On some mean value results involving |zeta(1/2+ it)|

A. Ivić

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We compute upper and lower bounds for the approximation of hyperbolic functions at points $1 / s$ $(s = 1, 2, \dots)$ by rationals $x / y$ , such that $x, y$ satisfy a quadratic equation. For instance, all positive integers $x, y$ with $y \equiv 0 (mod 2)$ solving the Pythagorean equation $x^{2} + y^{2} = z^{2}$ satisfy $| y sinh (1 / s) - x | ≫ \frac{log log y}{log y} .$ Conversely, for every $s = 1, 2, \dots$ there are infinitely many coprime integers $x, y$ , such that $| y sinh (1 / s) - x | ≪ \frac{log log y}{log y}$ and $x^{2} + y^{2} = z^{2}$ hold simultaneously for some integer $z$ . A generalization to the approximation of $h (e^{1 / s})$ for rational...