Ramanujan tau function

<h2 id="values">Values</h2>
The first few values of the tau function are given in the following table (sequence A000594 in the <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>):

<table><tbody><tr><th> n {\displaystyle n} </th><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><td>10</td><td>11</td><td>12</td><td>13</td><td>14</td><td>15</td><td>16</td></tr><tr><th> τ ( n ) {\displaystyle \tau (n)} </th><td>1</td><td>−24</td><td>252</td><td>−1472</td><td>4830</td><td>−6048</td><td>−16744</td><td>84480</td><td>−113643</td><td>−115920</td><td>534612</td><td>−370944</td><td>−577738</td><td>401856</td><td>1217160</td><td>987136</td></tr></tbody></table>
Calculating this function on an odd square number (i.e. a <a href="/facts/Centered_octagonal_number/ndKGGruE">centered octagonal number</a>) yields an odd number, whereas for any other number the function yields an even number.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>

<h2 id="ramanujans-conjectures">Ramanujan's conjectures</h2>
Ramanujan (1916) observed, but did not prove, the following three properties of 
 
 
 
 τ
 (
 n
 )
 
 
 {\displaystyle \tau (n)}
 
:

<ul><li>
 
 
 
 τ
 (
 m
 n
 )
 =
 τ
 (
 m
 )
 τ
 (
 n
 )
 
 
 {\displaystyle \tau (mn)=\tau (m)\tau (n)}
 
 if 
 
 
 
 gcd
 (
 m
 ,
 n
 )
 =
 1
 
 
 {\displaystyle \gcd(m,n)=1}
 
 (meaning that 
 
 
 
 τ
 (
 n
 )
 
 
 {\displaystyle \tau (n)}
 
 is a <a href="/facts/Multiplicative_function/4kfolbBl">multiplicative function</a>)</li>
<li>
 
 
 
 τ
 (
 
 p
 
 r
 +
 1
 
 
 )
 =
 τ
 (
 p
 )
 τ
 (
 
 p
 
 r
 
 
 )
 −
 
 p
 
 11
 
 
 τ
 (
 
 p
 
 r
 −
 1
 
 
 )
 
 
 {\displaystyle \tau (p^{r+1})=\tau (p)\tau (p^{r})-p^{11}\tau (p^{r-1})}
 
 for 
 
 
 
 p
 
 
 {\displaystyle p}
 
 prime and 
 
 
 
 r
 >
 0
 
 
 {\displaystyle r>0}
 
.</li>
<li>
 
 
 
 
 |
 
 τ
 (
 p
 )
 
 |
 
 ≤
 2
 
 p
 
 11
 
 /
 
 2
 
 
 
 
 {\displaystyle |\tau (p)|\leq 2p^{11/2}}
 
 for all <a href="/facts/Prime_number/L20FLkgn">primes</a> 
 
 
 
 p
 
 
 {\displaystyle p}
 
.</li></ul>
The first two properties were proved by Mordell (1917) and the third one, called the <a href="/facts/Ramanujan_conjecture/vfu3TYM6">Ramanujan conjecture</a>, was proved by <a href="/facts/Deligne/wYC8fhfq">Deligne</a> in 1974 as a consequence of his proof of the <a href="/facts/Weil_conjectures/JPSk0G5j">Weil conjectures</a> (specifically, he deduced it by applying them to a Kuga-Sato variety).

<h2 id="congruences-for-the-tau-function">Congruences for the tau function</h2>
For 
 
 
 
 k
 ∈
 
 Z
 
 
 
 {\displaystyle k\in \mathbb {Z} }
 
 and 
 
 
 
 n
 ∈
 
 N
 
 
 
 {\displaystyle n\in \mathbb {N} }
 
, the <a href="/facts/Divisor_function/rzvm1jw2">Divisor function</a> 
 
 
 
 
 σ
 
 k
 
 
 (
 n
 )
 
 
 {\displaystyle \sigma _{k}(n)}
 
 is the sum of the 
 
 
 
 k
 
 
 {\displaystyle k}
 
th powers of the divisors of 
 
 
 
 n
 
 
 {\displaystyle n}
 
. The tau function satisfies several congruence relations; many of them can be expressed in terms of 
 
 
 
 
 σ
 
 k
 
 
 (
 n
 )
 
 
 {\displaystyle \sigma _{k}(n)}
 
. Here are some:<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>

<ol><li>
 
 
 
 τ
 (
 n
 )
 ≡
 
 σ
 
 11
 
 
 (
 n
 )
  
 
 mod
 
  
 
 
 
 2
 
 11
 
 
 
  for 
 
 n
 ≡
 1
  
 
 mod
 
  
 
 
 8
 
 
 {\displaystyle \tau (n)\equiv \sigma _{11}(n)\ {\bmod {\ }}2^{11}{\text{ for }}n\equiv 1\ {\bmod {\ }}8}
 
<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a></li>
<li>
 
 
 
 τ
 (
 n
 )
 ≡
 1217
 
 σ
 
 11
 
 
 (
 n
 )
  
 
 mod
 
  
 
 
 
 2
 
 13
 
 
 
  for 
 
 n
 ≡
 3
  
 
 mod
 
  
 
 
 8
 
 
 {\displaystyle \tau (n)\equiv 1217\sigma _{11}(n)\ {\bmod {\ }}2^{13}{\text{ for }}n\equiv 3\ {\bmod {\ }}8}
 
<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a></li>
<li>
 
 
 
 τ
 (
 n
 )
 ≡
 1537
 
 σ
 
 11
 
 
 (
 n
 )
  
 
 mod
 
  
 
 
 
 2
 
 12
 
 
 
  for 
 
 n
 ≡
 5
  
 
 mod
 
  
 
 
 8
 
 
 {\displaystyle \tau (n)\equiv 1537\sigma _{11}(n)\ {\bmod {\ }}2^{12}{\text{ for }}n\equiv 5\ {\bmod {\ }}8}
 
<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a></li>
<li>
 
 
 
 τ
 (
 n
 )
 ≡
 705
 
 σ
 
 11
 
 
 (
 n
 )
  
 
 mod
 
  
 
 
 
 2
 
 14
 
 
 
  for 
 
 n
 ≡
 7
  
 
 mod
 
  
 
 
 8
 
 
 {\displaystyle \tau (n)\equiv 705\sigma _{11}(n)\ {\bmod {\ }}2^{14}{\text{ for }}n\equiv 7\ {\bmod {\ }}8}
 
<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a></li>
<li>
 
 
 
 τ
 (
 n
 )
 ≡
 
 n
 
 −
 610
 
 
 
 σ
 
 1231
 
 
 (
 n
 )
  
 
 mod
 
  
 
 
 
 3
 
 6
 
 
 
  for 
 
 n
 ≡
 1
  
 
 mod
 
  
 
 
 3
 
 
 {\displaystyle \tau (n)\equiv n^{-610}\sigma _{1231}(n)\ {\bmod {\ }}3^{6}{\text{ for }}n\equiv 1\ {\bmod {\ }}3}
 
<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a></li>
<li>
 
 
 
 τ
 (
 n
 )
 ≡
 
 n
 
 −
 610
 
 
 
 σ
 
 1231
 
 
 (
 n
 )
  
 
 mod
 
  
 
 
 
 3
 
 7
 
 
 
  for 
 
 n
 ≡
 2
  
 
 mod
 
  
 
 
 3
 
 
 {\displaystyle \tau (n)\equiv n^{-610}\sigma _{1231}(n)\ {\bmod {\ }}3^{7}{\text{ for }}n\equiv 2\ {\bmod {\ }}3}
 
<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a></li>
<li>
 
 
 
 τ
 (
 n
 )
 ≡
 
 n
 
 −
 30
 
 
 
 σ
 
 71
 
 
 (
 n
 )
  
 
 mod
 
  
 
 
 
 5
 
 3
 
 
 
  for 
 
 n
 ≢
 0
  
 
 mod
 
  
 
 
 5
 
 
 {\displaystyle \tau (n)\equiv n^{-30}\sigma _{71}(n)\ {\bmod {\ }}5^{3}{\text{ for }}n\not \equiv 0\ {\bmod {\ }}5}
 
<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a></li>
<li>
 
 
 
 τ
 (
 n
 )
 ≡
 n
 
 σ
 
 9
 
 
 (
 n
 )
  
 
 mod
 
  
 
 
 7
 
 
 {\displaystyle \tau (n)\equiv n\sigma _{9}(n)\ {\bmod {\ }}7}
 
<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a></li>
<li>
 
 
 
 τ
 (
 n
 )
 ≡
 n
 
 σ
 
 9
 
 
 (
 n
 )
  
 
 mod
 
  
 
 
 
 7
 
 2
 
 
 
  for 
 
 n
 ≡
 3
 ,
 5
 ,
 6
  
 
 mod
 
  
 
 
 7
 
 
 {\displaystyle \tau (n)\equiv n\sigma _{9}(n)\ {\bmod {\ }}7^{2}{\text{ for }}n\equiv 3,5,6\ {\bmod {\ }}7}
 
<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a></li>
<li>
 
 
 
 τ
 (
 n
 )
 ≡
 
 σ
 
 11
 
 
 (
 n
 )
  
 
 mod
 
  
 
 
 691.
 
 
 {\displaystyle \tau (n)\equiv \sigma _{11}(n)\ {\bmod {\ }}691.}
 
<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a></li></ol>
For 
 
 
 
 p
 ≠
 23
 
 
 {\displaystyle p\neq 23}
 
 prime, we have<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a><a class="footnote-ref" id="fnref:14" href="#fn:14">14</a>

<ol>
<li>
 
 
 
 τ
 (
 p
 )
 ≡
 0
  
 
 mod
 
  
 
 
 23
 
  if 
 
 
 (
 
 
 p
 23
 
 
 )
 
 =
 −
 1
 
 
 {\displaystyle \tau (p)\equiv 0\ {\bmod {\ }}23{\text{ if }}\left({\frac {p}{23}}\right)=-1}

</li><li>
 
 
 
 τ
 (
 p
 )
 ≡
 
 σ
 
 11
 
 
 (
 p
 )
  
 
 mod
 
  
 
 
 
 23
 
 2
 
 
 
  if 
 
 p
 
  is of the form 
 
 
 a
 
 2
 
 
 +
 23
 
 b
 
 2
 
 
 
 
 {\displaystyle \tau (p)\equiv \sigma _{11}(p)\ {\bmod {\ }}23^{2}{\text{ if }}p{\text{ is of the form }}a^{2}+23b^{2}}
 
<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a>
</li><li>
 
 
 
 τ
 (
 p
 )
 ≡
 −
 1
  
 
 mod
 
  
 
 
 23
 
  otherwise
 
 .
 
 
 {\displaystyle \tau (p)\equiv -1\ {\bmod {\ }}23{\text{ otherwise}}.}

</li></ol>
<h2 id="explicit-formula">Explicit formula</h2>
In 1975 Douglas Niebur proved an explicit formula for the Ramanujan tau function:<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a>

τ
        (
        n
        )
        =
        
          n
          
            4
          
        
        σ
        (
        n
        )
        −
        24
        
          ∑
          
            i
            =
            1
          
          
            n
            −
            1
          
        
        
          i
          
            2
          
        
        (
        35
        
          i
          
            2
          
        
        −
        52
        i
        n
        +
        18
        
          n
          
            2
          
        
        )
        σ
        (
        i
        )
        σ
        (
        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 
 
 
 
 σ
 (
 n
 )
 
 
 {\displaystyle \sigma (n)}
 
 is the <a href="/facts/Divisor_function/rzvm1jw2">sum of the positive divisors</a> of 
 
 
 
 n
 
 
 {\displaystyle n}
 
.

<h2 id="conjectures-on-------------------------------------n----------------------displaystyle-tau-n">Conjectures on 
 
 
 
 τ
 (
 n
 )
 
 
 {\displaystyle \tau (n)}
 
</h2>
Suppose that 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is a weight-
 
 
 
 k
 
 
 {\displaystyle k}
 
 integer <a href="/facts/Newform/KgNk1KCW">newform</a> and the Fourier coefficients 
 
 
 
 a
 (
 n
 )
 
 
 {\displaystyle a(n)}
 
 are integers. Consider the problem: 

Given that 
 
 
 
 f
 
 
 {\displaystyle f}
 
 does not have <a href="/facts/Complex_multiplication/YQNcbNXP">complex multiplication</a>, do almost all primes 
 
 
 
 p
 
 
 {\displaystyle p}
 
 have the property that 
 
 
 
 a
 (
 p
 )
 ≢
 0
 
 
 (
 mod
 
 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 
 
 
 
 a
 (
 n
 )
 
 
 (
 mod
 
 p
 )
 
 
 
 {\displaystyle a(n){\pmod {p}}}
 
 for 
 
 
 
 n
 
 
 {\displaystyle n}
 
 coprime to 
 
 
 
 p
 
 
 {\displaystyle p}
 
, it is unclear how to compute 
 
 
 
 a
 (
 p
 )
 
 
 (
 mod
 
 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 
 
 
 
 p
 
 
 {\displaystyle p}
 
 such that 
 
 
 
 a
 (
 p
 )
 =
 0
 
 
 {\displaystyle a(p)=0}
 
, which thus are congruent to 0 modulo 
 
 
 
 p
 
 
 {\displaystyle p}
 
. There are no known examples of non-CM 
 
 
 
 f
 
 
 {\displaystyle f}
 
 with weight greater than 2 for which 
 
 
 
 a
 (
 p
 )
 ≢
 0
 
 
 (
 mod
 
 p
 )
 
 
 
 {\displaystyle a(p)\not \equiv 0{\pmod {p}}}
 
 for infinitely many primes 
 
 
 
 p
 
 
 {\displaystyle p}
 
 (although it should be true for almost all 
 
 
 
 p
 
 
 {\displaystyle p}
 
. There are also no known examples with 
 
 
 
 a
 (
 p
 )
 ≡
 0
 
 
 (
 mod
 
 p
 )
 
 
 
 {\displaystyle a(p)\equiv 0{\pmod {p}}}
 
 for infinitely many 
 
 
 
 p
 
 
 {\displaystyle p}
 
. Some researchers had begun to doubt whether 
 
 
 
 a
 (
 p
 )
 ≡
 0
 
 
 (
 mod
 
 p
 )
 
 
 
 {\displaystyle a(p)\equiv 0{\pmod {p}}}
 
 for infinitely many 
 
 
 
 p
 
 
 {\displaystyle p}
 
. As evidence, many provided Ramanujan's 
 
 
 
 τ
 (
 p
 )
 
 
 {\displaystyle \tau (p)}
 
 (case of weight 12). The only solutions up to 
 
 
 
 
 10
 
 10
 
 
 
 
 {\displaystyle 10^{10}}
 
 to the equation 
 
 
 
 τ
 (
 p
 )
 ≡
 0
 
 
 (
 mod
 
 p
 )
 
 
 
 {\displaystyle \tau (p)\equiv 0{\pmod {p}}}
 
 are 2, 3, 5, 7, 2411, and 7758337633 (sequence A007659 in the <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>).<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a>
Lehmer (1947) conjectured that 
 
 
 
 τ
 (
 n
 )
 ≠
 0
 
 
 {\displaystyle \tau (n)\neq 0}
 
 for all 
 
 
 
 n
 
 
 {\displaystyle n}
 
, an assertion sometimes known as Lehmer's conjecture. Lehmer verified the conjecture for 
 
 
 
 n
 
 
 {\displaystyle n}
 
 up to 214928639999 (Apostol 1997, p. 22). The following table summarizes progress on finding successively larger values of 
 
 
 
 N
 
 
 {\displaystyle N}
 
 for which this condition holds for all 
 
 
 
 n
 ≤
 N
 
 
 {\displaystyle n\leq N}
 
.

<table><tbody><tr><th> N {\displaystyle N} </th><th>reference</th></tr><tr><td>3316799</td><td>Lehmer (1947)</td></tr><tr><td>214928639999</td><td>Lehmer (1949)</td></tr><tr><td>1000000000000000</td><td>Serre (1973, p. 98), Serre (1985)</td></tr><tr><td>1213229187071998</td><td>Jennings (1993)</td></tr><tr><td>22689242781695999</td><td>Jordan and Kelly (1999)</td></tr><tr><td>22798241520242687999</td><td>Bosman (2007)</td></tr><tr><td>982149821766199295999</td><td>Zeng and Yin (2013)</td></tr><tr><td>816212624008487344127999</td><td>Derickx, van Hoeij, and Zeng (2013)</td></tr></tbody></table>
<h2 id="ramanujans---------------------l--------------displaystyle-l---function">Ramanujan's 
 
 
 
 L
 
 
 {\displaystyle L}
 
-function</h2>
Ramanujan's <a href="/facts/L-function/REVCKvU7">
 
 
 
 L
 
 
 {\displaystyle L}
 
-function</a> is defined by

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

if 
 
 
 
 
 R
 e
 
 (
 s
 )
 >
 6
 
 
 {\displaystyle \mathrm {Re} (s)>6}
 
 and by <a href="/facts/Analytic_continuation/aBCuEgvt">analytic continuation</a> otherwise. It satisfies the functional equation

L
              (
              s
              )
              Γ
              (
              s
              )
            
            
              (
              2
              π
              
                )
                
                  s
                
              
            
          
        
        =
        
          
            
              L
              (
              12
              −
              s
              )
              Γ
              (
              12
              −
              s
              )
            
            
              (
              2
              π
              
                )
                
                  12
                  −
                  s
                
              
            
          
        
        ,
        
        s
        ∉
        
          
            Z
          
          
            0
          
          
            −
          
        
        ,
        
        12
        −
        s
        ∉
        
          
            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-s\notin \mathbb {Z} _{0}^{-}}

and has the <a href="/facts/Euler_product/Lvsv4U3M">Euler product</a>

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

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

<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Tom_M._Apostol/S6vGtJWN">Apostol, T. M.</a> (1997), "Modular Functions and Dirichlet Series in Number Theory", New York: Springer-Verlag 2nd Ed.</li>
<li>Ashworth, M. H. (1968), Congruence and identical properties of modular forms (D. Phil. Thesis, Oxford)</li>
<li><a href="/facts/Freeman_Dyson/kSRxTh2r">Dyson, F. J.</a> (1972), "Missed opportunities", Bull. Amer. Math. Soc., 78 (5): 635–652, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2FS0002-9904-1972-12971-9">10.1090/S0002-9904-1972-12971-9</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0271.01005">0271.01005</a></li>
<li>Kolberg, O. (1962), "Congruences for Ramanujan's function τ(n)", Arbok Univ. Bergen Mat.-Natur. Ser. (11), <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0158873">0158873</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0168.29502">0168.29502</a></li>
<li><a href="/facts/D._H._Lehmer/EI61SICa">Lehmer, D.H.</a> (1947), "The vanishing of Ramanujan's function τ(n)", Duke Math. J., 14 (2): 429–433, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1215%2Fs0012-7094-47-01436-1">10.1215/s0012-7094-47-01436-1</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0029.34502">0029.34502</a></li>
<li>Lygeros, N. (2010), <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL13/Lygeros/lygeros5.pdf">"A New Solution to the Equation τ(p) ≡ 0 (mod p)"</a> (PDF), Journal of Integer Sequences, 13: Article 10.7.4</li>
<li><a href="/facts/Louis_Mordell/L2Qpjnnz">Mordell, Louis J.</a> (1917), <a href="https://archive.org/stream/proceedingsofcam1920191721camb#page/n133">"On Mr. Ramanujan's empirical expansions of modular functions."</a>, <a href="/facts/Proceedings_of_the_Cambridge_Philosophical_Society/KUUzcS7S">Proceedings of the Cambridge Philosophical Society</a>, 19: 117–124, <a href="/facts/JFM_(identifier)/P6rFxKKx">JFM</a> <a href="https://zbmath.org/?format=complete&q=an:46.0605.01">46.0605.01</a></li>
<li>Newman, M. (1972), A table of τ (p) modulo p, p prime, 3 ≤ p ≤ 16067, National Bureau of Standards</li>
<li>Rankin, Robert A. (1988), "Ramanujan's tau-function and its generalizations", in Andrews, George E. (ed.), <a href="https://books.google.com/books?id=GJUEAQAAIAAJ">Ramanujan revisited (Urbana-Champaign, Ill., 1987)</a>, Boston, MA: <a href="/facts/Academic_Press/qxoKjzFy">Academic Press</a>, pp. 245–268, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-12-058560-1, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0938968">0938968</a></li>
<li><a href="/facts/Srinivasa_Ramanujan/HBPr4bzd">Ramanujan, Srinivasa</a> (1916), "On certain arithmetical functions", Trans. Camb. Philos. Soc., 22 (9): 159–184, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2280861">2280861</a></li>
<li><a href="/facts/Jean-Pierre_Serre/1xSr7BCh">Serre, J-P.</a> (1968), <a href="http://www.numdam.org/item?id=SDPP_1967-1968__9_1_A13_0">"Une interprétation des congruences relatives à la fonction 
 
 
 
 τ
 
 
 {\displaystyle \tau }
 
 de Ramanujan"</a>, Séminaire Delange-Pisot-Poitou, 14</li>
<li><a href="/facts/Peter_Swinnerton-Dyer/D66SILlt">Swinnerton-Dyer, H. P. F.</a> (1973), "On l-adic representations and congruences for coefficients of modular forms", in Kuyk, Willem; <a href="/facts/Jean-Pierre_Serre/1xSr7BCh">Serre, Jean-Pierre</a> (eds.), Modular Functions of One Variable III, Lecture Notes in Mathematics, vol. 350, pp. 1–55, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-540-37802-0">10.1007/978-3-540-37802-0</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-06483-1, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0406931">0406931</a></li>
<li>Wilton, J. R. (1930), "Congruence properties of Ramanujan's function τ(n)", Proceedings of the London Mathematical Society, 31: 1–10, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1112%2Fplms%2Fs2-31.1.1">10.1112/plms/s2-31.1.1</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn: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. <a href="/wiki/Neil_Sloane" target="_blank">/wiki/Neil_Sloane</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Page 4 of Swinnerton-Dyer 1973 - 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 <a href="https://doi.org/10.1007%2F978-3-540-37802-0" target="_blank">https://doi.org/10.1007%2F978-3-540-37802-0</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Due to Kolberg 1962 - Kolberg, O. (1962), "Congruences for Ramanujan's function τ(n)", Arbok Univ. Bergen Mat.-Natur. Ser. (11), MR 0158873, Zbl 0168.29502 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0158873" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0158873</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Due to Kolberg 1962 - Kolberg, O. (1962), "Congruences for Ramanujan's function τ(n)", Arbok Univ. Bergen Mat.-Natur. Ser. (11), MR 0158873, Zbl 0168.29502 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0158873" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0158873</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Due to Kolberg 1962 - Kolberg, O. (1962), "Congruences for Ramanujan's function τ(n)", Arbok Univ. Bergen Mat.-Natur. Ser. (11), MR 0158873, Zbl 0168.29502 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0158873" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0158873</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Due to Kolberg 1962 - Kolberg, O. (1962), "Congruences for Ramanujan's function τ(n)", Arbok Univ. Bergen Mat.-Natur. Ser. (11), MR 0158873, Zbl 0168.29502 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0158873" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0158873</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Due to Ashworth 1968 - Ashworth, M. H. (1968), Congruence and identical properties of modular forms (D. Phil. Thesis, Oxford) <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Due to Ashworth 1968 - Ashworth, M. H. (1968), Congruence and identical properties of modular forms (D. Phil. Thesis, Oxford) <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Due to Lahivi <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Due to D. H. Lehmer <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Due to D. H. Lehmer <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Due to Ramanujan 1916 - Ramanujan, Srinivasa (1916), "On certain arithmetical functions", Trans. Camb. Philos. Soc., 22 (9): 159–184, MR 2280861 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2280861" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=2280861</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Page 4 of Swinnerton-Dyer 1973 - 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 <a href="https://doi.org/10.1007%2F978-3-540-37802-0" target="_blank">https://doi.org/10.1007%2F978-3-540-37802-0</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Due to Wilton 1930 - 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 <a href="https://doi.org/10.1112%2Fplms%2Fs2-31.1.1" target="_blank">https://doi.org/10.1112%2Fplms%2Fs2-31.1.1</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Due to J.-P. Serre 1968, Section 4.5 <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Niebur, Douglas (September 1975). "A formula for Ramanujan's 
 
 
 
 τ
 
 
 {\displaystyle \tau }
 
-function". Illinois Journal of Mathematics. 19 (3): 448–449. doi:10.1215/ijm/1256050746. ISSN 0019-2082. <a href="https://doi.org/10.1215%2Fijm%2F1256050746" target="_blank">https://doi.org/10.1215%2Fijm%2F1256050746</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">N. Lygeros and O. Rozier (2010). "A new solution for the equation 
 
 
 
 τ
 (
 p
 )
 ≡
 0
 
 
 (
 mod
 
 p
 )
 
 
 
 {\displaystyle \tau (p)\equiv 0{\pmod {p}}}
 
" (PDF). Journal of Integer Sequences. 13: Article 10.7.4. <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Lygeros/lygeros5.pdf" target="_blank">https://cs.uwaterloo.ca/journals/JIS/VOL13/Lygeros/lygeros5.pdf</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
</ol>

Ramanujan tau function open-in-new

Ramanujan tau function