How to find the constant B in here?

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Because ##\zeta(2)^2=\left(\dfrac{\pi^2}{6}\right)^2=\dfrac{\pi^4}{6\cdot 6}.##In summary, the conversation is about determining the constant ##B## in the equation ##\sum_{n\leq x}\frac{d(n)}{n^2}=B-\frac{\log {x}}{x}+O(\frac{1}{x})## and whether ##\zeta(2)^2## is sufficient or if it needs to be simplified further. It is concluded that ##B = \zeta(2)^2## is sufficient and does not need to be simplified into ##\frac{\pi^{2
  • #1
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Homework Statement
Prove that, for ## x\geq 2 ##, ## \sum_{n\leq x}\frac{d(n)}{n^2}=B-\frac{\log x}{x}+O(\frac{1}{x}) ##, where ## B ## is a constant that you should determine.
Relevant Equations
If ## x\geq 2 ## and ## \alpha>0, \alpha\neq 1 ##, then ## \sum_{n\leq x}\frac{d(n)}{n^{\alpha}}=\frac{x^{1-\alpha}\log {x}}{1-\alpha}+\zeta (\alpha)^2+O(x^{1-\alpha}) ##.
Proof:

Let ## x\geq 2 ##.
Observe that
\begin{align*}
&\sum_{n\leq x}\frac{d(n)}{n^2}=\sum_{d\leq x}\frac{1}{d^2}\sum_{q\leq \frac{x}{d}}\frac{1}{q^2}\\
&=\sum_{d\leq x}\frac{1}{d^2}(\frac{(x/d)^{1-2}}{1-2}+\zeta {2}+O(\frac{1}{(x/d)^{2}}))\\
&=\frac{x^{1-2}}{1-2}\sum_{d\leq x}\frac{1}{d}+\zeta (2)\sum_{d\leq x}\frac{1}{d^2}+O(x^{1-2})\\
&=\frac{x^{1-2}}{1-2}(\log {x}+C+O(\frac{1}{x}))+\zeta {2}(\frac{x^{1-2}}{1-2}+\zeta {2}+O(x^{-2}))+O(x^{1-2}))\\
&=\frac{x^{1-2}\log {x}}{1-2}+C\cdot \frac{x^{1-2}}{1-2}+O(x^{-2})+\zeta (2)\cdot \frac{x^{1-2}}{1-2}+\zeta (2)^2+O(x^{-2})+O(x^{1-2})\\
&=\frac{x^{1-2}\log {x}}{1-2}+\zeta (2)^2+O(x^{1-2}).\\
\end{align*}
Thus ## \sum_{n\leq x}\frac{d(n)}{n^2}=B-\frac{\log {x}}{x}+O(\frac{1}{x}) ##, where ## B=\zeta (2)^2 ##.
Therefore, ## \sum_{n\leq x}\frac{d(n)}{n^2}=\zeta (2)^2-\frac{\log {x}}{x}+O(\frac{1}{x}) ## for ## x\geq 2 ##.
 
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  • #2
Is [itex]B = \zeta(2)^2[/itex] not sufficient, or do you need to use the standard result [tex]
\zeta(2) = \sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}?[/tex]
 
  • #3
pasmith said:
Is [itex]B = \zeta(2)^2[/itex] not sufficient, or do you need to use the standard result [tex]
\zeta(2) = \sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}?[/tex]
This was my question. How should I determine the constant ## B ## in here? Is ## \zeta (2)^2 ## really sufficient? Or do I need to simplify/evaluate this value more, into ## \frac{\pi^{2}}{6} ##?
 
  • #4
Math100 said:
This was my question. How should I determine the constant ## B ## in here? Is ## \zeta (2)^2 ## really sufficient? Or do I need to simplify/evaluate this value more, into ## \frac{\pi^{2}}{6} ##?
I would normally substitute it with ##\pi^2/6## in the final answer. But here we have a ##B## that swallows all constants wherever they come from, so there is no need for the substitution, a constant is a constant, no matter which. However, you should write ##-1## and not ##1-2.##
 
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  • #5
fresh_42 said:
I would normally substitute it with ##\pi^2/6## in the final answer. But here we have a ##B## that swallows all constants wherever they come from, so there is no need for the substitution, a constant is a constant, no matter which. However, you should write ##-1## and not ##1-2.##
So ## \zeta (2)^2=\zeta (4)=\sum_{n=1}^{\infty}\frac{1}{n^4}=\frac{\pi^{4}}{90} ##? And ## B=\frac{\pi^{4}}{90} ##?
 
  • #6
Math100 said:
So ## \zeta (2)^2=\zeta (4)=\sum_{n=1}^{\infty}\frac{1}{n^4}=\frac{\pi^{4}}{90} ##? And ## B=\frac{\pi^{4}}{90} ##?
## \zeta (2)^2=\dfrac{\pi^4}{36}\neq \dfrac{\pi^4}{90}=\zeta (4).##
 
  • #7
fresh_42 said:
## \zeta (2)^2=\dfrac{\pi^4}{36}\neq \dfrac{\pi^4}{90}=\zeta (4).##
Why ## \frac{\pi^{4}}{36} ##?
 
  • #8
Math100 said:
Why ## \frac{\pi^{4}}{36} ##?
Because ##\zeta(2)^2=\left(\dfrac{\pi^2}{6}\right)^2=\dfrac{\pi^4}{6\cdot 6}.##
 
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FAQ: How to find the constant B in here?

1. How do I determine the value of the constant B?

The value of the constant B can be found by using the formula B = y/x, where y is the dependent variable and x is the independent variable. This formula is commonly used in linear regression analysis to find the slope of a line.

2. What is the purpose of finding the constant B?

The constant B is used to represent the slope of a line in a linear equation. It helps to determine the relationship between the independent and dependent variables in a given dataset.

3. Can I use any method to find the constant B?

There are various methods that can be used to find the constant B, such as the least squares method, the method of moments, and the method of maximum likelihood. The choice of method depends on the specific data and the type of analysis being performed.

4. Is the constant B always a fixed value?

No, the value of the constant B can vary depending on the dataset and the chosen method of analysis. It is important to carefully consider the data and the method being used to ensure an accurate and meaningful value for B is obtained.

5. How can I interpret the value of the constant B?

The value of the constant B can be interpreted as the amount of change in the dependent variable for every one unit change in the independent variable. For example, if B is equal to 2, it means that for every one unit increase in the independent variable, the dependent variable will increase by 2 units.

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