I'd have another suggestion to replace your $Z(t/log(t))$ Z(t/\log(t))$: there's the Riemann-Siegel Theta function described in Harold Edwards' book,$theta(t)$. \theta(t)$. The Gram points satisfy: $theta(g_n) \theta(g_n) = npi$n\pi$, n$n$= 1, 2, 3, ... So the idea is to look at$W(alpha) W(\alpha) = Z(theta^{-1}(alpha))$Z(\theta^{-1}(\alpha))$ . That way, if $g_n$ is the nth $n$th Gram point, $theta^{-1}(npi) \theta^{-1}(n\pi) = g_n$ and
$W(npi) W(n\pi) = Z(theta^{-1}(npi)Z(\theta^{-1}(n\pi)) = Z(g_n) = (-1)^n \zeta(1/2 + i g_n)$ .
Cf. Gram's Law' at MathWorld: .
Perhaps there's a way to rescale $W(.)$ vertically from $Z(.)$ to get identical square-integrals over corresponding intervals say $[g_n, g_{n+1}]$ for Z and $[npi, [n\pi, (n+1)pi]$ n+1)\pi]$for W(.)$W(.)$... 1 [made Community Wiki] I'd have another suggestion to replace your$Z(t/log(t))$: there's the Riemann-Siegel Theta function described in Harold Edwards' book,$theta(t)$. The Gram points satisfy:$theta(g_n) = npi$, n = 1, 2, 3, ... So the idea is to look at$W(alpha) = Z(theta^{-1}(alpha))$. That way, if$g_n$is the nth Gram point,$theta^{-1}(npi) = g_n$and$W(npi) = Z(theta^{-1}(npi)) = Z(g_n) = (-1)^n zeta(1/2 + i g_n)$. Cf. Gram's Law' at MathWorld: . Perhaps there's a way to rescale$W(.)$vertically from$Z(.)$to get identical square-integrals over corresponding intervals say$[g_n, g_{n+1}]$for Z and$[npi, (n+1)pi]\$ for W(.) ...