On the number of primitive Pythagorean triangles

On the distribution of primitive Pythagorean triangles

“…Proof. This can be proved in the same way as Lemma 5.2 of Zhai [22] (quoted as Lemma 4.4 in this paper) with slight modifications only.…”

“…Therefore it is natural to search for stronger estimates under the Riemann Hypothesis (RH). The exponent 1/4 in R area (x) was improved by several authors under RH (see [11], [15], [14], [16], [22]). It is also of interest to consider the distributions of P (x), A(x) and H(x) unconditionally in short intervals.…”

Errata and Addenda to Mathematical Constants

“…Pythagorean Triple Constants. Improvements in estimates for P a (n) and P p (n) are found in [568,569]. Let P ℓ (n) denote the number of primitive Pythagorean triangles under the constraint that the two legs are both ≤ n; then [570]…”

On the number of representations of integers as differences between Piatetski-Shapiro numbers

“…For other asymptotic formulas, see [4,5] (primitive Pythagorean triples with perimeter less than x) and [4,[6][7][8] (primitive Pythagorean triples with area less than x).…”