[EpiMath]

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.

[somezero]

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.