In this paper, we consider the problem of zero-error network function computation. In a directed acyclic network, a single sink node requires to compute with zero error a function of source messages generated by multiple source nodes. We are interested in the information-theoretic computing capacity, which is defined as the average number of times that the function can be computed with zero error for one use of the network. The explicit characterization of the computing capacity in general is extremely challenging. The best known upper bound, applicable to arbitrary network topologies and arbitrary target functions, is the one proved by Guang et al. using the cut-set strong partition approach. This bound is tight for all previously considered network function computation problems whose computing capacities are known. In this paper, we focus on the model of computing the binary arithmetic sum over an asymmetric diamond network, which is of great importance to illustrate the combinatorial nature of network function computation problem. We first prove an upper bound of 1 on the computing capacity by using a linear programming approach, which rectifies an invalid upper bound previously proposed in the literature. However, this upper bound does not surpass the best known upper bound for this model, which is also equal to 1. Further, by developing a different graph coloring approach, we obtain an improved upper bound $\frac{1}{\log_3 2+\log3-1}$ (\approx 0.822). We thus show that the best known upper bound by Guang et al. is not tight for this model. On the other hand, we present an explicit code construction, which implies a lower bound $\frac{1}{2}\log_{3}6$ (\approx 0.815) on the computing capacity. Comparing the improved upper and lower bounds thus obtained, there exists a rough 0.007 gap between them.