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Journal of the London Mathematical Society (1)

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1 results

the sum-of-digits function of squares
michael drmota, joël rivat
Journal:

Journal of the London Mathematical Society / Volume 72 / Issue 2 / October 2005

Published online by Cambridge University Press:

04 October 2005, pp. 273-292

Print publication:

October 2005
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the set of squares $n^2$, $n<2^k$, is considered and the sum of binary digits $s(n^2)$ is split up into two parts $s_{[<k]}(n^2)+s_{[\ge k]}(n^2)$, where $s_{[<k]}(n^2) = s(n^2{\rm mod}2^k)$ collects the first $k$ digits and $s_{[\ge k]}(n^2) = s(\lfloor n^2/2^k\rfloor)$ collects the remaining digits. very precise results on the distribution of $s_{[<k]}(n^2)$ and $s_{[\ge k]}(n^2)$ are presented. for example, asymptotic formulae are provided for the numbers $\#\{n< 2^k{:} s_{[<k]}(n^2) = m\}$ and $\#\{n< 2^k {:} s_{[\ge k]}(n^2) = m\}$ and it is shown that these partial sum-of-digits functions are asymptotically equidistributed in residue classes. these results are prompted by a conjecture by gelfond saying that the (total) sum-of-digits function $s(n^2)$ is asymptotically equidistributed in residue classes.