15 The number of Pythagorean triples

Definition 15.1 Pythagorean triple exponent

Let $θ_{Pythag}$ be the least exponent for which one has

P (N) = c N^{1 / 2} - c^{'} N^{1 / 3} + N^{θ_{Pythag} + o (1)}

for unbounded $N$ and some fixed $c, c^{'}$ , where $P (N)$ is the number of primitive Pythagorean triples of area no greater than $N$ .

Lemma 15.2

One has $θ_{Pythag} \leq 1 / 4$ .

Proof ▶

See [ 229 , 49 ] . The previous bound $θ_{Pythag} \leq 1 / 3$ was obtained in [ 139 ] .

Lemma 15.3

If $(k, ℓ)$ is an exponent pair, and RH holds, then

θ_{Pythag} \leq max (\frac{1}{3} - \frac{5}{6} \frac{k + ℓ - 3 / 2}{4 (k + ℓ) - 7}, \frac{1}{2} - \frac{3}{2} \frac{k + ℓ - 3 / 2}{4 (k + ℓ) - 7})

Proof ▶

Lemma 15.4

Assuming RH, one has $θ_{Pythag} \leq 71 / 316$ .

Proof ▶