Ramanujan tau function

It appears in connection to an "error term" involved in counting the number of ways of expressing an integer as a sum of 24 squares.

>0, the Divisor function σk(n) is the sum of the kth powers of the divisors of n. The tau function satisfies several congruence relations; many of them can be expressed in terms of σk(n).

Here are some:[2] For p ≠ 23 prime, we have[2][8] In 1975 Douglas Niebur proved an explicit formula for the Ramanujan tau function:[10] where σ(n) is the sum of the positive divisors of n. Suppose that f is a weight-k integer newform and the Fourier coefficients a(n) are integers.

The only theorem in this regard is Elkies' famous result for modular elliptic curves, which guarantees that there are infinitely many primes p such that a(p) = 0, which thus are congruent to 0 modulo p. There are no known examples of non-CM f with weight greater than 2 for which a(p) ≢ 0 (mod p) for infinitely many primes p (although it should be true for almost all p).

There are also no known examples with a(p) ≡ 0 (mod p) for infinitely many p. Some researchers had begun to doubt whether a(p) ≡ 0 (mod p) for infinitely many p. As evidence, many provided Ramanujan's τ(p) (case of weight 12).

The only solutions up to 1010 to the equation τ(p) ≡ 0 (mod p) are 2, 3, 5, 7, 2411, and 7758337633 (sequence A007659 in the OEIS).

The following table summarizes progress on finding successively larger values of N for which this condition holds for all n ≤ N. Ramanujan's L-function is defined by if

It satisfies the functional equation and has the Euler product Ramanujan conjectured that all nontrivial zeros of