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We know that by Dirichlet's formula for the Divisor function $ \displaystyle \sum\limits_{n \leq x} d(n) = x \log{x} + (2C-1)x + \mathcal{O}(\sqrt{x})$.

What is the best approximation available till date for the given formula. I know that finding the infimum of the $\mathcal{O}'s$ is an unsolved problem, but would like to see the closest approximation.

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  • $\begingroup$ This question was already asked on MO, I'll try to find it (and close yours afterwards :-( ). You are very-very curious! :-) $\endgroup$
    – Wadim Zudilin
    Commented Aug 2, 2010 at 10:01
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    $\begingroup$ I don't know why should it stay open in this wrong form. ;-) BTW, all question lovers are advised to first visit the OEIS and Wikipedia, before posting their questions. You obviously missed this standard procedure... $\endgroup$
    – Wadim Zudilin
    Commented Aug 2, 2010 at 10:45
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    $\begingroup$ Please make some effort to solve your own curiosity questions. $\endgroup$
    – Tsuyoshi Ito
    Commented Aug 3, 2010 at 11:45

2 Answers 2

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Use Wikipedia: http://en.wikipedia.org/wiki/Dirichlet_divisor_problem .

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you can see also this paper:http://hkumath.hku.hk/~imr/IMRPreprintSeries/2010/IMR2010-10.pdf

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