View source for Ramanujan hypothesis

''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

<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>

<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.

====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>



====Comments====
See also [[Congruence equation|Congruence equation]].

====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>