05734021.png" /> is used, for example (see [[#References|[3]]]), in the problem of Hardy and Ramanujan on obtaining an asymptotic formula for the number of  "unbounded partitions"  of a natural number.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J.H. Lambert,   "Opera Mathematica" , '''1–2''' , O. Füssli  (1946–1948)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> G.M. Fichtenholz,   "Differential und Integralrechnung" , '''2''' , Deutsch. Verlag Wissenschaft.  (1964)</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></table>
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<table>
 
+
<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">[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 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>
 +
<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>
  
 
+
{{TEX|done}}
====Comments====
 
Lambert series also occur in the expansion of Eisenstein series, a particular kind of [[Modular form|modular form]]. See [[#References|[a1]]].
 
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.M. Apostol,  "Modular forms and Dirichlet series in analysis" , Springer  (1976)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Rademacher,  "Topics in analytic number theory" , Springer  (1973)</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>
 

Latest revision as of 09:27, 19 March 2023

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) 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)
[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=13504
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article