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Read G.H. Hardy's Introduction to Number Theory, Then possibly A Course of Pure Mathematics, also by him.

Note that the level for which you're aiming, is a level wherein such summation is much closer to triviality, than it is to `black magic` - whatever that means.

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If you mean Hardy and Wright's Introduction to the Theory of Numbers, I agree it is excellent, but you can solve the great majority of the project Euler problems without going to quite that level. I particularly enjoyed Daniel Shank's Solved and Unsolved Problems in Number Theory but even that goes beyond what is necessary.

I'll check Daniel Shanks' book out. But the OP was interested in reaching a level where the proof of such summation is not `black magic` - at least that was my impression - which implies reaching a level considerably higher than the minimum... Hardy would be helpful, and is at a sufficiently higher level than future Project Euler questions he might encounter e.g. those involving continued fractions, or Diophantine equations, or even modular arithmetic etc.

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