RamanujanTau

Ramanujan's tau function: the coefficients of the discriminant modular form $\Delta(q) = q\prod_{k\ge1}(1-q^k)^{24}$.

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RamanujanTau(n)the coefficient of $q^n$ in $\Delta(q)$

SageMathdelta_qexp(n + 1)[n]

Typed signatures

Type	Package	Applies
(integer) -> integer	number-theory	always

Domain: Number theory

Details

• Multiplicative: $\tau(mn)=\tau(m)\tau(n)$ for coprime $m,n$.
• Computed by truncated power-series multiplication of $\prod(1-q^k)^{24}$, exactly, not from a closed form.

Plot

Examples

Basic #

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In$\mathrm{RamanujanTau}(2)$Out$-24$

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In$\mathrm{RamanujanTau}(5)$Out$4\,830$

Scope #

Listable

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In$\mathrm{RamanujanTau}(\bigl\lbrack1, 2, 3, 4, 5, 6\bigr\rbrack)$Out$\bigl\lbrack1, -24, 252, -1\,472, 4\,830, -6\,048\bigr\rbrack$

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In$\mathrm{RamanujanTau}(12)$Out$-370\,944$

Exact value for a mid-size index; a 15-digit integer

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In$\mathrm{RamanujanTau}(485)$Out$362\,315\,536\,077\,180$

As a true matrix function, giving a lower-triangular result

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In$\mathrm{FullSimplify}(\mathrm{MatrixFunction}(\mathrm{RamanujanTau}, \bigl\lbrack\bigl\lbrack1, 0\bigr\rbrack, \bigl\lbrack-1, 2\bigr\rbrack\bigr\rbrack))$Out$\bigl\lbrack\bigl\lbrack1, 0\bigr\rbrack, \bigl\lbrack25, -24\bigr\rbrack\bigr\rbrack$

Threads over a list of primes

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In$\mathrm{RamanujanTau}(\bigl\lbrack2, 3, 5, 7, 11\bigr\rbrack)$Out$\bigl\lbrack-24, 252, 4\,830, -16\,744, 534\,612\bigr\rbrack$

Threads element-wise over a nested list; $\tau(-1)=0$

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In$\mathrm{RamanujanTau}(\bigl\lbrack\bigl\lbrack1, 0\bigr\rbrack, \bigl\lbrack-1, 2\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack1, 0\bigr\rbrack, \bigl\lbrack0, -24\bigr\rbrack\bigr\rbrack$

$\tau(0)=0$

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In$\mathrm{RamanujanTau}(0)$Out$0$

Properties #

Multiplicative: $\tau(6) = \tau(2)\tau(3)$

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In$\mathrm{RamanujanTau}(2)\mathrm{RamanujanTau}(3)$Out$-6\,048$

Coefficients of $q\prod(1-q^n)^{24}$, with sign changes

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In$\mathrm{RamanujanTau}(1..10)$Out$\bigl\lbrack1, -24, 252, -1\,472, 4\,830, -6\,048, -16\,744, 84\,480, -113\,643, -115\,920\bigr\rbrack$

$\tau(35)=\tau(5)\tau(7)$ since $\gcd(5,7)=1$

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In$\mathrm{RamanujanTau}(35)$Out$-80\,873\,520$

Product equals $\tau(35)$: multiplicative on coprime arguments

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In$\mathrm{RamanujanTau}(5)\mathrm{RamanujanTau}(7)$Out$-80\,873\,520$

The leading coefficient is $1$

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In$\mathrm{RamanujanTau}(1)$Out$1$

Known value · Wolfram MathematicalFunctionData, RamanujanTau

$\tau(3)=252$

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In$\mathrm{RamanujanTau}(3)$Out$252$

Known value · Wolfram MathematicalFunctionData, RamanujanTau

$\tau(4)=\tau(2)^2-2^{11}$

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In$\mathrm{RamanujanTau}(4)$Out$-1\,472$

Known value · Wolfram MathematicalFunctionData, RamanujanTau

$\tau(6)=\tau(2)\tau(3)$

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In$\mathrm{RamanujanTau}(6)$Out$-6\,048$

Known value · Wolfram MathematicalFunctionData, RamanujanTau

Negative at the prime 7

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In$\mathrm{RamanujanTau}(7)$Out$-16\,744$

Known value · Wolfram MathematicalFunctionData, RamanujanTau

Positive again at 8

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In$\mathrm{RamanujanTau}(8)$Out$84\,480$

Known value · Wolfram MathematicalFunctionData, RamanujanTau

$\tau(9)=\tau(3)^2-3^{11}$

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In$\mathrm{RamanujanTau}(9)$Out$-113\,643$

Known value · Wolfram MathematicalFunctionData, RamanujanTau

$\tau(10)=\tau(2)\tau(5)$

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In$\mathrm{RamanujanTau}(10)$Out$-115\,920$

Known value · Wolfram MathematicalFunctionData, RamanujanTau

Neat examples #

A sparse sum of tau values that cancels exactly to $0$

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In$\mathrm{RamanujanTau}(6)+\mathrm{RamanujanTau}(14)+\mathrm{RamanujanTau}(29)+\mathrm{RamanujanTau}(41)+\mathrm{RamanujanTau}(42)+\mathrm{RamanujanTau}(44)+\mathrm{RamanujanTau}(48)$Out$0$

About

Examples	22
Checked against	Wolfram Language 22 agreeSageMath 19 agree1 error
Crosswalk	Wolfram LanguageSageMath

Implementation

mappedsagepackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

SymPy has no Ramanujan tau function or weight-12 cusp form (no ramanujan_tau, no q-expansion of the modular discriminant); only divisor_sigma and relatives exist.

See also: PartitionsQ