Ramanujan’sτ -function

MichaelBennett

Values of the Ramanujan τ -function

Michael Bennett (joint with Adela Gherga, Vandita Pateland Samir Siksek)

University of British Columbia (and University of Manchester, WarwickUniversity)

Fields Institute : November 2020

Ramanujan’sτ -function

MichaelBennett The Ramanujan τ -function

The Ramanujan τ -function τ(n) is defined via the expansion

∆(z) = q

∞∏n=1

(1− qn)24 =

∞∑n=1

τ(n)qn, q = e2πiz.

Explicitly, we have

∆(z) = q − 24q2 + 252q3 − 1472q4 + 4830q5 − 6048q6 + · · ·

Ramanujan’sτ -function

MichaelBennett

The Ramanujan τ -function : multiplicativity

It was conjectured by Ramanujan and proved by Mordell thatτ(n) is a multiplicative function, i.e. that

τ(n1n2) = τ(n1)τ(n2),

for all coprime pairs of positive integers n1 and n2.

Ramanujan’sτ -function

MichaelBennett

The Ramanujan τ -function : parity

Further, we have

∞∑n=1

τ(n)qn ≡ q∞∏n=1

(1 + q8n)3 ≡ q∞∏n=1

(1− q8n)(1 + q8n)2

and so∞∑n=1

τ(n)qn ≡∞∑m=0

q(2m+1)2 mod 2,

via Jacobi’s triple product formula, whence τ(n) is oddprecisely when n is an odd square and, in particular, τ(p) iseven for every prime p.

Ramanujan’sτ -function

MichaelBennett

The Ramanujan τ -function : arithmetic

Amongst the many open questions about the possible values ofτ(n), the most notorious is an old conjecture of Lehmer to theeffect that τ(n) never vanishes.

There are many papers in the literature proving results aboutthe arithmetic nature of τ(n) under various hypotheses : onaverage, for most n, subject to GRH, etc. See e.g. work ofMurty-Murty, Garavev, Shparlinski, Konyagin, etc.

Ramanujan’sτ -function

MichaelBennett

The Ramanujan τ -function : upper bounds

In terms of the size of values of τ , one has the upper bound ofDeligne (originally conjectured by Ramanujan) :

|τ(p)| ≤ 2 · p11/2,

valid for prime p.

Ramanujan’sτ -function

MichaelBennett The Ramanujan τ -function : lower bounds

In the other direction, Atkin and Serre conjectured (as astrengthening of Lehmer’s conjecture) that, for ε > 0,

|τ(p)| �ε p9/2−ε,

so that, in particular, given a fixed integer a, there are at mostfinitely many primes p for which

τ(p) = a.

Ramanujan’sτ -function

MichaelBennett

The Ramanujan τ -function : odd values

While this problem remains open, in the special case where theinteger a is odd, Murty, Murty and Shorey (1987) proved thatthe equation

τ(n) = a,

has at most finitely many solutions in integers n (note that, inthis case, n is necessarily an odd square). More precisely, theydemonstrated the existence of an effectively computablepositive constant c such that if τ(n) is odd, then

|τ(n)| > (log(n))c.

Ramanujan’sτ -function

MichaelBennett

The Ramanujan τ -function : odd values continued

A number of recent papers have treated the problem ofexplicitly demonstrating that the equation

τ(n) = a

has, in fact, no solutions, for various odd values of a, including

a = ±1 (Lygeros and Rozier)

|a| < 100 an odd prime (Balakrishnan, Craig and Ono,Balakrishnan, Craig, Ono and Tsai, Dembner and Jain)

|a| < 100 an odd integer (Hanada and Madhukara)

Ramanujan’sτ -function

MichaelBennett

The Ramanujan τ -function : odd values continued

A number of recent papers have treated the problem ofexplicitly demonstrating that the equation

τ(n) = a

has, in fact, no solutions, for various odd values of a, including

a = ±1 (Lygeros and Rozier)

|a| < 100 an odd prime (Balakrishnan, Craig and Ono,Balakrishnan, Craig, Ono and Tsai, Dembner and Jain)

|a| < 100 an odd integer (Hanada and Madhukara)

Ramanujan’sτ -function

MichaelBennett

The Ramanujan τ -function : odd values continued

A number of recent papers have treated the problem ofexplicitly demonstrating that the equation

τ(n) = a

has, in fact, no solutions, for various odd values of a, including

a = ±1 (Lygeros and Rozier)

|a| < 100 an odd prime (Balakrishnan, Craig and Ono,Balakrishnan, Craig, Ono and Tsai, Dembner and Jain)

|a| < 100 an odd integer (Hanada and Madhukara)

Ramanujan’sτ -function

MichaelBennett

The Ramanujan τ -function : odd values continued

A number of recent papers have treated the problem ofexplicitly demonstrating that the equation

τ(n) = a

has, in fact, no solutions, for various odd values of a, including

a = ±1 (Lygeros and Rozier)

|a| < 100 an odd prime (Balakrishnan, Craig and Ono,Balakrishnan, Craig, Ono and Tsai, Dembner and Jain)

|a| < 100 an odd integer (Hanada and Madhukara)

Ramanujan’sτ -function

MichaelBennett

A non-Archimidean analogue of Murty-Murty-Shorey

Theorem (B., Gherga, Patel, Siksek 2020)

There exists an effectively computable constant κ > 0 suchthat if τ(n) is odd, with n ≥ 25, then either

P (τ(n)) > κlog log log n

log log log log n,

or there exists a prime p | n for which τ(p) = 0.

Ramanujan’sτ -function

MichaelBennett

A non-Archimidean analogue of Murty-Murty-Shorey

Recall that a powerful number (also known as squarefull or2-full) is defined to be an integer n with the property that if aprime p | n, then necessarily p2 | n.

Equivalently, we can write such an integer as n = a2b3, wherea and b are integers. Our techniques actually show thefollowing (from which the preceding Theorem is an immediateconsequence).

Ramanujan’sτ -function

MichaelBennett

A non-Archimidean analogue of Murty-Murty-Shorey

Theorem (B., Gherga, Patel, Siksek 2020)

We havelimn→∞

P (τ(n)) =∞,

where the limit is taken over powerful numbers n for whichτ(p) 6= 0 for each p | n. More precisely, there exists aneffectively computable constant κ > 0 such that if n ≥ 25 ispowerful, either

P (τ(n)) > κlog log log n

log log log log n,

or there exists a prime p | n for which τ(p) = 0.

Ramanujan’sτ -function

MichaelBennett

If Lehmer’s Conjecture is false.....

The restriction that n have no prime divisors p for whichτ(p) = 0 is, in fact, necessary if one wishes to obtain a lowerbound upon P (τ(n)) that tends to ∞ with n. Indeed, one mayobserve that, if τ(p) = 0, then

P(τ(p2k)

)= P

((−1)kp11k

)= p

is bounded, independent of k. While Lehmer’s conjectureremains unproven, we do know that if there is a prime p forwhich τ(p) = 0, then

p > 816212624008487344127999,

by work of Derickx, van Hoeij and Zeng.

Ramanujan’sτ -function

MichaelBennett

What we really prove....

We will actually show that, given m ≥ 2 and prime p for whichτ(p) 6= 0,

P (τ(pm))� log log(pm), for m ∈ {2, 3, 4, 5}, say,

and

P (τ(pm))� log log(pm)

log log log(pm),

more generally, where the implied constants are absolute.

Ramanujan’sτ -function

MichaelBennett

Explicit results

To demonstrate that these results and the techniquesunderlying them are somewhat practical, we prove the followingcomputational “coda” :

Theorem (B., Gherga, Patel, Siksek 2020)

If n is a powerful positive integer, then either n = 8, where wehave

τ(8) = 29 · 3 · 5 · 11,

orP (τ(n)) ≥ 13.

Ramanujan’sτ -function

MichaelBennett

Explicit results

Corollary (B., Gherga, Patel, Siksek 2020)

If n is a positive integer for which τ(n) is odd, then

P (τ(n)) ≥ 13.

In other words, the equation

τ(n) = ±3α5β7γ11δ

has no solutions in integers n ≥ 2 and α, β, γ, δ ≥ 0.

Ramanujan’sτ -function

MichaelBennett

The Ramanujan τ -function : prime values

It is conjectured that |τ(n)| takes on infinitely many primevalues, the smallest of which corresponds to

τ(2512) = −80561663527802406257321747.

Our arguments enable us to eliminate the possibility of powersof small primes arising as values of τ . By way of example, wehave the following.

Theorem (B., Gherga, Patel, Siksek 2020)

The equationτ(n) = ±qα

has no solutions in prime q with 3 ≤ q < 100 and α, n ∈ Z.

Ramanujan’sτ -function

MichaelBennett

Our techniques

Our proofs rely upon two quite different approaches. The first,to derive inequalities of the shape

P (τ(n)) > κlog log log n

log log log log n,

proceeds in the same way as the original work ofMurty-Murty-Shorey, appealing to bounds for linear forms inlogarithms (though we necessarily must use both bounds forcomplex and p-adic logarithms). The second (for our explicitresults), uses Frey curves and the modularity of associatedGalois representations.

Ramanujan’sτ -function

MichaelBennett

Generalities: coefficients of cuspidal newforms

It is worth observing that the techniques we employ are readilyextended to treat more generally coefficients λf (n) of cuspidalnewforms of (even) weight k ≥ 4 for Γ0(N), with trivialcharacter and λf (p) even for suitably large prime p; our resultscorrespond to the case of ∆(z), where k = 12 and N = 1. Werestrict our attention to this latter case for simplicity.

Ramanujan’sτ -function

MichaelBennett

Lucas sequences

A Lucas pair is a pair (α, β) of algebraic numbers such thatα+ β and αβ are non-zero coprime rational integers, and α/βis not a root of unity. In particular, associated to the Lucas pair(α, β) is a characteristic polynomial

X2 − (α+ β)X + αβ ∈ Z[X].

This polynomial has discriminant D = (α− β)2 ∈ Z \ {0}.Given a Lucas pair (α, β), the corresponding Lucas sequenceis given by

un =αn − βn

α− β, n = 0, 1, 2, . . . .

Ramanujan’sτ -function

MichaelBennett

Lucas sequences : primitive divisors

Let (α, β) be a Lucas pair. A prime ` is a primitive divisor ofthe n-th term of the corresponding Lucas sequence if ` dividesun but ` fails to divide (α− β)2 · u1u2 . . . un−1.

We shall make essential use of the celebrated Primitive DivisorTheorem of Bilu, Hanrot and Voutier.

Theorem (Bilu, Hanrot and Voutier)

Let (α, β) be a Lucas pair. If n ≥ 5 and n 6= 6 then un has aprimitive divisor.

Ramanujan’sτ -function

MichaelBennett

Lucas sequences : rank of apparition

Let ` be a prime. The smallest positive integer m such that` | um is called the rank of apparition of `; we denote this bym`.

The important result is that, if m` is finite, then either m` = `,or m` | `± 1.

Ramanujan’sτ -function

MichaelBennett From Ramanujan τ to Lucas sequences

Let us fix a prime p and consider the sequence{1, τ(p), τ(p2), τ(p3), . . .

}.

We will associate to this a Lucas pair and a correspondingLucas sequence. Our starting point is the identity

τ(pm) = τ(p)τ(pm−1)− p11τ(pm−2),

valid for all integer m ≥ 2. Once again, this was conjectured byRamanujan and proved by Mordell.

Ramanujan’sτ -function

MichaelBennett

The degenerate case

If Lehmer’s conjecture fails and there exists a prime p for whichτ(p) = 0, then our recursion implies that

τ(pn) = 0 for all odd n, and

τ(pn) = (−1)n/2p11n/2 for even n.

In particular, this does not correspond to a Lucas sequence.

Ramanujan’sτ -function

MichaelBennett

The degenerate case

If Lehmer’s conjecture fails and there exists a prime p for whichτ(p) = 0, then our recursion implies that

τ(pn) = 0 for all odd n, and

τ(pn) = (−1)n/2p11n/2 for even n.

In particular, this does not correspond to a Lucas sequence.

Ramanujan’sτ -function

MichaelBennett

The degenerate case

If Lehmer’s conjecture fails and there exists a prime p for whichτ(p) = 0, then our recursion implies that

τ(pn) = 0 for all odd n, and

τ(pn) = (−1)n/2p11n/2 for even n.

In particular, this does not correspond to a Lucas sequence.

Ramanujan’sτ -function

MichaelBennett

Otherwise

If we have that pk | τ(p), we may conclude that k ≤ 5, viaDeligne’s bounds. Supposing that pr ‖ τ(p), we write

un =τ(pn−1)

pr(n−1), n ≥ 1.

We may show that {un} is a Lucas sequence with

α =α′

pr, β =

β′

pr,

for α′ and β′ the roots of the quadratic equation

X2 − τ(p)X + p11 = 0.

Ramanujan’sτ -function

MichaelBennett

An aside : p | τ(p)

We note that p | τ(p) for

p = 2, 3, 5, 7, 2411, 7758337633, . . .

We expect that p | τ(p) for infinitely many primes p; see workof Lygeros and Rozier for a discussion of this problem andrelated computations.

Ramanujan’sτ -function

MichaelBennett Towards the proofs

We haveτ(p2) = τ2(p)− p11,

τ(p3) = τ(p)(τ2(p)− 2p11)

andτ(p4) = p22 − 3p11τ2(p) + τ4(p).

We can rewrite this last equation as

4τ(p4) =(2τ2(p)− 3p11

)2 − 5p22.

Ramanujan’sτ -function

MichaelBennett

τ(pm),m ∈ {2, 3, 4, 5}

In each case, we have that

P (τ(pm)) ≥ P (Y 2 −DX11),

for D ≤ 5, X and Y integers.

Appealing to bounds for linear forms in complex and q-adiclogarithms (say due to Bugeaud and Gyory), we find that

P (τ(pm)) ≥ c1 log log(pm), for m ∈ {2, 3, 4, 5},

for some positive constant c1.

Ramanujan’sτ -function

MichaelBennett

τ(pm),m ∈ {2, 3, 4, 5}

In each case, we have that

P (τ(pm)) ≥ P (Y 2 −DX11),

for D ≤ 5, X and Y integers.

Appealing to bounds for linear forms in complex and q-adiclogarithms (say due to Bugeaud and Gyory), we find that

P (τ(pm)) ≥ c1 log log(pm), for m ∈ {2, 3, 4, 5},

for some positive constant c1.

Ramanujan’sτ -function

MichaelBennett

τ(pm),m ≥ 6

More generally, for m ≥ 6, we can write

τ(pm) = τ δ(p)Fm(p11, τ2(p)),

where δ ∈ {0, 1}, δ ≡ m mod 2 and Fm(X,Y ) is a binaryform with integer coefficients and degree [m/2].Explicitly, expanding

1

1−√Y T +XT 2

=

∞∑m=0

Y δ/2 Fm(X,Y )Tm,

we find, for example, that

F6(X,Y ) = −X3 + 6X2Y − 5XY 2 + Y 3.

Ramanujan’sτ -function

MichaelBennett

τ(pm),m ≥ 6

Note that from the fact that the Lucas sequences {un} aredivisibility sequences, we have

Fm(X,Y ) | Fn(X,Y ) if n ≡ −1 mod m+ 1,

whence many of these forms are, in fact, reducible. Explicitly,we can write

Fm(X,Y ) =

[m/2]∑j=1

(−1)j(m− jj

)Y [m/2]−jXj

and so

Fm(X,Y ) =

[m/2]∏j=1

(Y − 4X cos2

(πj

m+ 1

)).

Ramanujan’sτ -function

MichaelBennett τ(pm),m ≥ 6

We observe that

Fm(X,Y ) = Gm(X,Y − 2X), (1)

where

Gm(X,Y ) =

[m/2]∏j=1

(Y − 2X cos

(2πj

m+ 1

)). (2)

Ramanujan’sτ -function

MichaelBennett

τ(pm),m ≥ 6 : Theorem 1

We write

τ(pm) = prmum+1 = prm · um+1

uqα· uq

α

uqα−1

· uqα−1 ,

where qα is the largest prime power divisor of m+ 1.

After ruling out some small cases, we have that the termuqα

uqα−1

is an irreducible binary form F (X,Y ) in X = p11−2r andY = p−2rτ2(p), of degree φ(qα)/2 ≥ 3, where φ denotes theEuler phi-function. Explicitly, we have

F (X,Y ) =

φ(qα)/2∏j=1

gcd(j,q)=1

(Y − 4X cos2

(πj

qα

)).

Ramanujan’sτ -function

MichaelBennett

τ(pm),m ≥ 6 : Theorem 1

We write

τ(pm) = prmum+1 = prm · um+1

uqα· uq

α

uqα−1

· uqα−1 ,

where qα is the largest prime power divisor of m+ 1.

After ruling out some small cases, we have that the termuqα

uqα−1

is an irreducible binary form F (X,Y ) in X = p11−2r andY = p−2rτ2(p), of degree φ(qα)/2 ≥ 3, where φ denotes theEuler phi-function. Explicitly, we have

F (X,Y ) =

φ(qα)/2∏j=1

gcd(j,q)=1

(Y − 4X cos2

(πj

qα

)).

Ramanujan’sτ -function

MichaelBennett

Thue-Mahler equations

We haveP (F (X,Y )) ≤ P (τ(pm)).

Putting this all together (using bounds for solutions ofThue-Mahler equations due to Bugeaud and Gyory, we find that

P (τ(pm))� log log(pm)

log log log(pm).

If we choose pm to be the largest prime power divisor of ourpowerful number n, then pm � log n, whereby

P (τ(n))� log log log n

log log log log n,

as claimed.

Ramanujan’sτ -function

MichaelBennett

P (τ(pm)) : explicit results

To deal with, for example, the equation

τ(pm) = ±2α3β5γ7δ11ω,

our approach is as follows

Use Ramanujan’s congruences for τ(n) modulo powers of2, 3, 5, 7 and 23 to reduce the possibilities for theexponents

Appeal to the Primitive Divisor Theorem to restrict m

Solve the Thue-Mahler equations associated with m ≥ 6

Treat the equations corresponding to τ(pm), m ≤ 5.

Ramanujan’sτ -function

MichaelBennett

P (τ(pm)) : explicit results

To deal with, for example, the equation

τ(pm) = ±2α3β5γ7δ11ω,

our approach is as follows

Use Ramanujan’s congruences for τ(n) modulo powers of2, 3, 5, 7 and 23 to reduce the possibilities for theexponents

Appeal to the Primitive Divisor Theorem to restrict m

Solve the Thue-Mahler equations associated with m ≥ 6

Treat the equations corresponding to τ(pm), m ≤ 5.

Ramanujan’sτ -function

MichaelBennett

P (τ(pm)) : explicit results

To deal with, for example, the equation

τ(pm) = ±2α3β5γ7δ11ω,

our approach is as follows

Use Ramanujan’s congruences for τ(n) modulo powers of2, 3, 5, 7 and 23 to reduce the possibilities for theexponents

Appeal to the Primitive Divisor Theorem to restrict m

Solve the Thue-Mahler equations associated with m ≥ 6

Treat the equations corresponding to τ(pm), m ≤ 5.

Ramanujan’sτ -function

MichaelBennett

P (τ(pm)) : explicit results

To deal with, for example, the equation

τ(pm) = ±2α3β5γ7δ11ω,

our approach is as follows

Use Ramanujan’s congruences for τ(n) modulo powers of2, 3, 5, 7 and 23 to reduce the possibilities for theexponents

Appeal to the Primitive Divisor Theorem to restrict m

Solve the Thue-Mahler equations associated with m ≥ 6

Treat the equations corresponding to τ(pm), m ≤ 5.

Ramanujan’sτ -function

MichaelBennett P (τ(p2)) :Frey curves

If we suppose that

τ(p2) = ±2α3β5γ7δ11ω,

for p odd, then we necessarily have α = γ = δ = 0 andβ ∈ {0, 1}.

We consider the Frey curves{Ep : Y 2 = X(X2 + 2τ(p)X + τ(p)2 − p11) if p ≡ 1 mod 4,

Ep : Y 2 = X(X2 + 2τ(p)X + p11) if p ≡ 3 mod 4.

Ramanujan’sτ -function

MichaelBennett P (τ(p2)) :Frey curves

If we suppose that

τ(p2) = ±2α3β5γ7δ11ω,

for p odd, then we necessarily have α = γ = δ = 0 andβ ∈ {0, 1}. We consider the Frey curves{Ep : Y 2 = X(X2 + 2τ(p)X + τ(p)2 − p11) if p ≡ 1 mod 4,

Ep : Y 2 = X(X2 + 2τ(p)X + p11) if p ≡ 3 mod 4.

Ramanujan’sτ -function

MichaelBennett

P (τ(p2)) :Frey curves

It follows from the modularity theorem and Ribet’s levellowering theorem that there is a normalized newform

f = q +

∞∑n=1

cnqn

of weight 2 and level 25 · 3δ111δ2 and a prime $ | 11 in theintegers of K = Q(c1, c2, . . . ) so that

ρEp,11 ∼ ρf,$.

In particular, we have

NormK/Q(a`(Ep)− c`) ≡ 0 mod 11.

Ramanujan’sτ -function

MichaelBennett

P (τ(p2)) :Frey curves

Recall that we had

τ2(p)− p11 = ±3β11ω, β ∈ {0, 1}.

Sieving using

NormK/Q(a`(Ep)− c`) ≡ 0 mod 11,

for ` - 66, ` < 200, we find that either

β = 1, ω = 0 or β = 0, 11 | ω.

Ramanujan’sτ -function

MichaelBennett P (τ(p2)) :Frey curves, final steps

We eliminate the first of these cases by reducing it to the Thueequation

U11 + 22U10V + 165U9V 2 + 990U8V 3 + 2970U7V 4

+ 8316U6V 5 + 12474U5V 6 + 17820U4V 7

+ 13365U3V 8 + 8910U2V 9 + 2673UV 10 + 486V 11 = 1.

The Magma Thue equation solver gives that the only solution is(U, V ) = (1, 0), corresponding to p = U2 − 3V 2 = 1, acontradiction.

Ramanujan’sτ -function

MichaelBennett

P (τ(p2)) :Frey curves, final steps

Writing ω = 11t, The second case reduces to

p11 ± (11t)11 = τ2(p),

which, via work of Darmon and Merel, has only trivial solutions.

Ramanujan’sτ -function

MichaelBennett P (τ(p4)) :Frey curves

We have Ep of the form Y 2 = F (X) for F (X) one of

X(X2 + (3p11 − 2τ(p)2)X + τ(p)4 − 3p11τ(p)2 + p22)

or

X(X2 + (2τ(p)2 − 3p11)X + τ(p)4 − 3p11τ(p)2 + p22).

The proof is similar, if somewhat more involved.

Ramanujan’sτ -function

MichaelBennett P (τ(p4)) :Frey curves

We have Ep of the form Y 2 = F (X) for F (X) one of

X(X2 + (3p11 − 2τ(p)2)X + τ(p)4 − 3p11τ(p)2 + p22)

or

X(X2 + (2τ(p)2 − 3p11)X + τ(p)4 − 3p11τ(p)2 + p22).

The proof is similar, if somewhat more involved.

Ramanujan’sτ -function

MichaelBennett

Some references : page 1

J. S. Balakrishnan, W. Craig and K. Ono,Variations of Lehmer’s conjecture for Ramanujan’stau-function,J. Number Theory, to appear.

J. S. Balakrishnan, W. Craig, K. Ono and W.-L. Tsai,Variants of Lehmer’s speculation for newforms,submitted for publication.

Y. Bilu, G. Hanrot and P. Voutier,Existence of primitive divisors of Luca and Lehmernumbers,J. Reine Angew. Math. 539 (2001), 75–122.

Ramanujan’sτ -function

MichaelBennett

Some references : page 2

P. Deligne,La conjecture de Weil I,Inst. Hautes Etudes Sci. Publ. Math. 43 (1974), 273–307.

S. Dembner and V. Jain,Hyperelliptic curves and newform coefficients,submitted for publication.

M. Derickx, M. Van Hoeij and Jinxiang Zeng,Computing Galois representations and equations formodular curves XH(`),arXiv:1312.6819v2.

Ramanujan’sτ -function

MichaelBennett

Some references : page 3

M. Hanada and R. Madhukara,Fourier coefficients of level 1 Hecke eigenforms,submitted for publication.

D. H. Lehmer,The vanishing of Ramaujan’s function τ(n),Duke Math. J. 14 (1947), 429–433.

F. Luca, S. Mabaso and P. Stanica,On the prime factors of the iterates of the Ramanujanτ -function,Proc. Edinburgh Math. Soc., to appear.

Ramanujan’sτ -function

MichaelBennett Some references : page 4

F. Luca and I. E. Shparlinski,Arithmetic properties of the Ramanujan function,Proc. Indian Acad. Sci. (Math. Sci.) 116 (2006), 1–8.

N. Lygeros and O. Rozier,A new solution to the equation τ(p) ≡ 0 mod p,J. Integer Sequences 13 (2010), Article 10.7.4.

N. Lygeros and O. Rozier,Odd prime values of the Ramanujan tau function,Ramanujan Journal 32 (2013), 269–280.

Ramanujan’sτ -function

MichaelBennett

Some references : page 5

L. J. Mordell,On Mr. Ramanujan’s empirical expansions of modularfunctions,Proc. Cambridge Phil. Soc. 19 (1917), 117–124.

M. Ram Murty and V. Kumar Murty,Odd values of Fourier coefficients of certain modular forms,

Int. J. Number Theory 3 (2007), 455-470.

Ramanujan’sτ -function

MichaelBennett

Some references : page 6

M. Ram Murty, V. Kumar Murty and T. N. Shorey,Odd values of the Ramanujan τ -function,Bulletin Soc. Math. France 115 (1987), 391–395.

S. Ramanujan,On certain arithmetical functions,Trans. Camb. Philos. Soc. 22 (1916), 159–184.