The Reimann Hypothesis is an interesting topic to me, but as a high school student, I can’t seem to find an understandable definition of a trivial zero. How could an infinite sum result in nothing?
In: Mathematics
It’s not an infinite sum. An easier example may come from geometric series, with sums of the form x^0 + x^1 + x^2 + x^3 + …. For any real number x strictly between -1 and +1, it turns out that this sum will evaluate to 1/(1-x): for example, 1 + 1/2 + 1/4 + 1/8 + … = 1/(1-1/2) = 1, and 1 – 1/3 + 1/9 – 1/27 + … = 1/(1-(-1/3)) = 3/4. Outside this range, the infinite sum does not converge to anything, and has no well-defined value in the real numbers. However, the function 1/(1-x) is defined almost everywhere outside this range (everywhere except at x = 1, in fact), so we could view this function as a sort of natural continuation of geometric series outside the range where they are actually well defined. The sum 1 + 2 + 4 + 8 + … diverges to infinity very quickly, but if we really wanted to assign it a value for some reason, our extension function would suggest a value of 1/(1-2) = -1. Of course it is not the case that 1 + 2 + 4 + 8 + … = -1, but the value -1 is associated in some way with this sum by the extension we are using.
The Riemann zeta function is a similar but much more complicated function extending sums of the form 1/1^x + 1/2^x + 1/3^x + 1/4^x + …. A sum of this form only converges when x > 1, and the Riemann zeta function agrees with the actual value of the sum on those inputs, but again the Riemann zeta function is also defined in most places outside this range. So, for example, zeta(-1) = -1/12. Bad pop-math articles will use this as a demonstration of the “fact” that 1 + 2 + 3 + 4 + … = -1/12, but of course this is not really true. The sum 1 + 2 + 3 + 4 + … diverges just as 1 + 2 + 4 + 8 + … does. It’s just that the zeta function takes on the value of -1/12 at -1, so -1/12 is associated with the infinite sum by our choice of extension function. Similarly, at any negative even integer (-2, -4, -6, etc.), the zeta function takes on a value of 0, but none of these represents the actual value of an infinite sum.
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