While I am studying the famous article [1], in English this is Andrew Granville and Greg Martin, *Prime Number Races*, The American Mathematical Monthly, vol. 113, (2006), I wondered what about a race of odd semiprimes.

A semiprime is a positive integer that is the product of two prime numbers (see the Wikipedia *Semiprime* or the article from the encyclopedia MathWorld).

As in the cited article we consider that our race the odd semiprimes separated in two teams, those semiprimes of the form $s=4n+1$ for some positive integer $m$, this is $s\equiv 1\text{ mod }4$ that are our Team 1

$$9,21,25,33,45,49,57,65,69,77,85,93,\ldots,$$

and those semiprimes of the form $s=4n+3$ for some positive integer $m$, that is $s\equiv 3\text{ mod }4$ are our Team 2

$$15 ,35, 39, 51, 55 ,63, 75, 87, 91 ,95, 99\ldots.$$ We denote the corresponding counting functions as $$\#\{\text{semiprimes }4n+1\leq x\}\tag{1}$$ and $$\#\{\text{semiprimes }4n+3\leq x\}.\tag{2}$$

Question.Is it possible to know which team will win the race? I am asking if it is possible to study in a similar way that shows the first section of the mentioned article our race of semiprimes, maybe it is possible to get a similar statement than Hardy's theorem about the difference of previous counting functions $(1)$ and $(2)$, or maybe one can to state a conjecture in the spirit of the conjecture due to Knapowski and Turán but now for the race of semiprimes of the form $4n+1$ versus $4n+3$.Many thanks.

I don't know if this problem is in the literature, feel free to refer the literature and I try to search and read it from the literature to know how does the race end.

## References:

[1] Andrew Granville and Greg Martin, *Carreras de números primos*, La Gaceta de la RSME, Volumen 8, Número 1 (enero-abril, 2005).

Real Sociedad Matemática Española$\endgroup$ – user142929 Aug 28 '19 at 21:35