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''Ramanujan conjecture''
 
''Ramanujan conjecture''
  
The conjecture, stated by S. Ramanujan [[#References|[1]]], that the Fourier coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077210/r0772101.png" /> of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077210/r0772102.png" /> (a cusp form of weight 12) satisfy the inequality
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The conjecture, stated by S. Ramanujan [[#References|[1]]], that the Fourier coefficients $\tau(n)$ of the function $\Delta$ (a [[cusp form]] of weight 12) satisfy the inequality
 
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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/r/r077/r077210/r0772103.png" /></td> </tr></table>
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| \tau(p) | \le 2 p^{11/2}\ \ \ \text{for}\,p\,\text{prime.}
 
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$$
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077210/r0772104.png" /> is also called the [[Ramanujan function|Ramanujan function]]. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077210/r0772105.png" /> is an eigen function of the Hecke operator, and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077210/r0772106.png" /> are the corresponding eigen values. H. Petersson generalized Ramanujan's hypothesis to the case of eigen values of the Hecke operators on modular forms of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077210/r0772107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077210/r0772108.png" /> an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077210/r0772109.png" /> (the Petersson conjecture). P. Deligne (see [[#References|[2]]]) reduced the Petersson conjecture to the Weil conjectures (cf. [[Zeta-function|Zeta-function]]), then proved the latter (1974). This also proved Ramanujan's hypothesis.
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$\tau(n)$ is also called the [[Ramanujan function]]. The function $\Delta$ is an eigen function of the [[Hecke operator]]s, and the $\tau(n)$ are the corresponding eigen values. H. Petersson generalized Ramanujan's hypothesis to the case of eigen values of the Hecke operators on [[modular form]]s of weight $k$, $k \ge 2$ an integer (the Petersson conjecture). P. Deligne (see [[#References|[2]]]) reduced the Petersson conjecture to the Weil conjectures (cf. [[Zeta-function]]), then proved the latter (1974). This also proved Ramanujan's hypothesis.
  
 
====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">  P. Deligne,  "La conjecture de Weil 1"  ''Publ. Math. IHES'' , '''43'''  (1974)  pp. 273–307</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>
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<table>
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<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>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  P. Deligne,  "La conjecture de Weil 1"  ''Publ. Math. IHES'' , '''43'''  (1974)  pp. 273–307</TD></TR>
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<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>
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</table>
  
  
  
 
====Comments====
 
====Comments====
See also [[Congruence equation|Congruence equation]].
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See also [[Congruence equation]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.M. Katz,  "An overview of Deligne's proof of the Riemann hypothesis for varieties over finite fields"  F.E. Browder (ed.) , ''Mathematical developments arising from Hilbert problems'' , ''Proc. Symp. Pure Math.'' , '''28''' , Amer. Math. Soc.  (1976)  pp. 275–305</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  N.M. Katz,  "An overview of Deligne's proof of the Riemann hypothesis for varieties over finite fields"  F.E. Browder (ed.) , ''Mathematical developments arising from Hilbert problems'' , ''Proc. Symp. Pure Math.'' , '''28''' , Amer. Math. Soc.  (1976)  pp. 275–305</TD></TR>
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</table>
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Revision as of 16:13, 11 October 2016

Ramanujan conjecture

The conjecture, stated by S. Ramanujan [1], that the Fourier coefficients $\tau(n)$ of the function $\Delta$ (a cusp form of weight 12) satisfy the inequality $$ | \tau(p) | \le 2 p^{11/2}\ \ \ \text{for}\,p\,\text{prime.} $$ $\tau(n)$ is also called the Ramanujan function. The function $\Delta$ is an eigen function of the Hecke operators, and the $\tau(n)$ are the corresponding eigen values. H. Petersson generalized Ramanujan's hypothesis to the case of eigen values of the Hecke operators on modular forms of weight $k$, $k \ge 2$ an integer (the Petersson conjecture). P. Deligne (see [2]) reduced the Petersson conjecture to the Weil conjectures (cf. Zeta-function), then proved the latter (1974). This also proved Ramanujan's hypothesis.

References

[1] S. Ramanujan, "On certain arithmetical functions" Trans. Cambridge Philos. Soc. , 22 (1916) pp. 159–184
[2] P. Deligne, "La conjecture de Weil 1" Publ. Math. IHES , 43 (1974) pp. 273–307
[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

See also Congruence equation.

References

[a1] N.M. Katz, "An overview of Deligne's proof of the Riemann hypothesis for varieties over finite fields" F.E. Browder (ed.) , Mathematical developments arising from Hilbert problems , Proc. Symp. Pure Math. , 28 , Amer. Math. Soc. (1976) pp. 275–305
How to Cite This Entry:
Ramanujan hypothesis. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ramanujan_hypothesis&oldid=39409
This article was adapted from an original article by K.Yu. Bulota (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article