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The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077200/r0772001.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077200/r0772002.png" /> is the coefficient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077200/r0772003.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077200/r0772004.png" />) in the expansion of the product
+
{{TEX|done}}{{MSC|11F}}
 
 
<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/r/r077/r077200/r0772005.png" /></td> </tr></table>
 
  
 +
The function $n \mapsto \tau(n)$, where $\tau(n)$ is the coefficient of $x^n$ ($n \ge 1$) in the expansion of the product
 +
$$
 +
D(x) = x \prod_{m=1}^\infty (1 - x^m)^{24}
 +
$$
 
as a power series:
 
as a power series:
 
+
$$
<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/r/r077/r077200/r0772006.png" /></td> </tr></table>
+
D(x) = \sum_{n=1}^\infty \tau(n) x^n \ .
 
+
$$
 
If one puts
 
If one puts
 +
$$
 +
\Delta(z) = D(\exp(2\pi i z))
 +
$$
 +
then the Ramanujan function is the $n$-th Fourier coefficient of the cusp form $\Delta(z)$, which was first investigated by S. Ramanujan [[#References|[1]]]. Certain values of the Ramanujan function: $\tau(1) = 1$, $\tau(2) = -24$, $\tau(3) = 252$, $\tau(4) = -1472$, $\tau(5) = 4830$, $\tau(6) = -6048$, $\tau(7) = -16744$, $\tau(30) = 9458784518400$. Ramanujan conjectured (and L.J. Mordell proved) that the following properties of the Ramanujan function are true:
 +
$$
 +
\tau(mn) = \tau(m) \tau(n) \ \text{if}\ (m,n) = 1 \,;
 +
$$
 +
$$
 +
\tau(p^{n+1}) = \tau(p^n)\tau(p) - p^{11} \tau(p^{n-1}) \ .
 +
$$
  
<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/r/r077/r077200/r0772007.png" /></td> </tr></table>
+
Consequently, the calculation of $\tau(n)$ reduces to calculating $\tau(p)$ when $p$ is prime. It is known that $|\tau(p)|  \le p^{11/2}$ (see [[Ramanujan hypothesis|Ramanujan hypothesis]]). It is known that the Ramanujan function satisfies many congruence relations. For example, Ramanujan knew the congruence
 
+
$$
then the Ramanujan function is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077200/r0772008.png" />-th Fourier coefficient of the cusp form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077200/r0772009.png" />, which was first investigated by S. Ramanujan [[#References|[1]]]. Certain values of the Ramanujan function: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077200/r07720010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077200/r07720011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077200/r07720012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077200/r07720013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077200/r07720014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077200/r07720015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077200/r07720016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077200/r07720017.png" />. Ramanujan conjectured (and L.J. Mordell proved) that the following properties of the Ramanujan function are true:
+
\tau(p) \equiv 1 + p^{11} \pmod{691} \ .
 
+
$$
<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/r/r077/r077200/r07720018.png" /></td> </tr></table>
 
 
 
<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/r/r077/r077200/r07720019.png" /></td> </tr></table>
 
 
 
Consequently, the calculation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077200/r07720020.png" /> reduces to calculating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077200/r07720021.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077200/r07720022.png" /> is prime. It is known that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077200/r07720023.png" /> (see [[Ramanujan hypothesis|Ramanujan hypothesis]]). It is known that the Ramanujan function satisfies many congruence relations. For example, Ramanujan knew the congruence
 
 
 
<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/r/r077/r077200/r07720024.png" /></td> </tr></table>
 
  
 
Examples of congruence relations discovered later are:
 
Examples of congruence relations discovered later are:
 
+
$$
<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/r/r077/r077200/r07720025.png" /></td> </tr></table>
+
\tau(n) \equiv \sigma_{11}(n) \pmod{2^{11}} \ \text{if}\ n \equiv 1 \pmod 8
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077200/r07720026.png" /></td> </tr></table>
+
$$
 
+
\tau(p) \equiv p + p^10 \pmod{25}
 +
$$
 
etc.
 
etc.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Ramanujan,  "On certain arithmetical functions"  ''Trans. Cambridge Philos. Soc.'' , '''22'''  (1916)  pp. 159–184</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.-P. Serre,  "Une interpretation des congruences relatives à la function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077200/r07720027.png" /> de Ramanujan"  ''Sém. Delange–Pisot–Poitou (Théorie des nombres)'' , '''9''' :  14  (1967/68)  pp. 1–17</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  O.M. Fomenko,  "Applications of the theory of modular forms to number theory"  ''J. Soviet Math.'' , '''14''' :  4  (1980)  pp. 1307–1362  ''Itogi Nauk. i Tekhn. Algebra Topol. Geom.'' , '''15'''  (1977)  pp. 5–91</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  S. Ramanujan,  "On certain arithmetical functions"  ''Trans. Cambridge Philos. Soc.'' , '''22'''  (1916)  pp. 159–184</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  J.-P. Serre,  "Une interpretation des congruences relatives à la function $\tau$ de Ramanujan"  ''Sém. Delange–Pisot–Poitou (Théorie des nombres)'' , '''9''' :  14  (1967/68)  pp. 1–17</TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top">  O.M. Fomenko,  "Applications of the theory of modular forms to number theory"  ''J. Soviet Math.'' , '''14''' :  4  (1980)  pp. 1307–1362  ''Itogi Nauk. i Tekhn. Algebra Topol. Geom.'' , '''15'''  (1977)  pp. 5–91</TD></TR>
 +
</table>
  
  
  
 
====Comments====
 
====Comments====
It is still (1990) not known whether there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077200/r07720028.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077200/r07720029.png" />. One believes that the answer is  "no" . For an elementary introduction to the background of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077200/r07720030.png" />, see [[#References|[a1]]].
+
It is still (1990) not known whether there exists an $n \in \mathbb{N}$ such that $\tau(n) = 0$. One believes that the answer is  "no" . For an elementary introduction to the background of $\Delta(z)$, see [[#References|[a1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.M. Apostol,  "Modular functions and Dirichlet series in number theory" , Springer  (1976)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.M. Apostol,  "Modular functions and Dirichlet series in number theory" , Springer  (1976)</TD></TR></table>

Revision as of 21:38, 27 November 2014

2020 Mathematics Subject Classification: Primary: 11F [MSN][ZBL]

The function $n \mapsto \tau(n)$, where $\tau(n)$ is the coefficient of $x^n$ ($n \ge 1$) in the expansion of the product $$ D(x) = x \prod_{m=1}^\infty (1 - x^m)^{24} $$ as a power series: $$ D(x) = \sum_{n=1}^\infty \tau(n) x^n \ . $$ If one puts $$ \Delta(z) = D(\exp(2\pi i z)) $$ then the Ramanujan function is the $n$-th Fourier coefficient of the cusp form $\Delta(z)$, which was first investigated by S. Ramanujan [1]. Certain values of the Ramanujan function: $\tau(1) = 1$, $\tau(2) = -24$, $\tau(3) = 252$, $\tau(4) = -1472$, $\tau(5) = 4830$, $\tau(6) = -6048$, $\tau(7) = -16744$, $\tau(30) = 9458784518400$. Ramanujan conjectured (and L.J. Mordell proved) that the following properties of the Ramanujan function are true: $$ \tau(mn) = \tau(m) \tau(n) \ \text{if}\ (m,n) = 1 \,; $$ $$ \tau(p^{n+1}) = \tau(p^n)\tau(p) - p^{11} \tau(p^{n-1}) \ . $$

Consequently, the calculation of $\tau(n)$ reduces to calculating $\tau(p)$ when $p$ is prime. It is known that $|\tau(p)| \le p^{11/2}$ (see Ramanujan hypothesis). It is known that the Ramanujan function satisfies many congruence relations. For example, Ramanujan knew the congruence $$ \tau(p) \equiv 1 + p^{11} \pmod{691} \ . $$

Examples of congruence relations discovered later are: $$ \tau(n) \equiv \sigma_{11}(n) \pmod{2^{11}} \ \text{if}\ n \equiv 1 \pmod 8 $$ $$ \tau(p) \equiv p + p^10 \pmod{25} $$ etc.

References

[1] S. Ramanujan, "On certain arithmetical functions" Trans. Cambridge Philos. Soc. , 22 (1916) pp. 159–184
[2] J.-P. Serre, "Une interpretation des congruences relatives à la function $\tau$ de Ramanujan" Sém. Delange–Pisot–Poitou (Théorie des nombres) , 9 : 14 (1967/68) pp. 1–17
[3] O.M. Fomenko, "Applications of the theory of modular forms to number theory" J. Soviet Math. , 14 : 4 (1980) pp. 1307–1362 Itogi Nauk. i Tekhn. Algebra Topol. Geom. , 15 (1977) pp. 5–91


Comments

It is still (1990) not known whether there exists an $n \in \mathbb{N}$ such that $\tau(n) = 0$. One believes that the answer is "no" . For an elementary introduction to the background of $\Delta(z)$, see [a1].

References

[a1] T.M. Apostol, "Modular functions and Dirichlet series in number theory" , Springer (1976)
How to Cite This Entry:
Ramanujan function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ramanujan_function&oldid=35020
This article was adapted from an original article by K.Yu. Bulota (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article