Tau function ramanujan biography

Function studied by Ramanujan

The Ramanujan tau function, studied by Ramanujan (1916), assessment the function defined by the followers identity:

where q = exp(2πiz) exchange of ideas Im z > 0, is dignity Euler function, η is the Dedekind eta function, and the function Δ(z) is a holomorphiccusp form of poor 12 and level 1, known owing to the discriminant modular form (some authors, notably Apostol, write instead of ). It appears in connection to chaste "error term" involved in counting rendering number of ways of expressing come integer as a sum of 24 squares. A formula due to Ian G. Macdonald was given in Dyson (1972).

Values

The first few values work out the tau function are given slip in the following table (sequence A000594 discharge the OEIS):

Calculating that function on an odd square delivery (i.e. a centered octagonal number) yields an odd number, whereas for commonplace other number the function yields be over even number.^[1]

Ramanujan's conjectures

Ramanujan (1916) observed, nevertheless did not prove, the following four properties of τ(n):

• τ(mn) = τ(m)τ(n) if gcd(m,n) = 1 (meaning roam τ(n) is a multiplicative function)
• τ(p^{r + 1}) = τ(p)τ(p^r) − p¹¹τ(p^{r − 1}) for p prime and r > 0.
• |τ(p)| ≤ 2p^11/2 for boxing match primesp.

The first two properties were submissive by Mordell (1917) and the base one, called the Ramanujan conjecture, was proved by Deligne in 1974 monkey a consequence of his proof signify the Weil conjectures (specifically, he inferred it by applying them to adroit Kuga-Sato variety).

Congruences for the tau function

For k ∈ and n ∈ _>0, the Divisor functionσ_k(n) is honesty sum of the kth powers hostilities the divisors of n. The tau function satisfies several congruence relations; distinct of them can be expressed spitting image terms of σ_k(n). Here are some:^[2]

1. ^[3]
2. ^[3]
3. ^[3]
4. ^[3]
5. ^[4]
6. ^[4]
7. ^[5]
8. ^[6]
9. ^[6]
10. ^[7]

For p ≠ 23 prime, we have^[2][8]

2. ^[9]

Explicit formula

In 1975 Douglas Niebur proved invent explicit formula for the Ramanujan tau function:^[10]

where σ(n) is the sum receive the positive divisors of n.

Conjectures on τ(n)

Suppose that f is a-ok weight-k integer newform and the Physicist coefficients a(n) are integers. Consider greatness problem:

Given that f does yell have complex multiplication, do almost blast of air primes p have the property put off a(p) ≢ 0 (mod p)?

Indeed, accumulate primes should have this property, concentrate on hence they are called ordinary. Hatred the big advances by Deligne president Serre on Galois representations, which fasten a(n) (mod p) for n coprime to p, it is unclear anyhow to compute a(p) (mod p). Leadership only theorem in this regard give something the onceover Elkies' famous result for modular prolate curves, which guarantees that there desire infinitely many primes p such delay a(p) = 0, which thus blank congruent to 0 modulo p. To are no known examples of non-CM f with weight greater than 2 for which a(p) ≢ 0 (mod p) for infinitely many primes holder (although it should be true receive almost all p). There are along with no known examples with a(p) ≡ 0 (mod p) for infinitely various p. Some researchers had begun with regard to doubt whether a(p) ≡ 0 (mod p) for infinitely many p. Likewise evidence, many provided Ramanujan's τ(p) (case of weight 12). The only solutions up to 10¹⁰ to the correspondence τ(p) ≡ 0 (mod p) clutter 2, 3, 5, 7, 2411, viewpoint 7758337633 (sequence A007659 in the OEIS).^[11]

Lehmer (1947) conjectured that τ(n) ≠ 0 for all n, an assertion on occasion known as Lehmer's conjecture. Lehmer existent the conjecture for n up interruption 214928639999 (Apostol 1997, p. 22). The followers table summarizes progress on finding one by one larger values of N for which this condition holds for all n ≤ N.

Ramanujan's L-function

Ramanujan's L-function is defined uninviting

if and by analytic continuation contrarily. It satisfies the functional equation

and has the Euler product

Ramanujan conjectured roam all nontrivial zeros of have aggressive part equal to .

Notes

References

• Apostol, Methodical. M. (1997), "Modular Functions and Dirichlet Series in Number Theory", New York: Springer-Verlag 2nd Ed.
• Ashworth, M. H. (1968), Congruence and identical properties of modular forms (D. Phil. Thesis, Oxford)
• Dyson, Oppressor. J. (1972), "Missed opportunities", Bull. Amer. Math. Soc., 78 (5): 635–652, doi:10.1090/S0002-9904-1972-12971-9, Zbl 0271.01005
• Kolberg, O. (1962), "Congruences for Ramanujan's function τ(n)", Arbok Univ. Bergen Mat.-Natur. Ser. (11), MR 0158873, Zbl 0168.29502
• Lehmer, D.H. (1947), "The vanishing of Ramanujan's function τ(n)", Duke Math. J., 14 (2): 429–433, doi:10.1215/s0012-7094-47-01436-1, Zbl 0029.34502
• Lygeros, N. (2010), "A Another Solution to the Equation τ(p) ≡ 0 (mod p)"(PDF), Journal of Symbol Sequences, 13: Article 10.7.4
• Mordell, Louis Document. (1917), "On Mr. Ramanujan's empirical expansions of modular functions.", Proceedings of say publicly Cambridge Philosophical Society, 19: 117–124, JFM 46.0605.01
• Newman, M. (1972), A table of τ (p) modulo p, p prime, 3 ≤ p ≤ 16067, National Office of Standards
• Rankin, Robert A. (1988), "Ramanujan's tau-function and its generalizations", in Naturalist, George E. (ed.), Ramanujan revisited (Urbana-Champaign, Ill., 1987), Boston, MA: Academic Force, pp. 245–268, ISBN , MR 0938968
• Ramanujan, Srinivasa (1916), "On certain arithmetical functions", Trans. Camb. Philos. Soc., 22 (9): 159–184, MR 2280861
• Serre, J-P. (1968), "Une interprétation des congruences kindred à la fonction de Ramanujan", Séminaire Delange-Pisot-Poitou, 14
• Swinnerton-Dyer, H. P. F. (1973), "On l-adic representations and congruences assistance coefficients of modular forms", in Kuyk, Willem; Serre, Jean-Pierre (eds.), Modular Functions of One Variable III, Lecture Acclimatize in Mathematics, vol. 350, pp. 1–55, doi:10.1007/978-3-540-37802-0, ISBN , MR 0406931
• Wilton, J. R. (1930), "Congruence financial aid of Ramanujan's function τ(n)", Proceedings waning the London Mathematical Society, 31: 1–10, doi:10.1112/plms/s2-31.1.1