Ramanujan tau function

The Ramanujan tau function, studied by Ramanujan (1916), is the function τ : N → Z {displaystyle au :mathbb {N} o mathbb {Z} } defined by the following identity: The Ramanujan tau function, studied by Ramanujan (1916), is the function τ : N → Z {displaystyle au :mathbb {N} o mathbb {Z} } defined by the following identity: where q = exp ⁡ ( 2 π i z ) {displaystyle q=exp(2pi iz)} with ℑ z > 0 {displaystyle Im z>0} and η {displaystyle eta } is the Dedekind eta function and the function Δ ( z ) {displaystyle Delta (z)} is a holomorphic cusp form of weight 12 and level 1, known as the discriminant modular form. 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. A formula due to Ian G. Macdonald was given in Dyson (1972). The first few values of the tau function are given in the following table (sequence A000594 in the OEIS): Ramanujan (1916) observed, but did not prove, the following three properties of τ ( n ) {displaystyle au (n)} : The first two properties were proved by Mordell (1917) and the third one, called the Ramanujan conjecture, was proved by Deligne in 1974 as a consequence of his proof of the Weil conjectures (specifically, he deduced it by applying them to a Kuga-Sato variety). For k ∈ Z and n ∈ Z>0, define σk(n) as the sum of the k-th 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: For p ≠ 23 prime, we have Suppose that f {displaystyle f} is a weight k {displaystyle k} integer newform and the Fourier coefficients a ( n ) {displaystyle a(n)} are integers. Consider the problem: If f {displaystyle f} does not have complex multiplication, prove that almost all primes p {displaystyle p} have the property that a ( p ) ≠ 0 mod p {displaystyle a(p) eq 0{mod {p}}} . Indeed, most primes should have this property, and hence they are called ordinary. Despite the big advances by Deligne and Serre on Galois representations, which determine a ( n ) mod p {displaystyle a(n){mod {p}}} for n {displaystyle n} coprime to p {displaystyle p} , we do not have any clue as to how to compute a ( p ) mod p {displaystyle a(p){mod {p}}} . The only theorem in this regard is Elkies' famous result for modular elliptic curves, which indeed guarantees that there are infinitely many primes p {displaystyle p} for which a ( p ) = 0 {displaystyle a(p)=0} , which in turn is obviously 0 mod p {displaystyle 0{mod {p}}} . We do not know any examples of non-CM f {displaystyle f} with weight > 2 {displaystyle >2} for which a ( p ) ≠ 0 {displaystyle a(p) eq 0} mod p {displaystyle p} for infinitely many primes p {displaystyle p} (although it should be true for almost all p {displaystyle p} ). We also do not know any examples where a ( p ) = 0 {displaystyle a(p)=0} mod p {displaystyle p} for infinitely many p {displaystyle p} . Some people had begun to doubt whether a ( p ) = 0 mod p {displaystyle a(p)=0{mod {p}}} indeed for infinitely many p {displaystyle p} . As evidence, many provided Ramanujan's τ ( p ) {displaystyle au (p)} (case of weight 12 {displaystyle 12} ). The largest known p {displaystyle p} for which τ ( p ) = 0 mod p {displaystyle au (p)=0{mod {p}}} is p = 7758337633 {displaystyle p=7758337633} . The only solutions to the equation τ ( p ) ≡ 0 mod p {displaystyle au (p)equiv 0{mod {p}}} are p = 2 , 3 , 5 , 7 , 2411 , {displaystyle p=2,3,5,7,2411,} and 7758337633 {displaystyle 7758337633} up to 10 10 {displaystyle 10^{10}} . Lehmer (1947) conjectured that τ ( n ) ≠ 0 {displaystyle au (n) eq 0} for all n {displaystyle n} , an assertion sometimes known as Lehmer's conjecture. Lehmer verified the conjecture for n < 214928639999 {displaystyle n<214928639999} (Apostol 1997, p. 22). The following table summarizes progress on finding successively larger values of N {displaystyle N} for which this condition holds for all n ≤ N {displaystyle nleq N} .