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Rafael-SOWNet merged 1 commit into from
Oct 28, 2026
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The Möbius, divisor and prime-counting functions #2188

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22 changes: 19 additions & 3 deletions BREAKING-CHANGES.md
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Expand Up @@ -440,8 +440,7 @@ the right shape, written out or once its operations are carried out, throws `Inv
**Not silent: these names followed by a bracket now throw `UnrecognizedFunctionParseException` where they were read as
products.** An unknown name followed by a bracket is a product, so `zeta(2)` was `zeta * 2` and simplified to
`2 * zeta`. The long names of the functions the library is to have are refused until each is implemented, as `trunc`
is: `zeta`, `riemannzeta`, `hurwitzzeta`, `lambertw`, `productlog`, `mobius`, `moebiusmu`, `divisorsigma`, `primenu`,
`primeomega`, `primepi`, `harmonicnumber`, `catalannumber`, `multinomial`, `fallingfactorial`, `factorialpower`,
is: `zeta`, `riemannzeta`, `hurwitzzeta`, `lambertw`, `productlog`, `harmonicnumber`, `catalannumber`, `multinomial`, `fallingfactorial`, `factorialpower`,
`risingfactorial`, `forwarddifference`, `laplacian`, `jacobian` and `hessian`. Names that will not be functions,
because what they mean differs between sources or is already written exactly, are refused with the spelling to use:
`heaviside`, `unitstep`, `isprime`, `pochhammer`, `doublefactorial`, `factorial2`, `fractionalpart` and `quotient`.
Expand All @@ -454,7 +453,24 @@ Each name without a bracket is still a variable, and short names such as `beta`
| `"zeta(2)".ToEntity().Simplify()` | `2 * zeta` | throws `UnrecognizedFunctionParseException` |
| `"isprime(7)".ToEntity()` | `isprime * 7` | throws `UnrecognizedFunctionParseException`: write `n in PP` |
| `"heaviside(x)".ToEntity()` | `heaviside * x` | throws `UnrecognizedFunctionParseException`: write `iverson(x > 0)`, `iverson(x >= 0)` or `(1 + sgn(x))/2` |
| `"divisorsigma(k, n)".ToEntity()` | throws `UnhandledParseException` | throws `UnrecognizedFunctionParseException` |
| `"multinomial(a, b, c)".ToEntity()` | throws `UnhandledParseException` | throws `UnrecognizedFunctionParseException` |

### The Möbius, divisor and prime-counting functions

**Not silent: `mobius(n)`, `divisorsigma(k, n)`, `primenu(n)`, `primeomega(n)` and `primepi(x)` are now functions, where
they were read as products or, with two arguments, were a parse error.** `mobius` (also `moebiusmu`) is the Möbius
function μ, `divisorsigma(k, n)` the sum of the `k`-th powers of the divisors of `n`, and `primenu` and `primeomega` the
number of prime factors without and with multiplicity, ω and Ω. Each is defined on the positive integers and is `NaN`
elsewhere. `primepi(x)` counts the primes at most the real number `x`
([#2182](https://github.com/asc-community/AngouriMath/issues/2182)).

| Input | Was (2.6.0) | Now |
|---|---|---|
| `"mobius(6)".ToEntity().Simplify()` | `6 * mobius` | `1` |
| `"divisorsigma(1, 6)".ToEntity().Simplify()` | throws `UnhandledParseException` | `12` |
| `"primeomega(12)".ToEntity().Simplify()` | `12 * primeomega` | `3` |
| `"primepi(10)".ToEntity().Evaled` | `primepi * 10` | `4` |
| `"mobius(0)".ToEntity().Simplify()` | `0` | `NaN` |
### A subscript may have a subscript

**Not silent: what was a parse error is a name.** A name took at most one `_`, so `x_1_2` and `X_a_beta` were lexer
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3 changes: 3 additions & 0 deletions Sources/.editorconfig
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Expand Up @@ -805,3 +805,6 @@ file_header_template=\nCopyright (c) 2019-2026 Angouri.\nAngouriMath is licensed

[Tests/UnitTests/Common/SimplifyOverASymbolicCommonDenominatorTest.cs]
file_header_template=\nCopyright (c) 2019-2026 Angouri.\nAngouriMath is licensed under MIT.\nDetails: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.\nWebsite: https://am.angouri.org.\n

[Tests/UnitTests/Convenience/NumberTheoryFunctionsTest.cs]
file_header_template=\nCopyright (c) 2019-2026 Angouri.\nAngouriMath is licensed under MIT.\nDetails: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.\nWebsite: https://am.angouri.org.\n
20 changes: 20 additions & 0 deletions Sources/AngouriMath/Convenience/MathS.cs
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Expand Up @@ -50,6 +50,26 @@ public static class NumberTheory
/// </example>
public static Entity Phi(Entity integer) => new Phif(integer);

/// <summary>The Möbius function <c>μ(n)</c> of a positive integer</summary>
/// <remarks>https://github.com/asc-community/AngouriMath/issues/2182</remarks>
public static Entity Mobius(Entity n) => new Mobiusf(n);

/// <summary>The divisor function <c>σ_k(n)</c>: the sum of the <c>k</c>-th powers of the divisors of <c>n</c></summary>
/// <remarks>https://github.com/asc-community/AngouriMath/issues/2182</remarks>
public static Entity DivisorSigma(Entity order, Entity n) => new DivisorSigmaf(order, n);

/// <summary>The number of distinct prime factors <c>ω(n)</c> of a positive integer</summary>
/// <remarks>https://github.com/asc-community/AngouriMath/issues/2182</remarks>
public static Entity PrimeNu(Entity n) => new PrimeNuf(n);

/// <summary>The number of prime factors <c>Ω(n)</c> of a positive integer, with multiplicity</summary>
/// <remarks>https://github.com/asc-community/AngouriMath/issues/2182</remarks>
public static Entity PrimeOmega(Entity n) => new PrimeOmegaf(n);

/// <summary>The prime-counting function <c>π(x)</c>: how many primes are at most <c>x</c></summary>
/// <remarks>https://github.com/asc-community/AngouriMath/issues/2182</remarks>
public static Entity PrimePi(Entity x) => new PrimePif(x);

/// <summary>
/// The <c>n</c>-th prime, <c>prime(n)</c>: <c>prime(1)</c> is <c>2</c> and
/// <c>prime(25)</c> is <c>97</c>. Defined on the positive whole numbers, and a
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12 changes: 6 additions & 6 deletions Sources/AngouriMath/Core/Antlr/AngouriMath.g
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Expand Up @@ -595,6 +595,12 @@ atom returns[Entity value]
| 'phi(' args = function_arguments ')' { Assert("phi", 1, $args.list.Count); $value = MathS.NumberTheory.Phi($args.list[0]); }
| 'prime(' args = function_arguments ')' { Assert("prime", 1, $args.list.Count); $value = MathS.NumberTheory.Prime($args.list[0]); }
| 'valuation(' args = function_arguments ')' { Assert("valuation", 2, $args.list.Count); $value = MathS.NumberTheory.Valuation($args.list[0], $args.list[1]); }
| 'mobius(' args = function_arguments ')' { Assert("mobius", 1, $args.list.Count); $value = MathS.NumberTheory.Mobius($args.list[0]); }
| 'moebiusmu(' args = function_arguments ')' { Assert("moebiusmu", 1, $args.list.Count); $value = MathS.NumberTheory.Mobius($args.list[0]); }
| 'divisorsigma(' args = function_arguments ')' { Assert("divisorsigma", 2, $args.list.Count); $value = MathS.NumberTheory.DivisorSigma($args.list[0], $args.list[1]); }
| 'primenu(' args = function_arguments ')' { Assert("primenu", 1, $args.list.Count); $value = MathS.NumberTheory.PrimeNu($args.list[0]); }
| 'primeomega(' args = function_arguments ')' { Assert("primeomega", 1, $args.list.Count); $value = MathS.NumberTheory.PrimeOmega($args.list[0]); }
| 'primepi(' args = function_arguments ')' { Assert("primepi", 1, $args.list.Count); $value = MathS.NumberTheory.PrimePi($args.list[0]); }
| 'card(' args = function_arguments ')' { Assert("card", 1, $args.list.Count); $value = MathS.Sets.Card($args.list[0]); }
| 'powerset(' args = function_arguments ')' { Assert("powerset", 1, $args.list.Count); $value = MathS.Sets.PowerSet($args.list[0]); }
| 'union(' args = function_arguments ')' { Assert("union", 2, $args.list.Count); $value = $args.list[1] is Entity.Set.Inf { Element: Variable } unionRange ? MathS.Sets.IndexedUnion($args.list[0], unionRange.Element, unionRange.SupSet) : throw new InvalidArgumentParseException("union expects its second argument to say which name ranges over which set, as in union(A_i, i in I)"); }
Expand Down Expand Up @@ -681,12 +687,6 @@ atom returns[Entity value]
| 'hurwitzzeta(' args = function_arguments ')' { $value = NotImplementedFunction("hurwitzzeta", "Hurwitz zeta function yet"); }
| 'lambertw(' args = function_arguments ')' { $value = NotImplementedFunction("lambertw", "Lambert W function yet"); }
| 'productlog(' args = function_arguments ')' { $value = NotImplementedFunction("productlog", "Lambert W function yet"); }
| 'mobius(' args = function_arguments ')' { $value = NotImplementedFunction("mobius", "Mobius function yet"); }
| 'moebiusmu(' args = function_arguments ')' { $value = NotImplementedFunction("moebiusmu", "Mobius function yet"); }
| 'divisorsigma(' args = function_arguments ')' { $value = NotImplementedFunction("divisorsigma", "divisor function yet"); }
| 'primenu(' args = function_arguments ')' { $value = NotImplementedFunction("primenu", "count of distinct prime factors yet"); }
| 'primeomega(' args = function_arguments ')' { $value = NotImplementedFunction("primeomega", "count of prime factors with multiplicity yet"); }
| 'primepi(' args = function_arguments ')' { $value = NotImplementedFunction("primepi", "prime-counting function yet"); }
| 'harmonicnumber(' args = function_arguments ')' { $value = NotImplementedFunction("harmonicnumber", "harmonic numbers yet"); }
| 'catalannumber(' args = function_arguments ')' { $value = NotImplementedFunction("catalannumber", "Catalan numbers yet"); }
| 'multinomial(' args = function_arguments ')' { $value = NotImplementedFunction("multinomial", "multinomial coefficients yet"); }
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12 changes: 6 additions & 6 deletions Sources/AngouriMath/Core/Antlr/AngouriMath.interp
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Expand Up @@ -150,6 +150,12 @@ null
'phi('
'prime('
'valuation('
'mobius('
'moebiusmu('
'divisorsigma('
'primenu('
'primeomega('
'primepi('
'card('
'powerset('
'union('
Expand Down Expand Up @@ -201,12 +207,6 @@ null
'hurwitzzeta('
'lambertw('
'productlog('
'mobius('
'moebiusmu('
'divisorsigma('
'primenu('
'primeomega('
'primepi('
'harmonicnumber('
'catalannumber('
'multinomial('
Expand Down
114 changes: 57 additions & 57 deletions Sources/AngouriMath/Core/Antlr/AngouriMath.tokens
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Expand Up @@ -385,63 +385,63 @@ WS=237
'phi('=148
'prime('=149
'valuation('=150
'card('=151
'powerset('=152
'union('=153
'intersection('=154
'complement('=155
'image('=156
'preimage('=157
'floor('=158
'ceil('=159
'ceiling('=160
'round('=161
'min('=162
'max('=163
'argmax('=164
'argmin('=165
'gcd('=166
'lcm('=167
'binomial('=168
'iverson('=169
'aleph('=170
'erf('=171
'erfc('=172
'erfi('=173
'Ei('=174
'li('=175
'polygamma('=176
'digamma('=177
'Si('=178
'Ci('=179
'Shi('=180
'Chi('=181
'det('=182
'transpose('=183
'dot('=184
'cross('=185
'resultant('=186
'conjugate('=187
'conj('=188
'arg('=189
'Re('=190
'Im('=191
're('=192
'real('=193
'im('=194
'imag('=195
'trunc('=196
'zeta('=197
'riemannzeta('=198
'hurwitzzeta('=199
'lambertw('=200
'productlog('=201
'mobius('=202
'moebiusmu('=203
'divisorsigma('=204
'primenu('=205
'primeomega('=206
'primepi('=207
'mobius('=151
'moebiusmu('=152
'divisorsigma('=153
'primenu('=154
'primeomega('=155
'primepi('=156
'card('=157
'powerset('=158
'union('=159
'intersection('=160
'complement('=161
'image('=162
'preimage('=163
'floor('=164
'ceil('=165
'ceiling('=166
'round('=167
'min('=168
'max('=169
'argmax('=170
'argmin('=171
'gcd('=172
'lcm('=173
'binomial('=174
'iverson('=175
'aleph('=176
'erf('=177
'erfc('=178
'erfi('=179
'Ei('=180
'li('=181
'polygamma('=182
'digamma('=183
'Si('=184
'Ci('=185
'Shi('=186
'Chi('=187
'det('=188
'transpose('=189
'dot('=190
'cross('=191
'resultant('=192
'conjugate('=193
'conj('=194
'arg('=195
'Re('=196
'Im('=197
're('=198
'real('=199
'im('=200
'imag('=201
'trunc('=202
'zeta('=203
'riemannzeta('=204
'hurwitzzeta('=205
'lambertw('=206
'productlog('=207
'harmonicnumber('=208
'catalannumber('=209
'multinomial('=210
Expand Down
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