Let \(f(s_ 1,s_ 2)=\sum^ \infty_{m,n=1}a_{mn}\exp(\lambda_ ms_ 1+\mu_ ns_ 2)\), \(s_ j=\sigma_ j+it_ j\), \(j=1,2\) complex numbers, be a double Dirichlet series. Conditions for \(f\) to be an entire function are well known. Assuming this to be the case, one defines the maximum modulus function \(M_ f\) to be \[ M_ f(\sigma_ 1,\sigma_ 2)=\sup\{| f(s_ 1,s_ 2)|:\text{Re} s_ j=\sigma_ j\}, \] the maximum term \(\mu_ f\) to be \[ \mu_ f(\sigma_ 1,\sigma_ 2)=\sup_{m,n\in\mathbb{N}^ 2}\{| a_{mn}|\exp(\lambda_ m\sigma_ 1=\mu_ n\sigma_ 2), \] the functions \[ \begin{aligned} I^ p_ f(\sigma_ 1,\sigma_ 2) & = \lim_{(T_ 1,T_ 2)\to\infty}(4T_ 1T_ 2)^{-1}\int^{T_ 1}_{-T_ 1}\int^{T_ 2}_{-T_ 2}| f(s_ 1,s_ 2)|^ pdt_ 1dt_ 2\text{ and } \\ m_ f^{p,q}(\sigma_ 1,\sigma_ 2) & = 4\exp(-q\sigma_ 1-q\sigma_ 2)\int^{\sigma_ 1}_ 0\int^{\sigma_ 2}_ 0I^ p_ f(x_ 1,x_ 2) e^{-q(x_ 1+x_ 2)}dx_ 1dx_ 2. \end{aligned} \] The author then shows that i) \(\log M_ f(\sigma_ 1,\sigma_ 2)\) is asymptotically equivalent to \(\log\mu_ f(\sigma_ 1,\sigma_ 2)\); ii) \(\log I^ p_ f(\sigma_ 1,\sigma_ 2)\) is asymptotically equivalent to \(p\) \(\log M_ f(\sigma_ 1,\sigma_ 2)\). He also studies asymptotic expressions for \(\log m_ f^{p,q}(\sigma_ 1,\sigma_ 2)\).