Proposition 8

Let $K$ be a valued field and $f_1, f_2, g_1, g_2$ four functions in $\mathscr{H}(\mathfrak{F}, K)$; then the relations $f_1 \sim g_1$ and $f_2 \sim g_2$ imply $f_1 f_2 \sim g_1 g_2$.

Indeed, we have $f_1 f_2 - g_1 g_2 = f_1 (f_2 - g_2) + (f_1 - g_1) g_2$; since $f_1 \preceq g_1$, $f_1 - g_1 \ll g_1$ and $f_2 - g_2 \ll g_2$, we have $f_1 f_2 - g_1 g_2 \ll g_1 g_2$ (V, p. 215, prop. 5 and 6).

In contrast, we have given an example in V, p. 214 where one has $f_1 = g_1$, $f_2 \sim g_2$ and yet the relation $f_1 + f_2 \asymp g_1 + g_2$ does not hold (so neither a fortiori does

2 See, for example, A. E. INGHAM, The distribution of prime numbers (Cambridge Tracts, n 30), Cambridge University Press, 1932.

The comparison relations $f \ll g, f \sim g$ are called strong relations. Two functions $f, g$ from $\mathscr{H}(\mathfrak{F}, V)$ are called comparable (or strongly comparable when one wants to avoid any possible confusion) if they satisfy one of the three relations: $f \ll g, f \gg g$, or "there exists a $\lambda \neq 0$ such that $f \sim g\lambda$".