Totient summatory: Φ(n) = φ(1)+φ(2)+...+φ(n). Asymptotically Φ(n) ≈ 3n²/π² ≈ 0.3040·n². Φ(n)+1 = number of Farey fractions F_n (fractions 0≤p/q≤1 with q≤n in lowest terms). Key identity: Σ_{d|n} φ(d) = n.
How to Use
Enter n:
- Φ(n): Cumulative sum
- φ(k): Individual values
- Asymptotic: Compare to 3n²/π²
Farey Connection
|F_n| = Φ(n)+1. The Farey sequence F_n contains all fractions p/q with 0≤p/q≤1, gcd(p,q)=1, q≤n. So Φ(n) counts the 'new' fractions (excluding 0/1). This links totient sums to fraction density.
Growth Rate
Φ(n) = (3/π²)n² + O(n log n). The density of coprime pairs approaches 6/π² ≈ 0.6079. This is the probability that two random integers are coprime!
Step-by-Step Instructions
- 1Enter n.
- 2Compute Φ(n).
- 3See φ(k) values.
- 4Compare asymptotic.
- 5Count Farey fractions.