View source for Ramanujan hypothesis

''Ramanujan conjecture''

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
$$
| \tau(p) | \le 2 p^{11/2}\ \ \ \text{for}\,p\,\text{prime.}
$$
$\tau(n)$ is also called the [[Ramanujan function]]. The function $\Delta$ is an eigen function of the [[Hecke 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.

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

====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>
<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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