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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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This question was already asked on MO, I'll try to find it (and close yours afterwards :-( ). You are very-very curious! :-) –  Wadim Zudilin Aug 2 '10 at 10:01
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Oh! No please dont end my curiosity by closing this thread. –  Chandrasekhar Aug 2 '10 at 10:11
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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... –  Wadim Zudilin Aug 2 '10 at 10:45
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Please make some effort to solve your own curiosity questions. –  Tsuyoshi Ito Aug 3 '10 at 11:45
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