Just one reason (of very many): it’s not an analytic question.
Here are the values of zeta on the 1/2-line (where at least 40% of the zeros are, all should be) and the 4/5-line, where none should be.
The latter gets arbitrarily close to 0, i.
o.
pic.
twitter.
com/qSHhwwCY6X — Alex Kontorovich (@AlexKontorovich) November 25, 2019 The critical strip of the Riemann zeta function is the portion of the complex plane with real part between 0 and 1.
The Riemann hypothesis conjectures that the zeros of the zeta function in this region all have real part 1/2.
To put it another way, the Riemann hypothesis says that ζ(x + it) is never zero unless x = 1/2.
However, according to Alex Kontorovich’s tweet, the function gets arbitrarily close to zero infinitely often for other values of x.
I wrote a little Mathematica code to produce graphs like the ones above.
Here is a plot of ζ(1/2 + it).
ParametricPlot[ {Re[Zeta[1/2 + I t]], Im[Zeta[1/2 + I t]]}, {t, 0, 100}, AspectRatio -> 1, PlotRange -> {{-2, 6}, {-3, 3}} ] At first I didn’t specify the aspect ratio and I got nearly circular arcs.
Then I realized that was an artifact of Mathematica’s default aspect ratio.
Also, I set the plot range so the image above would be comparable with the next image.
Notice all the lines crossing the origin.
These correspond to the zeros of ζ along the line Re(x) = 1/2.
Next I changed the 1/2 to 1/3.
If you look closely you’ll see a gap around the origin.
Here’s a closeup of the origin in both graphs.
First for x = 1/2 and now for x = 1/3.
I hadn’t seen the result in Kontorovich’s tweet before.
I don’t know whether it holds for all x or just for certain values of x.
I assume the former.
But apparently it at least holds for x = 4/5.
So we should be able to find values of |&zeta(4/5 + it)| as small as we like by taking t large enough, and we should always be able to find more such values by looking further out.
In practice, you might have to very far out before you find a small value.
Looking as far as 1000, for example, it seems the minimum is about 0.
2.
Plot[ Abs[Zeta[4/5 + I t]], {t, 0, 1000}, PlotRange -> {0, 1} ] If you ask Mathematica to find the minimum in the graph above, it won’t give a reasonable answer; there are too many local minima.
If instead you ask it to look repeatedly over small intervals, it does much better.
The following minimizes |ζ(4/5 + it)| over the interval [i, i+1].
f[i_] := FindMinimum[ {Abs[Zeta[4/5 + I t]], i 947.
}} As I write this I have a longer program running, and so far the smallest value Ive found is 0.
1809 which occurs at t = 1329.
13.
According to Kontorovich you can find as small a value as you like, say less than 0.
1 or 0.
001, but apparently it might take a very long time.
Update: The minimum over 0 ≤ t ≤ 10,000 is 0.
142734, which occurs at 7563.
56.
Related posts Distribution of the zeros of ζ Bernoulli numbers, Riemann zeta, and strange sums Probability of coprime sets.