Working definition
DLMF 27.2.1 states the fundamental theorem of arithmetic: every integer greater than one factors uniquely into prime powers. Three functions are defined on that footing. The totient at 27.2.7 counts the integers up to n that are coprime to n. The divisor function at 27.2.9 counts the divisors of n. The Mobius function at 27.2.12 is one at n equal to one, minus one to the number of prime factors when they are all distinct, and zero when any prime is repeated. Separately, 27.2.3 states the prime counting function as asymptotic to x over the natural logarithm of x, which is a statement about a ratio in a limit rather than a count at any particular x.
Notation
phi of n for the totientd of n for the divisor countmu of n for the Mobius function and pi of x for the prime countAssumptions
- Unique factorisation at 27.2.1 is stated for integers greater than one, so the functions defined on it inherit that domain.
- The Mobius function is zero exactly when a prime is repeated, which is a condition on the factorisation rather than on the size of n.
- The prime counting statement at 27.2.3 is asymptotic, so it constrains a ratio as x grows and says nothing at a fixed x.
Invariants
- Unique factorisation into prime powers, DLMF 27.2.1
- The prime counting function is asymptotic to x over log x, DLMF 27.2.3
- The Mobius function vanishes exactly when a prime is repeated, DLMF 27.2.12
Reproducible procedure
- Take a factorisation as the starting point, since the totient, divisor and Mobius functions are all read off it.
- Treat 27.2.3 as a limiting ratio and never as an estimate with a stated error at a particular x.
- Check for a repeated prime before using the Mobius function, because that case is zero rather than small.
Error and boundary controls
- The asymptotic law at 27.2.3 uses the relation defined at 2.1.1: the ratio tends to one. It supplies no bound on the difference at any finite x, and the difference is not small in the way the ratio suggests.
- Computing the totient or divisor function from a factorisation is only as reliable as the factorisation, which for large n is the hard part rather than the counting.
- The chapter gives definitions, not algorithms, and makes no claim about the cost or stability of obtaining a factorisation.
What this does not establish
The chapter defines the functions and states one asymptotic law. It settles no open question about how primes are distributed, and an asymptotic law is not a formula for the prime count.
Explicit applications
0 cross-domain bridges
This foundational concept currently supports related concepts; a direct domain bridge is scheduled for a later registry version.
Authoritative references
- [1]DLMF Chapter 27: Functions of Number Theory · National Institute of Standards and Technology
Establishes: Unique factorisation into prime powers, the asymptotic law for the prime counting function, and definitions for the totient, divisor and Mobius functions.
Boundary: The chapter states the definitions and the asymptotic law. An asymptotic statement about the prime count bounds a ratio in a limit and predicts no individual value, and the chapter settles no open question about the distribution of primes.
Direct answer
- DLMF 27.2.1 states the fundamental theorem of arithmetic: every integer greater than one factors uniquely into prime powers. Three functions are defined on that footing. The totient at 27.2.7 counts the integers up to n that are coprime to n. The divisor function at 27.2.9 counts the divisors of n. The Mobius function at 27.2.12 is one at n equal to one, minus one to the number of prime factors when they are all distinct, and zero when any prime is repeated. Separately, 27.2.3 states the prime counting function as asymptotic to x over the natural logarithm of x, which is a statement about a ratio in a limit rather than a count at any particular x.
Mechanism and method
- Take a factorisation as the starting point, since the totient, divisor and Mobius functions are all read off it.
- Treat 27.2.3 as a limiting ratio and never as an estimate with a stated error at a particular x.
- Check for a repeated prime before using the Mobius function, because that case is zero rather than small.
What is measured
- Unique factorisation into prime powers, DLMF 27.2.1
- The prime counting function is asymptotic to x over log x, DLMF 27.2.3
- The Mobius function vanishes exactly when a prime is repeated, DLMF 27.2.12
Limitations
- The asymptotic law at 27.2.3 uses the relation defined at 2.1.1: the ratio tends to one. It supplies no bound on the difference at any finite x, and the difference is not small in the way the ratio suggests.
- Computing the totient or divisor function from a factorisation is only as reliable as the factorisation, which for large n is the hard part rather than the counting.
- The chapter gives definitions, not algorithms, and makes no claim about the cost or stability of obtaining a factorisation.
- Unique factorisation at 27.2.1 is stated for integers greater than one, so the functions defined on it inherit that domain.
- The Mobius function is zero exactly when a prime is repeated, which is a condition on the factorisation rather than on the size of n.
- The prime counting statement at 27.2.3 is asymptotic, so it constrains a ratio as x grows and says nothing at a fixed x.
What this does not establish
- The chapter defines the functions and states one asymptotic law. It settles no open question about how primes are distributed, and an asymptotic law is not a formula for the prime count.