Tau Conjecture

Also known as Ramanujan's Hypothesis. Ramanujan proposed that

$$\tau(n)\sim{\mathcal O}(n^{11/2+\epsilon}),$$

where $\tau(n)$ is the Tau Function, defined by

$$\sum_{n=1}^\infty \tau(n)x^n = x(1-3x+5x^3-7x^6+\ldots)^8.$$

This was proven by Deligne (1974), who was subsequently awarded the Fields Medal for his proof.

See also Tau Function

References

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, 1959.