An additive problem of prime numbers. III (Q1185144)

In part II [ibid. 67, No. 7, 248--252 (1991; Zbl 0747.11040)], the author encountered the quantity \(G(X)=\text{Re}\left\{\sum_{\gamma>0}{X^{i\gamma}\over (1/2+i\gamma)(3/2+i\gamma)}\right\}\), where \(\gamma\) runs over imaginary parts of zeros of Riemann's zeta function. In this paper, the arithmetic and oscillatory nature of \(G(X)\) is investigated. In particular, it is shown that \(G(X^ +_ n)>0.012\) and \(G(X^ -_ n)<-0.012\) for some sequences \(X_ n^ +\) and \(X^ -_ n\) tending to infinity.