Ramanujan hypothesis

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.

Comments

See also Congruence equation.

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
[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