[For parts I and II see Acta Math. Acad. Sci. Hung. 37, 481-496 (1981; Zbl 0469.42009), and ibid. 39, 95-105 (1982; Zbl 0491.42030).] - The author continues his considerations on Lebesgue functions for d-multiple function series (*) \(\sum_{k\in Z^+_ d}a_ k\Phi(x)\) which he introduced in part I. Let \(K_ n(x,y) (n\in Z^+_ d)\) denote the kernels with respect to \(\{\Phi_ k(x)\); \(k\in Z^+_ d\}\) and let the Lebesgue functions be defined by \(L^*_ m(x)=\int \sum_{k\leq m}| \Phi_ k(x)\cdot \Phi_ k(y)| dy\) resp. with \(\epsilon =(\epsilon_ 1,...,\epsilon_ d), \epsilon_ i=0\quad or\quad =1, L_ m^{\epsilon}(x)=\int K_ m^{\epsilon}(x,y)dy,\) where \(K_ m^{\epsilon}(x,y)=\max \{| K_ n(x,y)|:n\leq m,n_ i=m_ i\quad if\quad \epsilon_ i=0\}.\) Theorem 1: If \(\sum_{k\in Z^+_ d}a^ 2_ k=\infty\) (resp. \(\sum_{k}a^ 2_ k=\infty\), \(k\in {\mathbb{Z}}^+_ d)\) and \(k\leq m)\) then there exists an orthonormal system with \(L^*_ m(x)\leq C\) such that the series (*) does not converge regularly (resp. (*) does not converge in Pringsheim's sense). In case \(L_ m(x)\leq C\cdot \lambda_ m\), the author then proves a sufficient condition such that (*) is regularly convergent (Theorem 2); this condition cannot be relaxed (Theorem 3). These theorems are extensions of results which \textit{K. Tandori} obtained for single series [Acta Sci. Math. 42, 171-173 (1980; Zbl 0436.42021); ibid. 42, 175-182 (1980; Zbl 0436.42022)].