The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation

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The Riemann zeta function $latex {zeta(s)}&fg=000000$ is defined in the region $latex {hbox{Re}(s)>1}&fg=000000$ by the absolutely convergent series

$latex displaystyle zeta(s) = sum_{n=1}^infty frac{1}{n^s} = 1 + frac{1}{2^s} + frac{1}{3^s} + ldots. (1)&fg=000000$

Thus, for instance, it is known that $latex {zeta(2)=pi^2/6}&fg=000000$, and thus

$latex displaystyle sum_{n=1}^infty frac{1}{n^2} = 1 + frac{1}{4} + frac{1}{9} + ldots = frac{pi^2}{6}. (2)&fg=000000$

For $latex {hbox{Re}(s) leq 1}&fg=000000$, the series on the right-hand side of (1) is no longer absolutely convergent, or even conditionally convergent. Nevertheless, the $latex {zeta}&fg=000000$ function can be extended to this region (with a pole at $latex {s=1}&fg=000000$) by analytic continuation. For instance, it can be shown that after analytic continuation, one has $latex {zeta(0) = -1/2}&fg=000000$, $latex {zeta(-1) = -1/12}&fg=000000$, and $latex {zeta(-2)=0}&fg=000000$, and more generally

$latex displaystyle zeta(-s) = – frac{B_{s+1}}{s+1} (3)&fg=000000$

for $latex {s=1,2,ldots}&fg=000000$, where $latex {B_n}&fg=000000$ are the Bernoulli numbers. If one formally applies

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