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Pillai’s arithmetical function (gcd-sum function) is defined by $$ P(n) = \sum_{k=1}^n\gcd(k,n) $$ Let $\sum_{n\leq x}P(n)$ be summation of all values of P for all $n$ up to given $x$. I dervied that $$ \sum_{n\leq x}P(n) = \sum_{d\leq x}\mu(d)\sum_{k\leq\frac{x}{d}}k\tau(k) $$ where $\mu$ is a Moebius function and $\tau$ is a divisor function. But it has at least an O(n) complexity. I wonder is it possible to get the summation in sublinear time?

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There is an approximation that may serve your purposes. First derive an alternate expression for $P(n):$

$$P(n) = \sum_{k=1}^n \gcd(k, n) = \sum_{d|n} \sum_{k=1 \atop \gcd(k, n)=d}^n d = \sum_{d|n} d \sum_{\gcd(k/d, n/d)=1} 1 = \sum_{d|n} d \times \varphi\left(\frac{n}{d}\right).$$

Now recall that $$\sum_{d|n} \varphi(d) = n$$ so that $$\sum_{n\ge 1} \frac{\varphi(n)}{n^s} \times \zeta(s) = \sum_{n\ge 1} \frac{n}{n^s}$$ which implies $$\sum_{n\ge 1} \frac{\varphi(n)}{n^s} = \frac{\zeta(s-1)}{\zeta(s)}.$$

Introduce $$L(s) = \sum_{n\ge 1} \frac{P(n)}{n^s}.$$ Using the convolution formula for $P(n)$ we thus have $$L(s) = \frac{\zeta(s-1)^2}{\zeta(s)}.$$

We can now use the Mellin-Perron formula to predict, but not quite prove, the asymptotics of the partial sums of $P(n).$ We have that $$\sum_{k=1}^n P(n) = \frac{1}{2} P(n) + \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} L(s) \frac{n^s}{s} ds$$ where $c>2.$

We shift this integral to the left. But the dominant pole at $s=2$ has residue $$\mathrm{Res}\left(L(s)\frac{n^s}{s}; s=2\right) = \frac{3}{\pi^2} n^2 \log n + n^2 \left(\frac{6}{\pi^2}\gamma - \frac{3}{2\pi^2} - \frac{18}{\pi^4} \zeta'(2) \right).$$

This yields the following approximation: $$\sum_{k=1}^n P(n) \sim \frac{1}{2} P(n) + \frac{3}{\pi^2} n^2 \log n + n^2 \left(\frac{6}{\pi^2}\gamma - \frac{3}{2\pi^2} - \frac{18}{\pi^4} \zeta'(2) \right).$$

Shifting the integral further to the left would pick up a contribution from the non-trivial zeros of the zeta function and hence is best left to professionals.

For your information the exact value for the sum up to $P(1000)$ is $$ 2475190 $$ while the approximation yields $$ 2476126.$$

The exact value for the sum up to $P(1200)$ is $$ 3649500$$ while the approximation yields $$ 3647105.$$

Remark. As the question text does not prove the equation for the sum that is quoted there I will include it here, so that it counts as verified.

Start with $$P(n) = \sum_{d|n} d \times \varphi\left(\frac{n}{d}\right) = n \sum_{d|n} \frac{\varphi(d)}{d}.$$

Now observe that $$\frac{\varphi(d)}{d} = \prod_p \left(1-\frac{1}{p}\right)$$ where the product ranges over the prime divisors $p$ of $n.$ This immediately implies that $$ \frac{\varphi(d)}{d} = \sum_{q|d} \frac{\mu(q)}{q}$$ by definition of the Mobius function.

Returning to the sum we obtain $$\sum_{n\le x} P(n) = \sum_{n\le x} n \sum_{d|n} \sum_{q|d} \frac{\mu(q)}{q}.$$ Switching the two inner sums and putting $d=qf$ yields $$\sum_{n\le x} n \sum_{q|n} \frac{\mu(q)}{q} \sum_{f|n/q} 1 = \sum_{n\le x} n \sum_{q|n} \frac{\mu(q)}{q} \tau(n/q)$$ which yields $$\sum_{n\le x} \sum_{q|n} \mu(q) \times n/q \times \tau(n/q).$$ Summing over all $q$ and putting $n=kq$ we finally obtain $$\sum_{q\le x} \mu(q) \sum_{k\le\lfloor x/q\rfloor} k \; \tau(k),$$ confirming the statement in the question.

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