The classical second integral moment of $\zeta(s)$ shows that the integral average of $|\zeta(\tfrac{1}{2}+it)|^2$ is $\log t$. Assuming the Riemann Hypothesis and letting $\gamma,\gamma^+$ be the imaginary parts of consecutive critical zeros of $\zeta(s)$, Conrey and Ghosh proved that the mean value of $|\zeta(\tfrac{1}{2}+it)|^2$ over the maxima between $\gamma, \gamma^+$ up to $T$ is asymptotic to $\frac{1}{2}(e^2-5)\frac{T}{2\pi}\log(\frac{T}{2\pi})^2$. In other words, the discrete mean of $|\zeta(\tfrac{1}{2}+it)|^2$ at a critical point is $\frac{1}{2}(e^2-5)\log t$, which is a constant factor larger.

In this talk, we will demonstrate that the analogous phenomenon does not exist for the $Z$-function associated to a Dirichlet $L$-functions. Specifically, we show that the discrete mean value of Hardy's $Z$-function over its local extrema has an asymptotic formula with a negative leading coefficient. In contrast, Korolev and Jutila have proven that the integral mean value of Hardy’s $Z$-function does not exhibit such behavior. By improving Conrey and Ghosh’s method, we can compute as many lower-order terms as desired.

This is joint work with Jonathan Bober (Bristol) and Micah Milinovich (Mississippi).