I would like to know if there is a good estimate for the sum which concerns all primes not exceeding $x$:
$$\sum\limits_{p\leq x}\frac{1}{p^s}$$$$0<s<1$$.
Only this.
Thanks in advance!
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Konstantinos Gaitanas
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Hint: $$\zeta(s) = \sum_{n\leq x} \frac{1}{n^s} - \frac{x^{1-s}}{1-x} + \mathcal{O}\left(\frac1{x^{\Re[s]}}\right)$$ Mimic the usual proof of Merten's theorem by logging the Euler product of $\zeta$ and expanding. – Balarka Sen Feb 18 '14 at 19:29
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@BalarkaSen is it possible to help me a little bit?I am having some trouble bounding the sum from above... – Konstantinos Gaitanas Feb 19 '14 at 21:19
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1Does this question differ from http://math.stackexchange.com/questions/49383/how-does-sum-px-p-s-grow-asymptotically-for-textres-1 ? – anon Feb 21 '14 at 22:08
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@anon i suppose not.Thank you so very much!! – Konstantinos Gaitanas Feb 21 '14 at 23:08