LaTeX Need help -- Having trouble with eqnarray*/gathered/aligned

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AI Thread Summary
The discussion centers around the mathematical expression Z(n) involving multiple integrals and logarithmic functions. The user is attempting to format complex equations in LaTeX, specifically using the eqnarray* environment for better alignment of equal signs. They express frustration with Overleaf's handling of their LaTeX code, noting that it runs off the page when using the aligned environment. A participant points out a potential issue with unmatched braces in the code, suggesting that this could affect the compilation. The overall focus is on achieving the correct formatting for mathematical expressions in LaTeX while addressing technical challenges in the process.
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I have a long chain of equalities that I trying to get to display correctly with looking ugly or going off the page. I got this to work on this forum (MathJax) but having various issues in Overleaf
Code:
\begin{gathered} Z(n) = \int_{C^n}  {\tfrac{{\prod\nolimits_{i = 1}^n {d{x_i}} }}{{1 - \prod\nolimits_{k = 1}^n {{x_k}} }} \\ &=& \int_{C^{n - 1}}  {\log \left( {\tfrac{1}{{1 - \prod\nolimits_{k = 1}^{n - 1} {{x_k}} }}} \right) \cdot \prod\limits_{i = 1}^{n - 1} {\tfrac{{d{x_i}}}{{{x_i}}}} }   \\ &=& \int_{C^{n - 1}}   {\log \left[ {\prod\limits_{q = 0}^\infty  {\left( {1 + \prod\limits_{k = 1}^{n - 1} {x_k^{{2^q}}} } \right)} } \right] \cdot \prod\limits_{i = 1}^{n - 1} {\tfrac{{d{x_i}}}{{{x_i}}}} }  \\  = \sum\limits_{q = 0}^\infty  \int_{C^{n - 1}}  {\log \left( {1 + \prod\limits_{k = 1}^{n - 1} {x_k^{{2^q}}} } \right) \cdot \prod\limits_{i = 1}^{n - 1} {\tfrac{{d{x_i}}}{{{x_i}}}} }   \\ = \mathop  \lim\limits_{N \to \infty } \,\sum\limits_{q = 0}^\infty  {\sum\limits_{k = 1}^\infty  {\tfrac{{{{\left( { - 1} \right)}^{k - 1}}}}{k}\int_{C_N^{n - 1}}  {\prod\limits_{k = 1}^{n - 1} {x_k^{k{2^q} - 1}d{x_k}} } } }   \\ &=& \mathop  \lim\limits_{N \to \infty } \,\sum\limits_{q = 0}^\infty  {\sum\limits_{k = 1}^\infty  {\tfrac{{{{\left( { - 1} \right)}^{k - 1}}}}{k}{{\left( {n - 2} \right)}^{\sum\limits_{p = 1}^n {\frac{{k{2^q}}}{{2N}}} }}{{\prod\limits_{i = 1}^{n - 1} {\left[ {\frac{1}{{2N}}\Gamma \left( {\frac{{k{2^q}}}{{2N}}} \right)} \right]} } \mathord{\left/ {\vphantom {{\prod\limits_{i = 1}^{n - 1} {\left[ {\frac{1}{{2N}}\Gamma \left( {\frac{{k{2^q}}}{{2N}}} \right)} \right]} } {\Gamma \left( {1 + \sum\limits_{j = 1}^{n - 1} {\frac{{k{2^q}}}{{2N}}} } \right)}}} \right. } {\Gamma \left( {1 + \sum\limits_{j = 1}^{n - 1} {\frac{{k{2^q}}}{{2N}}} } \right)}}} }  \\ &=& \sum\limits_{q = 0}^\infty  {\sum\limits_{k = 1}^\infty  {\tfrac{{{{\left( { - 1} \right)}^{k - 1}}}}{{{k^n}{2^{\left( {n - 1} \right)q}}}} \cdot \underbrace {\mathop  \lim\limits_{N \to \infty } \,{{\left( {n - 2} \right)}^{\frac{{\left( {n - 1} \right)k{2^q}}}{{2N}}}}}_{ = {{\left( {n - 2} \right)}^0}} \cdot \underbrace {\mathop  \lim\limits_{N \to \infty } \,\tfrac{{{\Gamma ^{n - 1}}\left( {1 + \frac{{k{2^q}}}{{2N}}} \right)}}{{\Gamma \left( {1 + \tfrac{{\left( {n - 1} \right)k{2^q}}}{{2N}}} \right)}}}_{ = 1{\text{ Limit 2.4}}}} } \\  = \sum\limits_{q = 0}^\infty  {\sum\limits_{k = 1}^\infty  {\tfrac{{{{\left( { - 1} \right)}^{k - 1}}}}{{{k^n}}} \cdot {{\left( {\tfrac{1}{{{2^{n - 1}}}}} \right)}^q}} }  = \sum\limits_{k = 1}^\infty  {\left[ {\tfrac{{{{\left( { - 1} \right)}^{k - 1}}}}{{{k^n}}}\sum\limits_{q = 0}^\infty  {{{\left( {\tfrac{1}{{{2^{n - 1}}}}} \right)}^q}} } \right]}  \\ = {\left( {1 - {2^{1 - n}}} \right)^{ - 1}}\sum\limits_{k = 1}^\infty  {\tfrac{{{{\left( { - 1} \right)}^{k - 1}}}}{{{k^n}}}} = \sum\limits_{k = 1}^\infty  {\tfrac{1}{{{k^n}}}}  = \zeta \left( n \right)  \\ \end{gathered}

Note there were $'s at the beginning and end of the above code. I can't get it to compile with eqnarray in Overleaf so I posted the gathered. When I use aligned it ignores the newline commands and just runs off the page in just a single row.

This is what Overleaf does with that code

AltZetaLatex.png


Ideally we will get this to work with eqnarray*, that is how I want it to display with aligned equal signs, on the forum it looks like this (except in latex I used only a single integral sign to denote the multiple integrals)

AltZeta.png

Note the Latex code above is slightly different. Thanks. BTW this is the first time using an actual LaTeX editor.[/code]
 
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Your code looks incomprehensible to me. But I can determine that the '{' in character 37 does not seem to be matched by a closing '}'. Does that make a difference?
 
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