Ramanujan tau function

The Ramanujan tau function, studied by Ramanujan (1916), is the function ${\displaystyle \tau :\mathbb {N} \to \mathbb {Z} }${\displaystyle \tau :\mathbb {N} \to \mathbb {Z} } defined by the following identity:

${\displaystyle \sum _{n\geq 1}\tau (n)q^{n}=q\prod _{n\geq 1}\left(1-q^{n}\right)^{24}=q\phi (q)^{24}=\eta (z)^{24}=\Delta (z),}${\displaystyle \sum _{n\geq 1}\tau (n)q^{n}=q\prod _{n\geq 1}\left(1-q^{n}\right)^{24}=q\phi (q)^{24}=\eta (z)^{24}=\Delta (z),}

where ${\displaystyle q=\exp(2\pi iz)}${\displaystyle q=\exp(2\pi iz)} with ${\displaystyle \mathrm {Im} (z)>0}${\displaystyle \mathrm {Im} (z)>0}, ${\displaystyle \phi }${\displaystyle \phi } is the Euler function, ${\displaystyle \eta }${\displaystyle \eta } is the Dedekind eta function, and the function ${\displaystyle \Delta (z)}${\displaystyle \Delta (z)} is a holomorphic cusp form of weight 12 and level 1, known as the discriminant modular form (some authors, notably Apostol, write ${\displaystyle \Delta /(2\pi )^{12}}${\displaystyle \Delta /(2\pi )^{12}} instead of ${\displaystyle \Delta }${\displaystyle \Delta }). 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):

Calculating this function on an odd square number (i.e. a centered octagonal number) yields an odd number, whereas for any other number the function yields an even number.^[1]

Ramanujan (1916) observed, but did not prove, the following three properties of ${\displaystyle \tau (n)}${\displaystyle \tau (n)}:

• ${\displaystyle \tau (mn)=\tau (m)\tau (n)}${\displaystyle \tau (mn)=\tau (m)\tau (n)} if ${\displaystyle \gcd(m,n)=1}${\displaystyle \gcd(m,n)=1} (meaning that ${\displaystyle \tau (n)}${\displaystyle \tau (n)} is a multiplicative function)
• ${\displaystyle \tau (p^{r+1})=\tau (p)\tau (p^{r})-p^{11}\tau (p^{r-1})}${\displaystyle \tau (p^{r+1})=\tau (p)\tau (p^{r})-p^{11}\tau (p^{r-1})} for ${\displaystyle p}${\displaystyle p} prime and ${\displaystyle r>0}${\displaystyle r>0}.
• ${\displaystyle |\tau (p)|\leq 2p^{11/2}}${\displaystyle |\tau (p)|\leq 2p^{11/2}} for all primes ${\displaystyle p}${\displaystyle p}.

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 ${\displaystyle k\in \mathbb {Z} }${\displaystyle k\in \mathbb {Z} } and ${\displaystyle n\in \mathbb {N} }${\displaystyle n\in \mathbb {N} }, the Divisor function ${\displaystyle \sigma _{k}(n)}${\displaystyle \sigma _{k}(n)} is the sum of the ${\displaystyle k}${\displaystyle k}th powers of the divisors of ${\displaystyle n}${\displaystyle n}. The tau function satisfies several congruence relations; many of them can be expressed in terms of ${\displaystyle \sigma _{k}(n)}${\displaystyle \sigma _{k}(n)}. Here are some:^[2]

1. ${\displaystyle \tau (n)\equiv \sigma _{11}(n){\pmod {2^{11}}}{\text{ for }}n\equiv 1{\pmod {8}}}${\displaystyle \tau (n)\equiv \sigma _{11}(n){\pmod {2^{11}}}{\text{ for }}n\equiv 1{\pmod {8}}}^[3]
2. ${\displaystyle \tau (n)\equiv 1217\sigma _{11}(n){\pmod {2^{13}}}{\text{ for }}n\equiv 3{\pmod {8}}}${\displaystyle \tau (n)\equiv 1217\sigma _{11}(n){\pmod {2^{13}}}{\text{ for }}n\equiv 3{\pmod {8}}}^[3]
3. ${\displaystyle \tau (n)\equiv 1537\sigma _{11}(n){\pmod {2^{12}}}{\text{ for }}n\equiv 5{\pmod {8}}}${\displaystyle \tau (n)\equiv 1537\sigma _{11}(n){\pmod {2^{12}}}{\text{ for }}n\equiv 5{\pmod {8}}}^[3]
4. ${\displaystyle \tau (n)\equiv 705\sigma _{11}(n){\pmod {2^{14}}}{\text{ for }}n\equiv 7{\pmod {8}}}${\displaystyle \tau (n)\equiv 705\sigma _{11}(n){\pmod {2^{14}}}{\text{ for }}n\equiv 7{\pmod {8}}}^[3]
5. ${\displaystyle \tau (n)\equiv n^{-610}\sigma _{1231}(n){\pmod {3^{6}}}{\text{ for }}n\equiv 1{\pmod {3}}}${\displaystyle \tau (n)\equiv n^{-610}\sigma _{1231}(n){\pmod {3^{6}}}{\text{ for }}n\equiv 1{\pmod {3}}}^[4]
6. ${\displaystyle \tau (n)\equiv n^{-610}\sigma _{1231}(n){\pmod {3^{7}}}{\text{ for }}n\equiv 2{\pmod {3}}}${\displaystyle \tau (n)\equiv n^{-610}\sigma _{1231}(n){\pmod {3^{7}}}{\text{ for }}n\equiv 2{\pmod {3}}}^[4]
7. ${\displaystyle \tau (n)\equiv n^{-30}\sigma _{71}(n){\pmod {5^{3}}}{\text{ for }}n\not \equiv 0{\pmod {5}}}${\displaystyle \tau (n)\equiv n^{-30}\sigma _{71}(n){\pmod {5^{3}}}{\text{ for }}n\not \equiv 0{\pmod {5}}}^[5]
8. ${\displaystyle \tau (n)\equiv n\sigma _{9}(n){\pmod {7}}}${\displaystyle \tau (n)\equiv n\sigma _{9}(n){\pmod {7}}}^[6]
9. ${\displaystyle \tau (n)\equiv n\sigma _{9}(n){\pmod {7^{2}}}{\text{ for }}n\equiv 3,5,6{\pmod {7}}}${\displaystyle \tau (n)\equiv n\sigma _{9}(n){\pmod {7^{2}}}{\text{ for }}n\equiv 3,5,6{\pmod {7}}}^[6]
10. ${\displaystyle \tau (n)\equiv \sigma _{11}(n){\pmod {691}}.}${\displaystyle \tau (n)\equiv \sigma _{11}(n){\pmod {691}}.}^[7]

For ${\displaystyle p\neq 23}${\displaystyle p\neq 23} prime, we have^{[2] [8]}

11. ${\displaystyle \tau (p)\equiv 0{\pmod {23}}{\text{ if }}\left({\frac {p}{23}}\right)=-1}${\displaystyle \tau (p)\equiv 0{\pmod {23}}{\text{ if }}\left({\frac {p}{23}}\right)=-1}
12. ${\displaystyle \tau (p)\equiv \sigma _{11}(p){\pmod {23^{2}}}{\text{ if }}p{\text{ is of the form }}a^{2}+23b^{2}}${\displaystyle \tau (p)\equiv \sigma _{11}(p){\pmod {23^{2}}}{\text{ if }}p{\text{ is of the form }}a^{2}+23b^{2}}^[9]
13. ${\displaystyle \tau (p)\equiv -1{\pmod {23}}{\text{ otherwise}}.}${\displaystyle \tau (p)\equiv -1{\pmod {23}}{\text{ otherwise}}.}

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

${\displaystyle \tau (n)=n^{4}\sigma (n)-24\sum _{i=1}^{n-1}i^{2}(35i^{2}-52in+18n^{2})\sigma (i)\sigma (n-i).}${\displaystyle \tau (n)=n^{4}\sigma (n)-24\sum _{i=1}^{n-1}i^{2}(35i^{2}-52in+18n^{2})\sigma (i)\sigma (n-i).}

where ${\displaystyle \sigma (n)}${\displaystyle \sigma (n)} is the sum of the positive divisors of ${\displaystyle n}${\displaystyle n}.

Suppose that ${\displaystyle f}${\displaystyle f} is a weight-${\displaystyle k}${\displaystyle k} integer newform and the Fourier coefficients ${\displaystyle a(n)}${\displaystyle a(n)} are integers. Consider the problem:

Given that ${\displaystyle f}${\displaystyle f} does not have complex multiplication, do almost all primes ${\displaystyle p}${\displaystyle p} have the property that ${\displaystyle a(p)\not \equiv 0{\pmod {p}}}${\displaystyle a(p)\not \equiv 0{\pmod {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 ${\displaystyle a(n){\pmod {p}}}${\displaystyle a(n){\pmod {p}}} for ${\displaystyle n}${\displaystyle n} coprime to ${\displaystyle p}${\displaystyle p}, it is unclear how to compute ${\displaystyle a(p){\pmod {p}}}${\displaystyle a(p){\pmod {p}}}. The only theorem in this regard is Elkies' famous result for modular elliptic curves, which guarantees that there are infinitely many primes ${\displaystyle p}${\displaystyle p} such that ${\displaystyle a(p)=0}${\displaystyle a(p)=0}, which thus are congruent to 0 modulo ${\displaystyle p}${\displaystyle p}. There are no known examples of non-CM ${\displaystyle f}${\displaystyle f} with weight greater than 2 for which ${\displaystyle a(p)\not \equiv 0{\pmod {p}}}${\displaystyle a(p)\not \equiv 0{\pmod {p}}} for infinitely many primes ${\displaystyle p}${\displaystyle p} (although it should be true for almost all ${\displaystyle p}${\displaystyle p}. There are also no known examples with ${\displaystyle a(p)\equiv 0{\pmod {p}}}${\displaystyle a(p)\equiv 0{\pmod {p}}} for infinitely many ${\displaystyle p}${\displaystyle p}. Some researchers had begun to doubt whether ${\displaystyle a(p)\equiv 0{\pmod {p}}}${\displaystyle a(p)\equiv 0{\pmod {p}}} for infinitely many ${\displaystyle p}${\displaystyle p}. As evidence, many provided Ramanujan's ${\displaystyle \tau (p)}${\displaystyle \tau (p)} (case of weight 12). The only solutions up to ${\displaystyle 10^{10}}${\displaystyle 10^{10}} to the equation ${\displaystyle \tau (p)\equiv 0{\pmod {p}}}${\displaystyle \tau (p)\equiv 0{\pmod {p}}} are 2, 3, 5, 7, 2411, and 7758337633 (sequence A007659 in the OEIS).^[11]

Lehmer (1947) conjectured that ${\displaystyle \tau (n)\neq 0}${\displaystyle \tau (n)\neq 0} for all ${\displaystyle n}${\displaystyle n}, an assertion sometimes known as Lehmer's conjecture. Lehmer verified the conjecture for ${\displaystyle n}${\displaystyle n} up to 214928639999 (Apostol 1997, p. 22). The following table summarizes progress on finding successively larger values of ${\displaystyle N}${\displaystyle N} for which this condition holds for all ${\displaystyle n\leq N}${\displaystyle n\leq N}.

Ramanujan's ${\displaystyle L}${\displaystyle L}-function is defined by

${\displaystyle L(s)=\sum _{n\geq 1}{\frac {\tau (n)}{n^{s}}}}${\displaystyle L(s)=\sum _{n\geq 1}{\frac {\tau (n)}{n^{s}}}}

if ${\displaystyle \mathrm {Re} (s)>6}${\displaystyle \mathrm {Re} (s)>6} and by analytic continuation otherwise. It satisfies the functional equation

${\displaystyle {\frac {L(s)\Gamma (s)}{(2\pi )^{s}}}={\frac {L(12-s)\Gamma (12-s)}{(2\pi )^{12-s}}},\quad s\notin \mathbb {Z} _{0}^{-},,12円-s\notin \mathbb {Z} _{0}^{-}}${\displaystyle {\frac {L(s)\Gamma (s)}{(2\pi )^{s}}}={\frac {L(12-s)\Gamma (12-s)}{(2\pi )^{12-s}}},\quad s\notin \mathbb {Z} _{0}^{-},,12円-s\notin \mathbb {Z} _{0}^{-}}

and has the Euler product

${\displaystyle L(s)=\prod _{p,円{\text{prime}}}{\frac {1}{1-\tau (p)p^{-s}+p^{11-2s}}},\quad \mathrm {Re} (s)>7.}${\displaystyle L(s)=\prod _{p,円{\text{prime}}}{\frac {1}{1-\tau (p)p^{-s}+p^{11-2s}}},\quad \mathrm {Re} (s)>7.}

Ramanujan conjectured that all nontrivial zeros of ${\displaystyle L}${\displaystyle L} have real part equal to ${\displaystyle 6}${\displaystyle 6}.

• Apostol, T. 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, F. 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 New Solution to the Equation τ(p) ≡ 0 (mod p)" (PDF), Journal of Integer Sequences, 13: Article 10.7.4
• Mordell, Louis J. (1917), "On Mr. Ramanujan's empirical expansions of modular functions.", Proceedings of the 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 Bureau of Standards
• Rankin, Robert A. (1988), "Ramanujan's tau-function and its generalizations", in Andrews, George E. (ed.), Ramanujan revisited (Urbana-Champaign, Ill., 1987), Boston, MA: Academic Press, pp. 245–268, ISBN 978-0-12-058560-1, 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 relatives à la fonction ${\displaystyle \tau }${\displaystyle \tau } de Ramanujan", Séminaire Delange-Pisot-Poitou, 14
• Swinnerton-Dyer, H. P. F. (1973), "On l-adic representations and congruences for coefficients of modular forms", in Kuyk, Willem; Serre, Jean-Pierre (eds.), Modular Functions of One Variable III, Lecture Notes in Mathematics, vol. 350, pp. 1–55, doi:10.1007/978-3-540-37802-0, ISBN 978-3-540-06483-1, MR 0406931
• Wilton, J. R. (1930), "Congruence properties of Ramanujan's function τ(n)", Proceedings of the London Mathematical Society, 31: 1–10, doi:10.1112/plms/s2-31.1.1

• Modular forms
• Multiplicative functions
• Srinivasa Ramanujan
• Zeta and L-functions

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