What Is The Ratio Of Sum Of All Prime under N to sum of all composite under N?

Question

The picture is of the graph of ratio of sum of all prime numbers under 2,000,000 to sum of all composite numbers under 2,000,000 ,which I created using python and matplotlib.

[ X Axis = Range of Numbers ]

[ Y Axis = Ratio]

It appears That the ratio is approaching 0 as we go to $\infty$.

Can anyone explain what it implies ? What does it say about the distribution of prime numbers ? What is the reason for such behaviours ? Is it okay to infer that for larger values , the curve will infinitely come closer to 0 without ever increasing?

Answer 1

The number of primes under $N$ is asymptotic to $N/(\log N)$, so the sum of those primes is no more than $N^2/(\log N)$. The sum of all numbers under $N$ is essentially $N^2/2$, so the primes make a negligible contribution to this sum (since $\log N$ well exceeds $2$).

Answer 2

Let $p_n$ be the largest prime not exceeding $N$ and let $p$ be the sequence of primes and $c$ be the sequence of composites. We have

$$ \sum_{p \le p_n} p = \frac{n^2\log n}{2} + \frac{n^2\log \log n}{2} - \frac{3n^2}{4} + o(n^2) $$

and $p_n \sim n\log n + n\log\log n - n$. Hence

$$ \sum_{c \le p_n}c = \sum_{r \le p_n}r - \sum_{p \le p_n} p \sim \frac{n^2\log^2 n}{2} $$

Hence $$ \frac{\sum_{p \le p_n} p}{\sum_{c \le p_n}c} \sim \frac{1}{\log n} $$