NT_MU Function (Gnumeric)

• Introduced: Gnumeric 0.57 release notes explicitly record its addition.
• Supported in: Gnumeric 1.12.62; no retirement is established.
• Scope: Desktop Gnumeric with its Number Theory plugin. English comma-separated formulas are shown.
• Checked: October 14, 2026. The tables and named advanced check use 1.12.57 ssconvert. Current 1.12.62 source was inspected; that release was not executed.

NT_MU returns the Möbius value μ(n). It is zero when a prime square divides n; otherwise it is −1 for an odd number of distinct prime factors and +1 for an even number. The special case μ(1)=1 uses the empty-product convention. This is an arithmetic function, not a mean or a probability. Versioned function help.

=NT_MU(n)

n is one positive integer. Positive fractions are floored before evaluation; validate integral input yourself when fractions indicate a data problem.

Current implementation.

The eight ordinary results below were independently checked using integer arithmetic.

For a list of positive integers in A2:A20, enter =NT_MU(A2) in B2 and fill down. Values with ABS(B2)=1 are square-free: no prime appears twice in their factorization. A zero is useful information, not an error. For example, 30=2×3×5 has μ(30)=−1, whereas 12=2²×3 has μ(12)=0.

Do not interpret −1 as “not prime.” Both a prime such as 97 and a square-free product of three distinct primes such as 30 return −1. Use a primality test for that separate question.

The independent divisor identity is Σ μ(d)=0 over all positive divisors d of any n>1. For n=12, those divisors are 1,2,3,4,6,12. The native advanced check =NT_MU(1)+NT_MU(2)+NT_MU(3)+NT_MU(4)+NT_MU(6)+NT_MU(12) returns 0: 1−1−1+0+1+0.

The function is multiplicative for coprime inputs. Since 2 and 3 are coprime, μ(6)=μ(2)μ(3)=1. Coprimality matters: μ(4)=0, while μ(2)²=1. This distinction is essential when using Möbius inversion or constructing arithmetic checks.

In the standard binary64 build, the 1.12.57 shared bound is 2^52 = 4503599627370496. The 1.12.62 source bound is 2^53 = 9007199254740992. The tested large-input rejection is an older-build observation; the newer range was source-checked only. An accepted number is not a guarantee that a resource-intensive prime calculation will complete.

Zero, negative integers and positive values below one produce #NUM!. F5 is empty and is converted to zero, which is also invalid here. Text that cannot be converted to a number gives #VALUE!; an omitted required argument gives #N/A in this tested build. A returned #LIMIT! would indicate the factorization machinery could not handle an otherwise accepted request, not a Möbius value.

F5 is an actually empty cell. Errors below are the tested build’s results, not zero-valued mathematical answers.

• Current worksheet implementation
• Tested-version worksheet implementation
• Current calculation helper

The retained workbook’s stale SUM cache and uncached SUM both recalculated correctly. Formula identities, typed inputs and native results were inspected. The importer reported unsupported style-extension elements; that warning is retained, and no all-formatting claim is made. GUI entry, other locales and later-release output were not tested.

ISPRIME, GCD, ABS.