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
Trivial is the term of art there, zero’s is just the result of the hypothesis in question.
Triviality in mathematics just means roughly what it does every where else; unimportant. From Wolfram: “Related to or being the mathematically most simple case. More generally, the word “trivial” is used to describe any result which requires little or no effort to derive or prove.”
So in the case of, “All non-trivial zeros of the Zeta function have real part one-half.” it’s just letting you know in a formal sort of way to exclude that set of trivial zeros the function spits out.
For a more technical, not ELI5 explanation the second answer here is a good one: https://math.stackexchange.com/questions/138112/what-does-it-really-mean-for-something-to-be-trivial
>The trivial map usually sends either everything to 0 or to 1 or to itself, depending on the context.
>The trivial solutions to xn+yn=zn
are when x=y=z=0, x=z=1 and y=0, or y=z=1 and x=0
>The trivial subgroups of a group are the whole group itself and the one-element subgroup.
>Or it means that the author doesn’t want to spell out something which is (in the author’s view) and easy/obvious fact. I think most commonly, though, it means a student is bluffing on his homework with something he or she can’t quite prove.
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.
“trivial”=”boring/obvious”, so this is not a special type of zero, but just the ones we don’t care about.
The Riemann hypothesis than simply reads “all other zeros than that have real part 1/2”.
> How could an infinite sum result in nothing?
Take an infinite sum that results in something, e.g. 1+1/2+1/4+ … =2.
Subtract that from the first term to get -1+1/2+1/4+…=0.
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