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The Möbius, divisor and prime-counting functions - #2188
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mobius(n), divisorsigma(k, n), primenu(n), primeomega(n) and primepi(x) were reserved by name in
built by MathS.NumberTheory.Mobius, DivisorSigma, PrimeNu, PrimeOmega and PrimePi.
mobius is the Möbius function: 0 where the square of a prime divides n, (-1)^k for a product of k
distinct primes, and 1 at 1. moebiusmu, Mathematica's name for it, reads as the same function.
divisorsigma(k, n) is sigma_k(n), the order first as the notation and Mathematica's
DivisorSigma[k, n] write it. It is the product over the prime powers p^a in n of
1 + p^k + ... + p^(ak), exact for a rational order and an expression for any other number:
divisorsigma(1/2, 4) is 3 + sqrt(2). primenu and primeomega are omega and Omega, the number of
prime factors without and with multiplicity. The four are defined on the positive integers, NaN at
any other number, and left as written for an expression.
The factorisation is trial division up to a million. That settles every number up to 10^12, and a
larger one whose part left after the small primes is below 10^12; anything else is left as written,
and 2^100 is factorised at once. primepi(x) counts the primes up to floor(x) by a sieve over the odd
numbers, for a real x up to ten million (94 ms there), and is NaN off the real line.
They print by name, draw as \operatorname{\mu}(n), \operatorname{\sigma}_k(n),
\operatorname{\omega}(n), \operatorname{\Omega}(n) and \operatorname{\pi}(x), so that a Greek
letter is not read back as a variable times a bracket, and export to SymPy's mobius,
divisor_sigma(n, k), primenu, primeomega and primepi. Their codomains are the integers, and the
non-negative integers for the three counts. The MCP server's syntax document and the syntax page
name them, and the native smoke test evaluates them.
Part of #2182.
Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
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The Kernel Benchmark gate failed twice today on unchanged library code: on #2187's first run, and on #2188 after its rebase. Both times it was SimplifyEasy, SolveEasy and EvalTrigPrecise, at 3-12x their time, and SimplifyEasy at 3% more allocation. Their re-runs matched the baseline to the byte. BenchmarkDotNet times one call before it decides how to measure, and where that call takes more than two thirds of its 500 ms iteration, it times one call per iteration from then on. The first call of those three took 349-362 ms on the failing runs and 252-300 ms on the passing ones. For a call of tens of microseconds, one call per iteration is another measurement. That first call paid for parsing every static expression of the class and for the library's own set-up: EvalEasy, a cache lookup, took 133 ms the first time on one machine. A [GlobalSetup] now touches the class and runs one simplification, one solve and one evaluation before the timed call. Measured on one machine, master against this, the first call drops from 172 to 8 ms (SimplifyEasy), 164 to 5 ms (SolveEasy) and 158 to 45 ms (EvalTrigPrecise). What the gated benchmarks allocate is unchanged: SimplifyEasy is 79,682 B/op on both, and the largest move is ParseEasy's +0.11% (21 B/op). The two compilation benchmarks, which the gate leaves out for their own spread, moved -0.5% and +2.0%. SolveHard, a call of 70 ms, is now timed one call per iteration rather than piloted, and allocates 9,832,912 B/op against 9,831,370. Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
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mobius(n),divisorsigma(k, n),primenu(n),primeomega(n)andprimepi(x)were reserved in #2184 and are now functions. Part of #2182.mobius(n), alsomoebiusmun,(-1)^kforkdistinct primesdivisorsigma(k, n)d^kover the divisorsdofn; any orderprimenu(n),primeomega(n)primepi(x)xThe first four are defined on the positive integers and give
NaNat any other number. Factorisation is trial division up to 10^6. That settles every number up to 10^12, and any larger one whose cofactor after the small primes is below 10^12; anything else is left as written.primepiuses a sieve up to 10^7, which takes 94 ms there."mobius(6)".ToEntity().Simplify()6 * mobius1"divisorsigma(1, 6)".ToEntity().Simplify()UnhandledParseException12"primeomega(12)".ToEntity().Simplify()12 * primeomega3"primepi(10)".ToEntity().Evaledprimepi * 104"mobius(0)".ToEntity().Simplify()0NaNLaTeX output is
\operatorname{\mu}(n),\operatorname{\sigma}_k(n),\operatorname{\omega},\operatorname{\Omega}and\operatorname{\pi}. SymPy output ismobius,divisor_sigma(n, k),primenu,primeomegaandprimepi.🤖 Generated with Claude Code
https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura