Convergence and the Riemann hypothesis

For $1 < p \leq 2$ it is shown that a certain sequence of functions converges to -1 in $L^{p-\varepsilon}(0, 1)$ for any small $\varepsilon > 0$ if and only if the Riemann zeta function satisfies $\zeta(s) \neq 0$ for $\sigma = Re s > 1/p$.