17 The number of Pythagorean triples
Let \(\theta _{\mathrm{Pythag}}\) be the least exponent for which one has
\[ P(N) = c N^{1/2} - c' N^{1/3} + N^{\theta _{\mathrm{Pythag}}+o(1)} \]
for unbounded \(N\) and some fixed \(c,c'\), where \(P(N)\) is the number of primitive Pythagorean triples of area no greater than \(N\).
One has \(\theta _{\mathrm{Pythag}}\leq 1/4\).
Proof
If \((k,\ell )\) is an exponent pair, and RH holds, then
\[ \theta _{\mathrm{Pythag}}\leq \max ( \frac{1}{3} - \frac{5}{6} \frac{k+\ell -3/2}{4(k+\ell )-7}, \frac{1}{2} - \frac{3}{2}\frac{k+\ell -3/2}{4(k+\ell )-7} ) \]
Assuming RH, one has \(\theta _{\mathrm{Pythag}}\leq 71/316\).
Proof
See [ 279 , Section 5.10 ] .