Prime reciprocal series is diverging

We know that Harmonic series $\sum \frac{1}{n}$ is diverging. The proof is very simple by grouping the terms. Whereas for geometric sequence $\frac{1}{2^n}$ is converging. Our question is what happens when we have $\sum_{p \in prime} \frac{1}{p}$ .

We will prove this by contradiction. Assuming that the sum of reciprocals of prime $S_p$ is a constant value. As $\frac{1}{p_i}$ are positive constants. That means for some $\frac{1}{p_1}+\frac{1}{p_2}+...+\frac{1}{p_{n-1}} < S_n -\frac{1}{2}$ and

$\frac{1}{p_1}+\frac{1}{p_2}+...+\frac{1}{p_{n-1}}+\frac{1}{p_{n}} \le S_n - \frac{1}{2}$ .

Thus $\frac{1}{p_{n+1}}+\frac{1}{p_{n+2}}+ ... < \frac{1}{2}$ and multiplying both sides by positive $x$ we get $\frac{x}{p_{n+1}}+\frac{x}{p_{n+2}}+ ... < \frac{x}{2}$

To prove this we first define a function $N(x)$ which counts the number of numbers whose prime factor is among the first $k$ primes. For example if $k = 4$ the primes are $\{ 2,3,5,7 \}$ . So the function $N(10) = 10, N(15) = 13, N(27) = 20$ . ie for $10$ all the factors are from prime numbers $\{ 2,3,5,7 \}$ . Let $k$ be the number of first k primes and $x$ be any number and we try to find $N(x)$ . We see that any number $y \le x$ can be written as $y = {p_1}^{a_1}{p_2}^{a_2}...{p_t}^{a_t}$ . We have to count all those $y$ whose factors are among the first $k$ primes. We realise that any number can be written as $uv$ where $w^2v$ or $u = w^2$ is a square number and so $v$ is a product of primes less than $k^{th}$ prime and has the form $p_1^{\alpha_1}p_2^{\alpha_2}p_3^{\alpha_3}...p_k^{\alpha_k}$ where $\alpha_1,\alpha_2 ...\alpha_k \in \{0,1\}$ . Thus $N(x) \le \sqrt{x}2^{k}$ and the numbers which are not divisible by first k primes are $x-N(x) \le \frac{x}{p_{k+1}}+\frac{x}{p_{k+2}}+\frac{x}{p_{k+3}}+....$ .

So there are two equations as below

$\frac{x}{p_{n+1}}+\frac{x}{p_{n+2}}+ ... < \frac{x}{2}$

$x-N(x) \le \frac{x}{p_{k+1}}+\frac{x}{p_{k+2}}+\frac{x}{p_{k+3}}+....$ .

Combining we have $x-N(x) < \frac{x}{2} \Rightarrow \frac{x}{2} < N(x)$ and holds for all values of $x$ . Let $k = n$ then $N(x) \le 2^n\sqrt{x}$ . Therefore we havce $\frac{x}{2} < N(x) \le 2^n\sqrt{x}$ . When $n = 2^{2n+2}$ we get $\frac{1}{2}2^{2n+1} < N(x) \le 2^{2n+1}$ . This contradiction proves that the series P diverges. ( A similar contradiction is obtained for $x$ equal to any value greater than $2^{2n+2}$ .

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