Dear Bahattin,
Sorry to have taken so long to get back to you. I think I will have a bit more time for this from here on. I've noticed a few oddities and I'm afraid that I have some bad news.
The bad news first: the sum, counting up to N, for the large values of N that you want, is going to take too long for a single machine, especially since it involves a reciprocal and a power. Consider this: if incrementing n takes 1 clock; testing it against N another clock, raising n to the s'th another two clocks, taking the reciprocal another clock and adding the result to an accumulator yet another clock - that's six clocks per iteration, assuming we're using ordinary ints and doubles. A fast machine these days doesn't run much faster than 4GHz, so our loop takes 1.5e-9 seconds per iteration. For your lowest N, 1.0e14, it would take 1.5e5 seconds, or about 2500 minutes, or 41 hours 40 minutes. You can expect the times to be hundreds of times worse for 500-digit precision. It will also be hundreds of times worse for 1.0e16, ditto for N==1.0e18:-(
Cloud computing might just save you here, and I understand that Mathematica can do some cloudy stuff.
I noticed that you originally had a graphic containing two equations (real & imaginary parts?) in your first post here; now it's changed to rendered postscript or something. Anyhow, as I recall it there was an integral expression on the right. Now there's a dx on the right which doesn't seem to refer to anything.
Regards, Graham.
