Approximaths

1 + 2 + 3 + 4 + …

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To calculate the value of the Riemann zeta function at s = -1, we use the functional equation (presented here):

\displaystyle \zeta(s) = \pi^{s/2} \frac{\xi(s)}{\Gamma(s/2)}

The Dirichlet series for the Riemann zeta function is defined for \Re(s) > 1 as:

\displaystyle \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}

For s = -1, the formal series (which is divergent in the classical sense) is:

\displaystyle \zeta(-1) = \sum_{n=1}^{\infty} \frac{1}{n^{-1}}= \sum_{n=1}^{\infty} n = 1 + 2 + 3 + 4 + \dots

Substituting s = -1 into the formula, we obtain:

\displaystyle \zeta(-1) = \pi^{-1/2} \frac{\xi(-1)}{\Gamma(-1/2)}

Using the recurrence property of the Gamma function, \Gamma(z) = \frac{\Gamma(z+1)}{z}, and the fact that \Gamma(1/2) = \sqrt{\pi}:

\displaystyle \Gamma(-1/2) = \frac{\Gamma(-1/2+1)}{-1/2} = \frac{\Gamma(1/2)}{-1/2} = -2\sqrt{\pi}

We exploit the functional equation \xi(s) = \xi(1-s), which implies:

\displaystyle \xi(-1) = \xi(1 - (-1)) = \xi(2)

From the definition of \xi(s), we have:

\displaystyle \xi(2) = \pi^{-2/2} \Gamma(2/2) \zeta(2)

Using \Gamma(1) = 1 and the solution of the Basel problem presented in the previous post \zeta(2) = \frac{\pi^2}{6}:

\displaystyle \xi(-1) = \pi^{-1} \cdot 1 \cdot \frac{\pi^2}{6} = \frac{\pi}{6}

Combining these results into our original expression:

\displaystyle \zeta(-1) =\frac{1}{\sqrt{\pi}} \cdot \frac{\frac{\pi}{6}}{-2\sqrt{\pi}}
\displaystyle = \frac{\pi}{-12 \cdot (\sqrt{\pi} \cdot \sqrt{\pi})}
\displaystyle = \frac{\pi}{-12\pi}
\displaystyle \boxed{\zeta(-1) = -\frac{1}{12}}

Though surprising at first, turns out to be extremely useful in physics — most notably in string theory and the calculation of the Casimir effect, where the infinite sum 1 + 2 + 3 + 4 + … = −1/12 naturally appears when regularizing the vacuum energy between two conducting plates.

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