Few weeks ago an user from Mathematics Stack Exchange answered my question *On an inequality that involves products and sums related to the sequence of semiprimes* (asked May 26). It seems that for disproving my conjecture was enough to use a *cheap* statements in analytic number theory (more cheap than the prime number theorem). If my post is good I would like dedicate it to the excellence in (real and complex analysis, functional analysis, topology and other subjects as) analytic number theory of the user who refuted my conjecture.

A semiprime $s$ is a positive integer that is the product of two prime numbers, see *Semiprine* from the encyclopedia Wikipedia, thus corresponding to the sequence *A001358* of the OEIS. I wondered if it is possible to deduce a statement at research level of the asymptotic

$$\Bigl(\prod_{\substack{1\leq s\leq X\\s\text{ semiprime}}}s\Bigr)\Bigl(\sum_{\substack{1\leq s\leq X\\s\text{ semiprime}}}\frac{1}{s}\Bigl)=\text{main term}+\text{error term},\tag{1}$$ where $\text{main term}=\text{main term}(X)$ is a function of the real variable $X$, let's say $X\geq 1$, and $\text{error term}=\text{error term}(X)$ is also a function of $X$ and represents a suitable error term in our asymptotic formula $(1)$ as $X\to\infty$.

Question.I would like to know what work can be done with the purpose to get a statement at research level for $(1)$ as $X\to\infty$, for a suitable error term expressed in big-O notation or little-o notation as you want (if it is feasible, you can to express your answer as an asymptotic identity $\sim$).Many thanks.

I don't know if this question concerning $(1)$ is in the literature, to ask this question I was inspired in a statement from [1]. If there is literature that provide a explicit answer for my question, then refer it answering my question as a reference request and I try to search an read those statements from the literature. I think that this post can be interesting to me as companion of the post of MSE, and I don't know if similar expressions as $(1)$ (I mean inequalities as the **Lemma** from [1] or asymptotics as our **Question**) are in the literature for other constellations of primes, or if it is interesting for other prime constellations as for example Ramanujan primes, primes in arithmetic progressions,... I evoke these problems if you want to explore some in your home, our case study here are the semiprimes.

## References:

[1] Takashi Agoh, Paul Erdös and Andrew Granville, *Primes at a (Somewhat Lengthy) Glance*, The American Mathematical Monthly, Vol. 104, No. 10 (December, 1997), pp. 943-945.