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

In summary, the conversation discusses the functions F(X,n) and Q(n), where F(X,n) is defined as the sum of logarithms of primes p that satisfy certain conditions, and Q(n) is defined as the sum of logarithms of primes p without any restrictions on the values of k and p. The question posed is how to prove the relationship between these two functions using partial summation. Assistance is requested in solving this problem.
  • #1
flouran
64
0
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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  • #2
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?
 

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

What is partial summation?

Partial summation is a mathematical technique used to calculate the sum of a series by breaking it into smaller parts and then adding them together. It is similar to the concept of partitioning in division.

What is the purpose of partial summation?

The purpose of partial summation is to simplify the calculation of a series by breaking it into smaller parts. This can make the calculation more manageable and easier to solve.

How do I use partial summation?

To use partial summation, you need to first identify a pattern or formula for the series. Then, you can break the series into smaller parts and apply the formula to each part. Finally, you add all the parts together to get the total sum of the series.

What types of series can be solved using partial summation?

Partial summation can be used to solve arithmetic, geometric, and telescoping series. It can also be applied to infinite series, as long as the series converges.

Are there any limitations to using partial summation?

Partial summation may not work for series that do not have a clear pattern or formula. It also may not work for series that do not converge. Additionally, partial summation may not provide an exact solution for some series, but rather an approximation.

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