Franklin wrote: << Well, to start with, it's called the "prime zeta function" (what a coincidence!). See MathWorld: http://mathworld.wolfram.com/PrimeZetaFunction.html for a start. I don't know anything about your alternating prime zeta function, but the specific sum you mention is A078437; also mentioned in http://mathworld.wolfram.com/PrimeSums.html, equation (7).

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Thanks, Franklin. (Sometimes it gets very tiresome to learn of all the mathematicians who have pre-plagiarized me!) --Dan