As you might have noticed i considered in 2 previous questions sums of the form $f(p_i x)$ where the sum is over the primes $p_i$ ( between some integer bounds : $a \leqslant p_i \leqslant b$ ) , $x$ some real positive number and $f$ is some entire function.
One of the ideas i had deserves a new seperate question , hence i ask it here rather than add it as a comment , also because it is more general.
The idea is that i consider $F(a,b,x)$ where $x$ is again the real number and $a$ and $b$ are integer bounds. In particular $F(2,b,x)$ and $F(sqrt(b),b,x)$.
$F(a,b,x)$ is defined as the sum : $f(a x) + f((a+1) x) + f((a+2) x) +...+ f(b x)$.
The idea is now to approximate $Q(2,b,x)$ = the sum $f(p_i x)$ for $2 =< p_i =< b$ by the sieve-like :
$Q'(2,b,x) = F(2,b,x) ( 1 - F(2,b,x/2)/F(2,b,x) ) ( 1 - F(2,b,x/3)/F(2,b,x) ) ... ( 1 - F(2,b,x/p_k)/F(2,b,x) )$
Where $p_k$ is the biggest prime $<$ sqrt($b$).
How good is this approximation ? When is $(Q(2,b,x) / Q'(2,b,x))^2 \leqslant$ (Fixed Constant) for all $x$ and $b$ ?
Can we express the error term ?
Likewise i could try to approximate $Q(a,b,x)$ with for example $a$ = sqrt($b$). And i might even use other sieves such as for example also including $\frac{df}{dx}$ or $\frac{dF}{db}$ etc.
However these methods seem dubious , on the other hand they seem appealing and clever ?
although i could probably find a sieve that works for almost every entire $f$ , they might depend on unproven conjectures and i wonder if there exists an unconditional sieve that works well for almost ALL $f$ ?
I also wonder when this also works if we replace real $x$ with complex $z$.