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On the Wolfram Mathworld page for the Riemann zeta function (here), just after equation $20$, we have the sentence:

While this formula defines $\zeta(s)$ for only the right half-plane $\Re[s]>0$, equation (◇) can be used to analytically continue it to the rest of the complex plane.

What is "equation (◇)"? Does it mean the last equation mentioned? It's used at several points in the article, and my mathematical intuition isn't strong enough to to figure out the meaning from the context.

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    $\begingroup$ I would not be surprised if it was used (four times) as a placeholder with the intent to fill in the number manually later, and that was never done. $\endgroup$
    – Mark S.
    Commented Sep 25, 2020 at 12:15
  • $\begingroup$ Some software cannot translate properly the expression (n) where n is not only a number, but a reference (a pointer) to equation n. This reference number will automatically be updated if an expression with a reference number is added or taken out of the text. So a text which has migrated from Maple to Word could lose its equation reference numbers and all be replaced by a marker, in this case (◇). $\endgroup$
    – MasB
    Commented Sep 25, 2020 at 12:35
  • $\begingroup$ Really interesting. I messaged the Mathworld editor a couple of weeks ago about it, heard nothing. Ah well, It's still an amazing resource. $\endgroup$ Commented Sep 25, 2020 at 15:22

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In this case it isn't the last equation mentioned, because we got equation (20) from (19) by rearranging the terms. The equation (◇) is:

$$\Gamma\left(\frac{s}{2}\right)\pi^{-\frac{s}{2}}\zeta(s)=\Gamma\left(\frac{1-s}{2}\right)\pi^{-\frac{-1-s}{2}}\zeta(1-s)$$ Which is the reflection formula for $\zeta(s)$.

Why? well, we can define a function $f(s)$ as the left hand side, and then $f(s)=f(1-s)$. So, if we know enough about the Gamma function (which we do), using this functional equation we can analytically continue $\zeta(s)$ to the complex plane for $s\neq 1.$

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  • $\begingroup$ Hi @David Kipper. Thanks for this. Is it always the case that the phrase means something like “the same equation” or “that last equation”? $\endgroup$ Commented Sep 25, 2020 at 11:59
  • $\begingroup$ It really depends on the context. In this case it didn't, while in other cases, it might be the previous equation, and in such a case I would expect to see some equation marked with (◇) appear right before the reference to it. $\endgroup$
    – user764256
    Commented Sep 25, 2020 at 13:08

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