Tau Conjecture
The tau conjecture, also known as Ramanujan's hypothesis after its proposer, states that
where 
is the tau function. This was proven by Deligne (1974) in the course of proving the more general Petersson conjecture. Deligne was awarded the Fields medal for his proof.
REFERENCES:
Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 136 and 140, 1997.
Deligne, P. "La conjecture de Weil. I." Inst. Hautes Études Sci. Publ. Math. 43, 273-307, 1974.
Deligne, P. "La conjecture de Weil. II." Inst. Hautes Études Sci. Publ. Math. 52, 137-252, 1980.
Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 169, 1999.
