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

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

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

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

Any help is appreciated!

Thanks.
 
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flouran said:
Let [tex]k[/tex] and [tex]n \le X[/tex] be large positive integers, and [tex]p[/tex] is a prime. Define

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


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

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

Any form of help is appreciated!

Thanks.

BUMP


Anyone?