[utreht (OP)]

hello, please tell me what is the definition of the zeros of the Riemann zeta function - I don't understand why its zeros are -2, -4, etc., because in spite of the substitution of s into the well-known functional equation, on the other hand it turns out, for example, for -2 that 1 + 4 + 9 + ... + n * n = 0, which is absurd for an unboundedly increasing n. Thanks!

[evan_au]

I suggest that you watch the series of videos about the Rieman Zeta function created by Numberphile.
- I understood it a lot better after I watched these, in the past year.
- Infinite sums are hard to get your head around, even when they do converge (eg Zeno's paradox of Achilles & the Tortoise)
- When they diverge (eg the sum of the integers or the sum of the squares), it's hard to say anything about them except + or - ∞
- The radical solution to the Rieman Zeta function for imaginary numbers < 0.5 (as I recall) is creating a mapping from the imaginary numbers > 0.5 onto the other half of the plane.
- This produces valid answers for imaginary numbers
- But produces some shockers like the sum of the positive integers = -1/12
- In some areas of maths and quantum physics, it really behaves like this is a valid answer
- This has been used to eliminate some annoying infinities in quantum theory (and still get answers that agree with the real world!)
- So it acts as if it is a "correct" answer, even though it is counter-intuitive...

For example:

...there are more - just Google Numberphile and Riemann or Zeta (there is even a video to repudiate it!)