On a problem of Yamamoto concerning biquadratic Gauss sums. II (Q1113932)

For a prime number \(p=a^ 2+4b^ 2\equiv 5 mod 8\) consider the Gauss sum \[ \sum^{p-1}_{m=1}(m/\omega)_ 4 e^{2\pi im/p}=\epsilon_ p\omega^{1/2}p^{1/4},\quad 0<\arg \omega^{1/2}<\pi /4, \] where \(\omega =a+2bi\) and \((m/\omega)_ 4\) denotes the biquadratic residue symbol in \({\mathbb{Q}}(i)\). It is known that \(\epsilon^ 4_ p=1\). In 1965 \textit{K. Yamamoto} [J. Reine Angew. Math. 219, 200-213 (1965; Zbl 0133.293)] observed that the inequality \[ (*)\quad Im(\epsilon_ p\sum^{(p-1)/2}_{m=1}(m/\omega)_ 4)>0 \] holds for \(p<4000\). He proposed the question whether this is always true. In a previous paper [Proc. Japan Acad., Ser. A 63, 35-38 (1987; Zbl 0624.10030)], the author reported a counter-example for (*). The purpose of the present paper is to explain the tendency of the inequality (*) to be satisfied: The limit \[ \lim_{x\to \infty}\frac{card\{p\leq x;\quad p\equiv 5 mod 8,(*)\quad holds\quad for\quad p\}}{card\{p\leq x;\quad p\equiv 5 mod 8\}} \] exists and lies between 0,9997 and 0,9998. For the proof inequality (*) is reduced to a relation of type \(Re(L(1,\chi_{\omega})/\omega^{1/2})>0.\)