Approximation of partial sum over prime omega function

Question

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The prime omega function $\omega(n)$ counts the number of distinct prime factors of a natural number $n$, and can be defined as $\omega(n)=\sum_{p \mid n}1$. Let $S(N)=\sum_{n=1}^{N}n\omega(n)$. Let $\pi(n)$ denote the number of primes up to $n$.

$$\eqalign{S(N) &=\dfrac{1}{2}\sum_{\scriptstyle p{\text{ prime}} \atop \scriptstyle p\leq N} p\left(\left\lfloor\dfrac{N}{p}\right\rfloor ^2+\left\lfloor\dfrac{N}{p}\right\rfloor\right)\cr &\approx \dfrac{N\pi(N)}{2}+\dfrac{N^2}{2}\sum_{\scriptstyle p{\text{ prime}} \atop \scriptstyle p\leq N} \dfrac{1}{p}\cr &\approx \dfrac{N\pi(N)}{2}+\dfrac{N^2}{2}\left[\log \log N+M+o(1)\right],\cr}$$ where $M$ is the Meissel–Mertens constant.

The error term of this approximation is quite large. Is it possible to find a better approximation for this sum?

A better approximation for the sum of reciprocal primes (see https://arxiv.org/pdf/1703.08032.pdf) does not help to improve the overall sum.

The following approximation seems to work well. $$\tilde{S}(N,\epsilon)=\left[\dfrac{N\pi(N)}{2}+\dfrac{N^2}{2}\left(\log \log N+M\right)\right]\cdot \left(1+\dfrac{1}{(\log N)^{1+\epsilon}}\right)^{-1}$$, where $\epsilon \in (0,1)$. Define the percentage error as $E(N,\epsilon)=\left |\dfrac{S(N)-\tilde{S}(N,\epsilon)}{S(N)}\right |\times 100$. The following table gives an idea as to how well the approximation works for $\epsilon=\dfrac{2}{5}$.

$N$	$S(N)$	$E\left(N,\frac{2}{5}\right)$
$10^3$	$1104960$	$0.192894816907$
$10^4$	$124547620$	$0.0961202670514$
$10^5$	$13558020500$	$0.0234415251444$
$10^6$	$1446379298344$	$0.00741488319826$
$10^7$	$152305298038685$	$0.0118235266249$
$10^8$	$15895296473923817$	$0.00880016331673$
$10^9$	$1648198262567978862$	$0.00327960320611$
$10^{10}$	$170070011690546177056$	$0.00270740071241$

Answer 1

By partial summation, one has $$ S(N) = N\sum_{n=1}^N \omega(n) - \int_{1}^N \bigg(\sum_{n\leq t} \omega(n)\bigg) dt. $$ Using Mertens' theorem with the classical error term in the prime number theorem, one has, for some constant $c > 0$, $$ \begin{aligned} \sum_{n=1}^N \omega(n) = \sum_{p\leq N} \Big\lfloor \frac{N}{p}\Big\rfloor &= N \sum_{p\leq N} \frac{1}{p} - \sum_{p\leq N} \left\{ \frac{N}{p}\right\} \\ &= N \Big(\log\log N + M + O\Big(\exp\big(-c\sqrt{\log N}\big) \Big) \Big)- \sum_{p\leq N} \left\{ \frac{N}{p}\right\} \end{aligned} $$ where $\{\cdot\}$ denotes the fractional part. The sum of fractional parts can evaluated using Proposition 3 of this paper, and we have $$ \sum_{p\leq N} \left\{ \frac{N}{p}\right\} = \int_{2}^N \frac{\left\{N/t \right\}}{\log t} dt + O\Big(\exp\big(-c(\log N)^{3/5-\varepsilon}\big) \Big). $$ By Proposition 2 of the above paper, the integral can be expanded asymptotically into a sum with descending powers of $\log N$: for any $k\geq 1$, $$ \int_{2}^N \frac{\left\{N/t \right\}}{\log t} dt = c_0 \frac{N}{\log N} + c_1 \frac{N}{(\log N)^2} + \cdots + c_k \frac{N}{(\log N)^k} + O_k\left( \frac{N}{(\log N)^{k+1}}\right). $$ I leave the remaining details of the calculations to you, but this roughly shows that you should not hope for an error term better than $O_k\left( \frac{N^2}{(\log N)^k}\right)$ in your asymptotic expansion.

Answer 2

As Prof. Tao suggested, we can estimate $S(n)$ using Summation by parts method. $$S(n)\sim nf(n) - \sum_{k=2}^{n-1}f(k),$$ where $f(n)$ approximates the sum $\sum_{k=1}^{n}\omega(k)$. $$\sum_{k=1}^{n}\omega(k) \sim f(n) = n \left[ \log \log n + M + \sum _{k\geq 1}\left(\sum _{j=0}^{k-1}\frac{\gamma_j}{j!} - 1\right)\frac{(k-1)!}{(\log n)^k}\right],$$ where $M$ is the Meissel-Mertens constant and $\gamma_j$s are Stieltjes constants.

Let $T(n)=\sum_{m=2}^{n}f(m)$. We can estimate $T(n)$ using Euler-Maclaurin formula. $$T(n)\sim \int_{2}^{n}f(x) dx + \frac{1}{2}\left(f(n)+f(2)\right) + \sum_{k=1}^{\infty}\frac{B_{2k}}{(2k)!}\left(f^{(2k-1)}(n)-f^{(2k-1)}(2)\right),$$ where $B_k$ is the $k^{th}$ Bernoulli number (with $B_1=\frac{1}{2}$).

Following this approach we get the following estimates (denoted by $\hat{S}(n)$) for $S(n)$ using $5$ terms in the summatory part of $f(.)$.

$N$	$S(N)$	$\hat{S}(N)$	Percentage error
$10^3$	$1104960$	$1104744.867385$	$0.019470$
$10^4$	$124547620$	$124548871.265378$	$0.001005$
$10^5$	$13558020500$	$13559354294.383795$	$0.009838$
$10^6$	$1446379298344$	$1446466799145.233356$	$0.006050$
$10^7$	$152305298038685$	$152311928057871.725336$	$0.004353$
$10^8$	$15895296473923817$	$15895763172409493.954120$	$0.002936$
$10^9$	$1648198262567978862$	$1648231275071905930.713382$	$0.002003$
$10^{10}$	$170070011690546177056$	$170072396183761800450$	$0.001402$
$10^{11}$	$17482087123791069603605$	$17482264536463768138654$	$0.001015$