Summatory arithmetic function ψ(x)=Σ_{n≤x}Λ(n)=Σ_{p^k≤x}log p that sums log p over all prime powers p^k not exceeding x; it counts primes with multiplicity according to their prime-power contributions.

Principle

Aggregate prime-power logarithmic weights (via the von Mangoldt function) so that ψ captures both primes and their powers and is closely linked to analytic properties of the zeta function.

Demonstration

Compute ψ(10). Include Λ(2)=log2, Λ(3)=log3, Λ(4)=log2, Λ(5)=log5, Λ(7)=log7, Λ(8)=log2, Λ(9)=log3; summing these yields ψ(10)=log2+log3+log2+log5+log7+log2+log3 = 3 log2 + 2 log3 + log5 + log7 ≈ 7.960.

Misapplication

Interpreting ψ(x) as simply π(x) (the count of primes) neglects multiplicity from prime powers and leads to underestimates of weighted sums and misreading of explicit formulas.

Consequence

ψ(x) is the standard object in many prime distribution results; asymptotic ψ(x) ~ x is equivalent to the prime number theorem and precise estimates of ψ control error terms in prime counting.

Reversal

The reversal is to ignore prime powers and sum only over primes (θ or π), which removes multiplicity information and alters the connections to logarithmic derivatives of multiplicative generating functions.

Boundary

Defined for real x≥2 and built from Λ; includes every prime power contribution up to x but excludes composite numbers that are not prime powers; sensitive to the exact placement of prime powers.

Semantic Tension

Tension with θ(x) and π(x): ψ includes prime powers and therefore often behaves smoother for analytic purposes than θ, but both serve different analytic and combinatorial roles.

Synthesis

Chebyshev's ψ(x) is the cumulative von Mangoldt weight up to x, summing log p over all prime powers p^k≤x; it is the central weighted summatory function that ties prime-power contributions to analytic objects like the zeta function.