If 1+2+3+... can be "regularised" to -1/12, does it follow that 1+4+9+... can be 0 or that 1+8+27+... can be 1/120?

Reddit - AskScience: Got Questions? Get Answers. 01/21/18

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0 and 1/120 being the zeta function at -2 and -3? What about 1+1+1+... being replaced by ζ(0)=-1/2? Have these ones ever come up in some useful application?

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