Naked Science Forum
General Science => General Science => Topic started by: utreht on 05/10/2020 20:51:27

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 wellknown 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!

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