Editing Ramanujan tau function

{{Short description|Function studied by Ramanujan}}
[[Image:Absolute Tau function for x up to 16,000 with logarithmic scale.JPG|thumbnail|upright=1.64|Values of <math>|\tau(n)|</math>
for <math>n<16,000</math> with a logarithmic scale. The blue line picks only the values of <math>n</math> that are multiples of 121.]]

The '''Ramanujan tau function''', studied by {{harvs|txt|authorlink=Srinivasa Ramanujan|last=Ramanujan|year=1916}}, is the function 
<math>\tau : \mathbb{N}\to\mathbb{Z}</math> defined by the following identity:
:<math>\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),</math>
where <math>q=\exp(2\pi iz)</math> with <math>\mathrm{Im}(z)>0</math>, <math>\phi</math> is the [[Euler function]], <math>\eta</math> is the [[Dedekind eta function]], and the function <math>\Delta(z)</math> is a [[Holomorphic function|holomorphic]] [[cusp form]] of weight 12 and level 1, known as the [[Modular discriminant|discriminant modular form]] (some authors, notably [[Tom M. Apostol|Apostol]], write <math>\Delta/(2\pi)^{12}</math> instead of <math>\Delta</math>). 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 {{harvtxt|Dyson|1972}}.

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

{| class="wikitable" style="text-align:center"
|-
! <math>n</math>
|1||2||3||4||5||6||7||8||9||10||11||12||13||14||15||16
|-
! <math>\tau(n)</math>
|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.<ref>{{Cite OEIS|A016754|name=Odd squares: (2n-1)^2. Also centered octagonal numbers.}}</ref>

==Ramanujan's conjectures==
{{harvtxt|Ramanujan|1916}} observed, but did not prove, the following three properties of <math>\tau(n)</math>:

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

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).

==Congruences for the tau function==
For <math>k\in\mathbb{Z}</math> and <math>n\in\mathbb{N}</math>, the [[Divisor function]] <math>\sigma_k(n)</math> is the sum of the <math>k</math>th powers of the divisors of <math>n</math>. The tau function satisfies several congruence relations; many of them can be expressed in terms of <math>\sigma_k(n)</math>. Here are some:<ref name=swd>Page 4 of {{harvnb|Swinnerton-Dyer|1973}}</ref>
#<math>\tau(n)\equiv\sigma_{11}(n)\ \bmod\ 2^{11}\text{ for }n\equiv 1\ \bmod\ 8</math><ref name=kolberg>Due to {{harvnb|Kolberg|1962}}</ref>
#<math>\tau(n)\equiv 1217 \sigma_{11}(n)\ \bmod\  2^{13}\text{ for } n\equiv 3\ \bmod\ 8</math><ref name=kolberg/>
#<math>\tau(n)\equiv 1537 \sigma_{11}(n)\ \bmod\ 2^{12}\text{ for }n\equiv 5\ \bmod\ 8</math><ref name=kolberg/>
#<math>\tau(n)\equiv 705 \sigma_{11}(n)\ \bmod\ 2^{14}\text{ for }n\equiv 7\ \bmod\ 8</math><ref name=kolberg/>
#<math>\tau(n)\equiv n^{-610}\sigma_{1231}(n)\ \bmod\ 3^{6}\text{ for }n\equiv 1\ \bmod\ 3</math><ref name=ashworth>Due to {{harvnb|Ashworth|1968}}</ref>
#<math>\tau(n)\equiv n^{-610}\sigma_{1231}(n)\ \bmod\ 3^{7}\text{ for }n\equiv 2\ \bmod\ 3</math><ref name=ashworth/>
#<math>\tau(n)\equiv n^{-30}\sigma_{71}(n)\ \bmod\ 5^{3}\text{ for }n\not\equiv 0\ \bmod\ 5</math><ref>Due to Lahivi</ref>
#<math>\tau(n)\equiv n\sigma_{9}(n)\ \bmod\ 7</math><ref name=Lehmer>Due to D. H. Lehmer</ref>
#<math>\tau(n)\equiv n\sigma_{9}(n)\ \bmod\ 7^2\text{ for }n\equiv 3,5,6\ \bmod\ 7</math><ref name=Lehmer/>
#<math>\tau(n)\equiv\sigma_{11}(n)\ \bmod\ 691.</math><ref>Due to {{harvnb|Ramanujan|1916}}</ref>

For <math>p\neq 23</math> prime, we have<ref name=swd/><ref>Due to {{harvnb|Wilton|1930}}</ref>
<ol start=11>
<li><math>\tau(p)\equiv 0\ \bmod\ 23\text{ if }\left(\frac{p}{23}\right)=-1</math>
<li><math>\tau(p)\equiv \sigma_{11}(p)\ \bmod\ 23^2\text{ if } p\text{ is of the form } a^2+23b^2</math><ref>Due to J.-P. Serre 1968, Section 4.5</ref>
<li><math>\tau(p)\equiv -1\ \bmod\ 23\text{ otherwise}.</math>
</ol>

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

:<math>\tau(n)=n^4\sigma(n)-24\sum_{i=1}^{n-1}i^2(35i^2-52in+18n^2)\sigma(i)\sigma(n-i).</math>

where <math>\sigma(n)</math> is the [[Divisor function|sum of the positive divisors]] of <math>n</math>.

==Conjectures on the tau function==
Suppose that <math>f</math> is a weight-<math>k</math> integer [[newform]] and the Fourier coefficients <math>a(n)</math> are integers. Consider the problem: 
: Given that <math>f</math> does not have [[complex multiplication]], do almost all primes <math>p</math> have the property that <math>a(p)\not\equiv 0\pmod{p}</math> ?
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 <math>a(n)\pmod{p}</math> for <math>n</math> coprime to <math>p</math>, it is unclear how to compute <math>a(p)\pmod{p}</math>. The only theorem in this regard is Elkies' famous result for modular elliptic curves, which guarantees that there are infinitely many primes <math>p</math> such that <math>a(p)=0</math>, which thus are congruent to 0 modulo <math>p</math>. There are no known examples of non-CM <math>f</math> with weight greater than 2 for which <math>a(p)\not\equiv 0\pmod{p}</math> for infinitely many primes <math>p</math> (although it should be true for almost all <math>p</math>. There are also no known examples with <math>a(p)\equiv 0 \pmod{p}</math> for infinitely many <math>p</math>. Some researchers had begun to doubt whether <math>a(p)\equiv 0 \pmod{p}</math> for infinitely many <math>p</math>. As evidence, many provided Ramanujan's <math>\tau(p)</math> (case of weight 12). The only solutions up to <math>10^{10}</math> to the equation <math>\tau(p)\equiv 0\pmod{p}</math> are 2, 3, 5, 7, 2411, and {{val|7758337633}} {{OEIS|A007659}}.<ref name=Lygeros>{{cite journal |author=N. Lygeros and O. Rozier |year=2010 |title=A new solution for the equation <math>\tau(p)\equiv 0 \pmod{p}</math> |journal=Journal of Integer Sequences |volume=13 |pages=Article 10.7.4 |url=https://cs.uwaterloo.ca/journals/JIS/VOL13/Lygeros/lygeros5.pdf}}</ref>

{{harvtxt|Lehmer|1947}} conjectured that <math>\tau(n)\neq 0</math> for all <math>n</math>, an assertion sometimes known as Lehmer's conjecture. Lehmer verified the conjecture for <math>n</math> up to {{val|214928639999}} (Apostol 1997, p.&nbsp;22). The following table summarizes progress on finding successively larger values of <math>N</math> for which this condition holds for all <math>n\leq N</math>.
{| class="wikitable"
|-
! <math>N</math> !! reference
|-
|align="right"| {{val|3316799}} || Lehmer (1947)
|-
|align="right"| {{val|214928639999}} || Lehmer (1949)
|-
|align="right"| {{val|1000000000000000}} || Serre (1973, p.&nbsp;98), Serre (1985)
|-
|align="right"| {{val|1213229187071998}} || Jennings (1993)
|-
|align="right"| {{val|22689242781695999}} || Jordan and Kelly (1999)
|-
|align="right"| {{val|22798241520242687999}} || Bosman (2007)
|-
|align="right"| {{val|982149821766199295999}} || Zeng and Yin (2013)
|-
|align="right"| {{val|816212624008487344127999}} || Derickx, van Hoeij, and Zeng (2013)
|}

==Ramanujan's L-function==
'''Ramanujan's [[L-function|<math>L</math>-function]]''' is defined by
:<math>L(s)=\sum_{n\ge 1}\frac{\tau (n)}{n^s}</math>
if <math>\mathrm{Re}(s)>6</math> and by [[analytic continuation]] otherwise. It satisfies the functional equation
:<math>\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^{-}</math>
and has the [[Euler product]]
:<math>L(s)=\prod_{p\,\text{prime}}\frac{1}{1-\tau (p)p^{-s}+p^{11-2s}},\quad \mathrm{Re}(s)>7.</math>
Ramanujan conjectured that all nontrivial zeros of <math>L</math> have real part equal to <math>6</math>.

==Notes==
{{reflist|30em}}

==References==
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*{{Citation
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*{{Citation
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[[Category:Modular forms]]
[[Category:Multiplicative functions]]
[[Category:Srinivasa Ramanujan]]
[[Category:Zeta and L-functions]]