Riemann Zeta function of even numbers

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SUMMARY

The Riemann Zeta function for even integers is expressed as \(\zeta(2n) = \frac{\pi^{2n}}{m}\), where \(m\) varies based on the natural number \(n\). The values of \(m\) for \(n\) from 1 to 10 are: 6, 90, 945, 9450, 93555, \(\frac{638512875}{691}\), \(\frac{18243225}{2}\), \(\frac{325641566250}{3617}\), \(\frac{38979295480125}{43867}\), and \(\frac{1531329465290625}{174611}\). No discernible pattern in the values of \(m\) was identified in the discussion. The user suggested consulting Wikipedia for further insights.

PREREQUISITES
  • Understanding of the Riemann Zeta function
  • Familiarity with mathematical notation and series
  • Knowledge of natural numbers and their properties
  • Basic understanding of mathematical constants, particularly \(\pi\)
NEXT STEPS
  • Research the derivation of the Riemann Zeta function values for even integers
  • Explore the relationship between \(\zeta(2n)\) and Bernoulli numbers
  • Study the implications of the Riemann Hypothesis on even and odd zeta values
  • Investigate the historical context and significance of the values of \(m\) in number theory
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Mathematicians, number theorists, and students studying advanced calculus or complex analysis will benefit from this discussion on the Riemann Zeta function of even numbers.

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Given that

\zeta (2n)=\frac{{\pi}^{2n}}{m}

Then how do you find m with respect to n where n is a natural number.

For

n=1, m=6
n=2, m=90
n=3, m=945
n=4, m=9450
n=5, m=93555
n=6, m=\frac{638512875}{691}
n=7, m=\frac{18243225}{2}
n=8, m=\frac{325641566250}{3617}
n=9, m=\frac{38979295480125}{43867}
n=10, m=\frac{1531329465290625}{174611}

But I don't see any pattern.

Thanks.
 
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I would find it by consulting wikipedia.
 
Thanks.
 

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