How Do F(X,n) and Q(n) Relate Through Partial Summation?

[flouran]

Let k and n \le X be large positive integers, and p is a prime. Define

F(X,n) := \sum_{\substack{k^2+p = n\\X/2\le p<X\\\sqrt{X}/2 \le k < \sqrt{X}}}\log p
Q(n) := \sum_{k^2+p = n}\log p.Note that in Q(n), the ranges of k and p are unrestricted.

My question is:
I know that F(X,n) and Q(n) can be related by partial summation, but how do I prove this?

Any help is appreciated!

Thanks.

 

[flouran]

Let k and n \le X be large positive integers, and p is a prime. Define

F(X,n) := \sum_{\substack{k^2+p = n\\X/2\le p<X\\\sqrt{X}/2 \le k < \sqrt{X}}}\log p
Q(n) := \sum_{k^2+p = n}\log p.


Note that in Q(n), the ranges of k and p are unrestricted.

My question is:
I know that F(X,n) and Q(n) can be related by partial summation, but how do I prove this?

Any form of help is appreciated!

Thanks.

BUMP


Anyone?