How to write this in terms of ##\zeta (x)##?

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In summary, the conversation discusses how to re-write the expression $$\displaystyle\sum_{n=1}^\infty \frac{1}{n^x}*(\displaystyle\sum_{k \in S, \mathbb{Z}\S =n} \frac{1}{k^{yi}})$$ in terms of ##\zeta (x)## and how simplifying it in this way would avoid repetition. However, before simplifying the second term, it is necessary to specify what is being summed.
  • #1
MevsEinstein
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TL;DR Summary
I wrote $$\zeta (x+yi)$$ as ##\zeta(x)\zeta(yi) - \displaystyle\sum_{n=1}^\infty \frac{1}{n^x} *(\displaystyle\sum_{k \in S, \mathbb{Z}\S =n})##. I want to simplify the second term in terms of the zeta function so I can solve for ##\zeta (x)##.
How do I re-write $$\displaystyle\sum_{n=1}^\infty \frac{1}{n^x} *(\displaystyle\sum_{k \in S, \mathbb{Z}\S =n})$$ in terms of ##\zeta (x)## ? I want to solve for ##\zeta (x)## and simplifying the above expression in terms of ##\zeta (x)## would avoid repetition.
 
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  • #2
You have to fix the latex in your post. You can use dollar signs instead of hashes to make the whole line display style which might make it look better.
 
  • #3
Office_Shredder said:
You have to fix the latex in your post.
It took me forever to find the error. Turns out it was just a parentheses >D.
 
  • #4
Oh wait instead of \ it gives me ##\S=##
 
  • #5
MevsEinstein said:
How do I re-write $$\displaystyle\sum_{n=1}^\infty \frac{1}{n^x} *(\displaystyle\sum_{k \in S, \mathbb{Z}\S =n})$$
Before you can simplify the 2nd term, you need to say what is being summed.
 
  • #6
Mark44 said:
Before you can simplify the 2nd term, you need to say what is being summed.
OOPS! $$\displaystyle\sum_{n=1}^\infty \frac{1}{n^x}*(\displaystyle\sum_{k \in S, \mathbb{Z} \S =n} \frac{1}{k^{yi}})$$ There you go. Note that ##\mathbb{Z} \S =n## is actually ##\mathbb{Z}##\S = n, I couldn't fix the bug.
 

FAQ: How to write this in terms of ##\zeta (x)##?

1. How do I write an expression in terms of ##\zeta (x)##?

To write an expression in terms of ##\zeta (x)##, you need to replace any variables or constants in the expression with ##\zeta (x)##. For example, if the expression is ##x^2 + 3##, you can write it in terms of ##\zeta (x)## as ##\zeta (x)^2 + 3##.

2. What is the purpose of writing an expression in terms of ##\zeta (x)##?

Writing an expression in terms of ##\zeta (x)## can help simplify and generalize the expression. It can also make it easier to manipulate and solve for different values of ##x##.

3. Can any expression be written in terms of ##\zeta (x)##?

No, not all expressions can be written in terms of ##\zeta (x)##. It depends on the complexity and type of the expression. Some expressions may not have a simple form in terms of ##\zeta (x)##.

4. How do I know if I have written an expression correctly in terms of ##\zeta (x)##?

You can check if you have written an expression correctly in terms of ##\zeta (x)## by substituting different values for ##x## and comparing the results to the original expression. If they are equivalent, then you have written it correctly.

5. Can I use any other variable besides ##x## when writing an expression in terms of ##\zeta (x)##?

Yes, you can use any variable besides ##x## when writing an expression in terms of ##\zeta (x)##. Just make sure to replace all instances of the variable with ##\zeta (x)## in the expression.

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