Ramanujan tau function

The first few values of the tau function are given in the following table {{OEIS|id=A000594}}:

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.{{Cite OEIS|A016754|name=Odd squares: (2n-1)^2. Also centered octagonal numbers.}}

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

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

The first two properties were proved by {{harvtxt|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 $k\backslash in\backslash mathbb\{Z\}$ and $n\backslash in\backslash mathbb\{N\}$, the Divisor function $\backslash sigma\_k(n)$ 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 $\backslash sigma\_k(n)$. Here are some:Page 4 of {{harvnb|Swinnerton-Dyer|1973}}

1. $\backslash tau(n)\backslash equiv\backslash sigma\_\{11\}(n)\backslash \; \backslash bmod\backslash \; 2^\{11\}\backslash text\{\; for\; \}n\backslash equiv\; 1\backslash \; \backslash bmod\backslash \; 8$Due to {{harvnb|Kolberg|1962}}
2. $\backslash tau(n)\backslash equiv\; 1217\; \backslash sigma\_\{11\}(n)\backslash \; \backslash bmod\backslash \; 2^\{13\}\backslash text\{\; for\; \}\; n\backslash equiv\; 3\backslash \; \backslash bmod\backslash \; 8$
3. $\backslash tau(n)\backslash equiv\; 1537\; \backslash sigma\_\{11\}(n)\backslash \; \backslash bmod\backslash \; 2^\{12\}\backslash text\{\; for\; \}n\backslash equiv\; 5\backslash \; \backslash bmod\backslash \; 8$
4. $\backslash tau(n)\backslash equiv\; 705\; \backslash sigma\_\{11\}(n)\backslash \; \backslash bmod\backslash \; 2^\{14\}\backslash text\{\; for\; \}n\backslash equiv\; 7\backslash \; \backslash bmod\backslash \; 8$
5. $\backslash tau(n)\backslash equiv\; n^\{-610\}\backslash sigma\_\{1231\}(n)\backslash \; \backslash bmod\backslash \; 3^\{6\}\backslash text\{\; for\; \}n\backslash equiv\; 1\backslash \; \backslash bmod\backslash \; 3$Due to {{harvnb|Ashworth|1968}}
6. $\backslash tau(n)\backslash equiv\; n^\{-610\}\backslash sigma\_\{1231\}(n)\backslash \; \backslash bmod\backslash \; 3^\{7\}\backslash text\{\; for\; \}n\backslash equiv\; 2\backslash \; \backslash bmod\backslash \; 3$
7. $\backslash tau(n)\backslash equiv\; n^\{-30\}\backslash sigma\_\{71\}(n)\backslash \; \backslash bmod\backslash \; 5^\{3\}\backslash text\{\; for\; \}n\backslash not\backslash equiv\; 0\backslash \; \backslash bmod\backslash \; 5$Due to Lahivi
8. $\backslash tau(n)\backslash equiv\; n\backslash sigma\_\{9\}(n)\backslash \; \backslash bmod\backslash \; 7$Due to D. H. Lehmer
9. $\backslash tau(n)\backslash equiv\; n\backslash sigma\_\{9\}(n)\backslash \; \backslash bmod\backslash \; 7^2\backslash text\{\; for\; \}n\backslash equiv\; 3,5,6\backslash \; \backslash bmod\backslash \; 7$
10. $\backslash tau(n)\backslash equiv\backslash sigma\_\{11\}(n)\backslash \; \backslash bmod\backslash \; 691.$Due to {{harvnb|Ramanujan|1916}}

For $p\backslash neq\; 23$ prime, we haveDue to {{harvnb|Wilton|1930}}

11. $\backslash tau(p)\backslash equiv\; 0\backslash \; \backslash bmod\backslash \; 23\backslash text\{\; if\; \}\backslash left(\backslash frac\{p\}\{23\}\backslash right)=-1$

12. $\backslash tau(p)\backslash equiv\; \backslash sigma\_\{11\}(p)\backslash \; \backslash bmod\backslash \; 23^2\backslash text\{\; if\; \}\; p\backslash text\{\; is\; of\; the\; form\; \}\; a^2+23b^2$Due to J.-P. Serre 1968, Section 4.5

13. $\backslash tau(p)\backslash equiv\; -1\backslash \; \backslash bmod\backslash \; 23\backslash text\{\; otherwise\}.$

In 1975 Douglas Niebur proved an explicit formula for the Ramanujan tau function:{{Cite journal |last=Niebur |first=Douglas |date=September 1975 |title=A formula for Ramanujan's $\backslash tau$-function |journal=Illinois Journal of Mathematics |volume=19 |issue=3 |pages=448–449 |doi=10.1215/ijm/1256050746 |issn=0019-2082|doi-access=free }}

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

where $\backslash sigma(n)$ is the sum of the positive divisors of $n$.

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 $a(p)\backslash not\backslash equiv\; 0\backslash pmod\{p\}$ ?

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 $a(n)\backslash pmod\{p\}$ for $n$ coprime to $p$, it is unclear how to compute $a(p)\backslash pmod\{p\}$. 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 $a(p)\backslash not\backslash equiv\; 0\backslash pmod\{p\}$ for infinitely many primes $p$ (although it should be true for almost all $p$. There are also no known examples with $a(p)\backslash equiv\; 0\; \backslash pmod\{p\}$ for infinitely many $p$. Some researchers had begun to doubt whether $a(p)\backslash equiv\; 0\; \backslash pmod\{p\}$ for infinitely many $p$. As evidence, many provided Ramanujan's $\backslash tau(p)$ (case of weight 12). The only solutions up to $10^\{10\}$ to the equation $\backslash tau(p)\backslash equiv\; 0\backslash pmod\{p\}$ are 2, 3, 5, 7, 2411, and {{val|7758337633}} {{OEIS|A007659}}.{{cite journal |author=N. Lygeros and O. Rozier |year=2010 |title=A new solution for the equation $\backslash tau(p)\backslash equiv\; 0\; \backslash pmod\{p\}$ |journal=Journal of Integer Sequences |volume=13 |pages=Article 10.7.4 |url=https://cs.uwaterloo.ca/journals/JIS/VOL13/Lygeros/lygeros5.pdf}}

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

Ramanujan's $L$-function is defined by

:$L(s)=\backslash sum\_\{n\backslash ge\; 1\}\backslash frac\{\backslash tau\; (n)\}\{n^s\}$

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

:$\backslash frac\{L(s)\backslash Gamma\; (s)\}\{(2\backslash pi)^s\}=\backslash frac\{L(12-s)\backslash Gamma(12-s)\}\{(2\backslash pi)^\{12-s\}\},\backslash quad\; s\backslash notin\backslash mathbb\{Z\}\_0^-,\; \backslash ,12-s\backslash notin\backslash mathbb\{Z\}\_0^\{-\}$

and has the Euler product

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

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

{{reflist|30em}}

• {{Citation

| last=Apostol

| first=T. M.

| authorlink=Tom M. Apostol

| title=Modular Functions and Dirichlet Series in Number Theory

| year=1997

| journal=New York: Springer-Verlag 2nd Ed.

}}

• {{Citation

| last=Ashworth

| first=M. H.

| title=Congruence and identical properties of modular forms (D. Phil. Thesis, Oxford)

| year=1968

}}

• {{citation | last1=Dyson | first1=F. J. | author1-link=Freeman Dyson | title=Missed opportunities | zbl=0271.01005 | journal=Bull. Amer. Math. Soc. | volume=78 | issue=5 | pages=635–652 | year=1972 | doi=10.1090/S0002-9904-1972-12971-9| doi-access=free }}
• {{Citation

| last=Kolberg

| first=O.

| title=Congruences for Ramanujan's function τ(n)

| journal=Arbok Univ. Bergen Mat.-Natur. Ser.

| issue=11

| year=1962

| mr=0158873 | zbl=0168.29502

}}

• {{citation | last1=Lehmer | first1=D.H. | author1-link=D. H. Lehmer | title=The vanishing of Ramanujan's function τ(n) | zbl=0029.34502 | journal=Duke Math. J. | volume=14 | pages=429–433 | year=1947 | issue=2 | doi=10.1215/s0012-7094-47-01436-1}}
• {{Citation

| last=Lygeros

| first=N.

| title=A New Solution to the Equation τ(p) ≡ 0 (mod p)

| url=http://www.cs.uwaterloo.ca/journals/JIS/VOL13/Lygeros/lygeros5.pdf

| year=2010

| journal=Journal of Integer Sequences

| volume=13

| pages=Article 10.7.4

}}

• {{Citation | last1=Mordell | first1=Louis J. | author1-link=Louis Mordell | title=On Mr. Ramanujan's empirical expansions of modular functions. | url=https://archive.org/stream/proceedingsofcam1920191721camb#page/n133 | jfm=46.0605.01 | year=1917 | journal=Proceedings of the Cambridge Philosophical Society | volume=19 | pages=117–124}}
• {{Citation

| last=Newman

| first=M.

| title=A table of τ (p) modulo p, p prime, 3 ≤ p ≤ 16067

| year=1972

| publisher=National Bureau of Standards

}}

• {{Citation | last1=Rankin | first1=Robert A. | editor1-last=Andrews | editor1-first=George E. | title=Ramanujan revisited (Urbana-Champaign, Ill., 1987) | url=https://books.google.com/books?id=GJUEAQAAIAAJ | publisher=Academic Press | location=Boston, MA | isbn=978-0-12-058560-1 | mr=938968 | year=1988 | chapter=Ramanujan's tau-function and its generalizations | pages=245–268}}
• {{Citation

| last=Ramanujan

| first=Srinivasa

| author-link=Srinivasa Ramanujan

| title=On certain arithmetical functions

| journal=Trans. Camb. Philos. Soc.

| year=1916

| volume=22

| issue=9

| pages=159–184

| mr=2280861

}}

• {{Citation

| last=Serre

| first=J-P.

| title=Une interprétation des congruences relatives à la fonction $\backslash tau$ de Ramanujan

| journal=Séminaire Delange-Pisot-Poitou

| volume=14

| year=1968

| author-link=Jean-Pierre Serre

| url = http://www.numdam.org/item?id=SDPP_1967-1968__9_1_A13_0

}}

• {{Citation

| last=Swinnerton-Dyer

| first=H. P. F.

| author-link=Peter Swinnerton-Dyer

| title=Modular Functions of One Variable III

| contribution=On l-adic representations and congruences for coefficients of modular forms

| year=1973

| isbn=978-3-540-06483-1

| series=Lecture Notes in Mathematics

| volume=350

| mr=0406931

| pages=1–55

| doi=10.1007/978-3-540-37802-0

| editor1-last=Kuyk

| editor1-first=Willem

| editor2-last=Serre

| editor2-first=Jean-Pierre

| editor2-link=Jean-Pierre Serre

}}

• {{Citation

| last=Wilton

| first=J. R.

| title=Congruence properties of Ramanujan's function τ(n)

| year=1930

| journal=Proceedings of the London Mathematical Society

| volume=31

| pages=1–10

| doi=10.1112/plms/s2-31.1.1

}}

Category:Modular forms

Category:Multiplicative functions

Category:Srinivasa Ramanujan

Category:Zeta and L-functions