In mathematics, the Ramanujan tau function, studied by Srinivasa Ramanujan, is the function defined by
where is the Euler function, is the Dedekind eta function, is the modular discriminant, and with .
Values The first few values of the tau function are given in the following table :
12345678910111213141516 1−24252−14724830−6048−1674484480−113643−115920534612−370944−5777384018561217160987136
Calculating this function on an odd square number yields an odd number, whereas for any other number the function yields an even number.
Main properties Ramanujan conjectured two properties of , which can be rephrased equivalently as: if and are coprime (that is, is a multiplicative function) for prime and . Together, these two properties are equivalent to the identity
They were proved by Louis Mordell using what is now understood as the theory of Hecke operators. Ramanujan also conjectured the third propertyfor all primes , which is called the Ramanujan conjecture. Assuming the first two properties, Ramanujan noted that his conjecture is equivalent to the inequality
for all , where is the number-of-divisors function. Note that the weaker estimate is much easier to show. Ramanujan's conjecture was proved by Pierre Deligne in 1974 as a consequence of his proof of the Weil conjectures.
Ramanujan's L-function Because the modular discriminant is a cusp form of weight 12, it gives rise to an -function, called Ramanujan's -function. It is defined for via the Dirichlet series
and due to the first two properties of above, it has an Euler product
valid for . It satisfies the functional equation
which enables the analytic continuation of to all complex numbers , making it an entire function. Ramanujan conjectured that all nontrivial zeros of have real part equal to . This is a special case of the Grand Riemann hypothesis. Robert Rankin showed that has no zeros on the line .
Congruences for the tau function For and , the divisor function is the sum of the th powers of the divisors of . The tau function satisfies several congruence relations; many of them can be expressed in terms of . Here are some: Due to
Due to
Due to Lahivi Due to D. H. Lehmer
Due to
For primes , we have <li> <li> if is of the form Due to J.-P. Serre 1968, Section 4.5 <li>
Explicit formulas In 1972, Ian G. Macdonald proved an explicit formula for the Ramanujan tau functionIn 1975, Douglas Niebur proved the formula
where is the sum-of-divisor function.
Conjectures on the tau function Suppose that is a weight- integer newform whose Fourier coefficients are integers. Consider the problem: Given that does not have complex multiplication, do almost all primes have the property that ? 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 for coprime to , it is unclear how to compute . The only theorem in this regard is Elkies' famous result for modular elliptic curves, which guarantees that there are infinitely many primes such that , which thus are congruent to 0 modulo . There are no known examples of non-CM with weight greater than 2 for which for infinitely many primes (although it should be true for almost all . There are also no known examples with for infinitely many . Some researchers had begun to doubt whether for infinitely many . As evidence, many provided Ramanujan's (case of weight 12). The only solutions up to to the equation are 2, 3, 5, 7, 2411, and .
conjectured that for all , an assertion sometimes known as Lehmer's conjecture. Lehmer verified the conjecture for up to (Apostol 1997, p. 22). The following table summarizes progress on finding successively larger values of for which this condition holds for all . reference Lehmer (1947) Lehmer (1949) Serre (1973, p. 98), Serre (1985) Jennings (1993) Jordan and Kelly (1999) Bosman (2007) Zeng and Yin (2013) Derickx, van Hoeij, and Zeng (2013)
Notes
References