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Anonymous
The "sum" of all natural numbers 2017-06-12 06:51:04 Post No. 8971231
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The "sum" of all natural numbers 2017-06-12 06:51:04 Post No. 8971231
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Hello /sci/,
Riddle me this. If ζ(-1) = 1 + 2 + 3 + ... is -1/12, and ζ(0) = 1 + 1 + 1 ... is -1/2, then wouldn't ζ(-1) - ζ(0) = 0 + 1 + 2 + 3 + ... also equal -1/12? But clearly -1/12 - (-1/2) ≠ -1/12, right?
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>>8971231
Rieman zeta is an analytical continuation. The definitions are defined by a unique continuation from the valid range, not by addition and algebra.
So no
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>>8971239
Then why is it that in physics ζ(-1) is used in place of 1 + 2 + 3 ... ?
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>>8971239
By the way, to be clear I already know that the values of ζ(x + iy) with x < 1 are uniquely defined due to the properties of meromorphic functions. I'm questioning why those values are being used to define the value of series like 1 + 2 + 3 ...
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>>8971247
Because physicists are mathematically illiterate.
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>>8971392
Actually, mathematicians are physically illiterate. They create mathematics that does not represent the logic of this universe.
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>>8971231
The answer is -1/12 - (-1/2) = 5/12
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