Let 𝑆 S be the set of all integers 𝑛 n with 1 ≤ 𝑛 ≤ 10...

Prompt

Let 𝑆 S be the set of all integers 𝑛 n with 1 ≤ 𝑛 ≤ 10 12 1≤n≤10 12 such that: 𝑛 n is square‑free, and for every prime 𝑝 ∣ 𝑛 p∣n, we have 𝑝 ≡ 1 ( m o d 4 ) p≡1(mod4). (a) Give an asymptotic formula for ∣ 𝑆 ∣ ∣S∣ as 𝑥 → ∞ x→∞, in the form ∣ 𝑆 ∩ [ 1 , 𝑥 ] ∣ ∼ 𝐶 𝑥 ( log ⁡ 𝑥 ) 𝛼 ∣S∩[1,x]∣∼C (logx) α x ​ determining the constants 𝐶 > 0 C>0 and 𝛼 α. (b) Using your formula, estimate ∣ 𝑆 ∣ ∣S∣ for 𝑥 = 10 12 x=10 12 up to a relative error of at most a factor of 2. Justify your steps and indicate any deep theorems you are implicitly using (e.g. prime number theorem in arithmetic progressions, Siegel–Walfisz, etc.). Answer in a self‑consistent way, clearly separating the heuristic from the rigorous parts.