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The Riemann Zeta Function is defined as $ \displaystyle \zeta(s) = \sum\limits_{n=1}^{\infty} \frac{1}{n^s}$. It is not absolutely convergent or conditionally convergent for $\text{Re}(s) \leq 1$. Using analytic continuation, one can derive the fact that $\displaystyle \zeta(-s) = -\frac{B_{s+1}}{s+1}$ where $B_{s+1}$ are the Bernoulli numbers. Can one obtain this result without resorting to analytic continuation?

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    What is $B_s$?.2010-11-03
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    I suppose Trevor means the Bernoulli numbers.2010-11-03
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    So you don't want to use the [reflection relation](http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta/17/01/01/0002/)?2010-11-03
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    If you do not do analytic continuation, what do you mean by $\zeta(s)$ for negative integer $s$?2010-11-03
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    Yep. Even $\eta$, whose Dirichlet series has a wider region of convergence, has to be continued to be well-defined in the left half of the complex plane.2010-11-03
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    Your formula for zeta at -s should not use the variable s. There you *specifically* mean s is a positive integer. Try zeta(-k) = -B_{k + 1}/(k + 1) where k is a positive integer. To answer your main question, you simply need some way to construct an analytic formula for the zeta-function that makes sense beyond the half-plane where the series is defined. There may be ways to do this without explicitly using the magic words "analytic continuation", but unless you're Euler any method you use is likely to provide an analytic continuation.2010-11-03
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    Related: http://mathoverflow.net/questions/13130/historical-question-in-analytic-number-theory/13417#134172013-05-31

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