Bj"orner and others have been very successful in using a mixture of combinatorics and topology to do things such as evaluating alternating sums $\sum (-1)^i a_i $ by first finding a simplicial complex with $a_i$ counting the number of $i$-dimensional faces for each $i$, so that the alternating sum is the Euler characteristic. Topology then can help in that this alternating sum also then equals the alternating sum of ranks of homology groups, which sometimes is a much simpler expression. For instance, in many cases of interest the homology is concentrated in a single degree, e.g. for pure, shellable complexes; in the paper of Bj"orner linked above, the complexes are homotopy equivalent to wedges of spheres. If one wants a purely combinatorial approach, there is the related notion of sign-reversing involution which can be used to cancel pairs of faces, one of which contributes positively to the sum and the other of which contributes negatively; discrete Morse theory is a way to interpret such cancellations topologically in terms of elementary collapses, unifying the combinatorial and topological approaches.

One usually needs a pretty good description of the faces to do the sort of cancellation I mention in an effective manner; to use topology, one needs to know a lot about which faces are incident to each other. If the simplicial complex (or cell complex) comes from number theory, it seems likely one would then need a good bit of number theoretic information as input to this sort of process.

If I understand correctly Vel Nias's recent MO question, that seems to be going after not only numerical data, but also viewing topology of face incidences as a language in which to encode the structure of how different arithmetic progressions of primes may overlap with each other.

Björner and others have been very successful in using a mixture of combinatorics and topology to do things such as evaluating alternating sums $\sum (-1)^i a_i $ by first finding a simplicial complex with $a_i$ counting the number of $i$-dimensional faces for each $i$, so that the alternating sum is the Euler characteristic. Topology then can help in that this alternating sum also then equals the alternating sum of ranks of homology groups, which sometimes is a much simpler expression. For instance, in many cases of interest the homology is concentrated in a single degree, e.g. for pure, shellable complexes; in the paper of Björner linked above, the complexes are homotopy equivalent to wedges of spheres. If one wants a purely combinatorial approach, there is the related notion of sign-reversing involution which can be used to cancel pairs of faces, one of which contributes positively to the sum and the other of which contributes negatively; discrete Morse theory is a way to interpret such cancellations topologically in terms of elementary collapses, unifying the combinatorial and topological approaches.

One usually needs a pretty good description of the faces to do the sort of cancellation I mention in an effective manner; to use topology, one needs to know a lot about which faces are incident to each other. If the simplicial complex (or cell complex) comes from number theory, it seems likely one would then need a good bit of number theoretic information as input to this sort of process.

If I understand correctly Vel Nias's recent MO question, that seems to be going after not only numerical data, but also viewing topology of face incidences as a language in which to encode the structure of how different arithmetic progressions of primes may overlap with each other.