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?