Some Four Dimensional Definite Integrals Arising from Zeta-Function Theory

Abstract

The integral

$$\mathop \int\int\int\limits_\Delta\int (\alpha + \beta)^{k} (\gamma + \delta)^{l}(\alpha + \gamma)^{m}(\beta+\delta)^{n}\ d\alpha \ d\beta \ d\gamma \ d\delta$$

in which $\Delta=\lbrace (\alpha,\beta,\gamma,\delta)\in [0,1]^{4}:\alpha + \beta + \gamma + \delta \leq 1\rbrace$ occurs in a formula of J. B. Conrey. We evaluate this integral in closed form in some special cases and show how the results would apply to the fourth moment of $|\zeta^{(N)}(1/2+it)|$. A similar case involving the Nth derivative of Hardy’s function is worked out in detail.