Arithmetic function Λ(n) defined by Λ(n)=log p if n is a positive integer equal to p^k for some prime p and integer k≥1, and Λ(n)=0 otherwise; used to weight prime powers in explicit formulas.

Principle

Isolate prime powers by assigning them the logarithm of the underlying prime so that summatory transforms of Λ encode primes and prime powers in analytic identities.

Demonstration

Values: Λ(8)=Λ(2^3)=log 2, Λ(9)=log 3, Λ(12)=0 because 12 is not a pure prime power. The Chebyshev ψ(x) is the summatory ψ(x)=Σ_{n≤x}Λ(n).

Misapplication

Treating Λ(n) as log n for all n (instead of only for prime powers) destroys its selectivity and spoils identities that extract prime information from logarithmic derivatives of zeta-like functions.

Consequence

Λ appears in explicit formulas connecting prime counting to zeros of zeta-type functions; weighted sums of Λ convey prime distribution information and are central to analytic number theory methods.

Reversal

The opposite would be assigning nonzero weight to composite numbers that are not prime powers or distributing log factors evenly; such a reversal blurs the link between Λ and underlying primes.

Boundary

Defined on positive integers; Λ singles out exact prime powers and gives zero elsewhere—does not measure multiplicity beyond taking log p on any p^k and is not a multiplicative function in the usual sense.

Semantic Tension

Often conflated with the logarithm function or with indicator functions of primes; tension arises because Λ behaves like log p on prime powers but is zero elsewhere, unlike continuous log or prime characteristic functions.

Synthesis

The von Mangoldt function is the discrete weight that places log p at each prime power p^k and zero elsewhere; it concentrates prime information in sums and transforms that bridge additive summation and multiplicative structure.