# sum over prime index done by a weird sieve?

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 $z$.

• When you write "$=<$", do you actually mean "$\le$"? – celtschk Sep 12 '12 at 15:02
• Yes , sorry im not good at tex. – mick Sep 12 '12 at 15:03
• You might consider Detexify useful. – celtschk Sep 12 '12 at 15:14
• Thanks , i did look at it. – mick Sep 12 '12 at 15:22