Ramanujan tau function

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

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

Values

The first few values of the tau function are given in the following table (sequence A000594 in the OEIS):

$n$	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
$τ (n)$	1	−24	252	−1472	4830	−6048	−16744	84480	−113643	−115920	534612	−370944	−577738	401856	1217160	987136

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's conjectures

Ramanujan (1916) observed, but did not prove, the following three properties of $τ (n)$ :

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

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).

Congruences for the tau function

For $\mathbb {Z}$ and $\mathbb {Z}$ , the Divisor function $σ k (n)$ is 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 $σ k (n)$ . Here are some:^[2]

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

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

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

Explicit formula

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

\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 $σ(n)$ is the sum of the positive divisors of $n$ .

Conjectures on τ(n)

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

Given that

f

does not have complex multiplication, do almost all primes

p

have the property that

a (p) ≢ 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 $a (n) (mod p)$ for $n$ coprime to $p$ , it is unclear how to compute $a (p) (mod p)$ . 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 $a (p) \equiv 0 (mod p)$ for infinitely many $p$ . Some researchers had begun to doubt whether $a (p) \equiv 0 (mod p)$ 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 $τ (p) \equiv 0 (mod p)$ are 2, 3, 5, 7, 2411, and 7758337633 (sequence A007659 in the OEIS).^[11]

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

$N$	reference
3316799	Lehmer (1947)
214928639999	Lehmer (1949)
1000000000000000	Serre (1973, p. 98), Serre (1985)
1213229187071998	Jennings (1993)
22689242781695999	Jordan and Kelly (1999)
22798241520242687999	Bosman (2007)
982149821766199295999	Zeng and Yin (2013)
816212624008487344127999	Derickx, van Hoeij, and Zeng (2013)

Ramanujan's L-function

Ramanujan's L-function is defined by

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

if $\Re s>6$ and by analytic continuation otherwise. It satisfies the functional equation

{\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-s\notin \mathbb {Z} _{0}^{-}

and has the Euler product

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

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

Notes

^ Sloane, N. J. A. (ed.). "Sequence A016754 (Odd squares: (2n-1)^2. Also centered octagonal numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^ ^a ^b Page 4 of Swinnerton-Dyer 1973
^ ^a ^b ^c ^d Due to Kolberg 1962
^ ^a ^b Due to Ashworth 1968
^ Due to Lahivi
^ ^a ^b Due to D. H. Lehmer
^ Due to Ramanujan 1916
^ Due to Wilton 1930
^ Due to J.-P. Serre 1968, Section 4.5
^ Niebur, Douglas (September 1975). "A formula for Ramanujan's $\tau$-function". Illinois Journal of Mathematics. 19 (3): 448–449. doi:10.1215/ijm/1256050746. ISSN 0019-2082.
^ N. Lygeros and O. Rozier (2010). "A new solution for the equation $\tau (p)\equiv 0{\pmod {p}}$ " (PDF). Journal of Integer Sequences. 13: Article 10.7.4.

References

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 $\tau$ 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

[1] Sloane, N. J. A. (ed.). "Sequence A016754 (Odd squares: (2n-1)^2. Also centered octagonal numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

[swd-2] Page 4 of Swinnerton-Dyer 1973

[kolberg-3] Due to Kolberg 1962

[ashworth-4] Due to Ashworth 1968

[5] Due to Lahivi

[Lehmer-6] Due to D. H. Lehmer

[7] Due to Ramanujan 1916

[8] Due to Wilton 1930

[9] Due to J.-P. Serre 1968, Section 4.5

[10] Niebur, Douglas (September 1975). "A formula for Ramanujan's $\tau$-function". Illinois Journal of Mathematics. 19 (3): 448–449. doi:10.1215/ijm/1256050746. ISSN 0019-2082.

[Lygeros-11] N. Lygeros and O. Rozier (2010). "A new solution for the equation $\tau (p)\equiv 0{\pmod {p}}$ " (PDF). Journal of Integer Sequences. 13: Article 10.7.4.

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