simplemind
JF-Expert Member
- Apr 10, 2009
- 16,418
- 9,209
Riemann's proof of the prime number theorem uses integration of 1/zeta along a contour that looks like a vertical line in the complex plane. He got an error term of size n^(real part of the contour).
As we know what's going on with zeroes outside the critical strip his result is an error term of size n. The Riemann Hypothesis is the best possible error term, yielding n^.5. Analytic number theory is sensitive enough to such computations that one can squeeze a lot of results out of a tighter error bound, but I am not competent to describe them.
I can describe some compelling evidence for why the problem is interesting: the best results I know of for zero free regions are due to Selberg. He proved there is a zero free region that looks roughly like a Gaussian pushing into the critical strip, but the boundary of this region tends to the line real part = 1 as the imaginary part gets very large! So even our best results after years of trying hard aren't enough to improve the error term in Riemann's original proof! Often times when one discovers that the best methods one has get tantalizingly close to an improvement it indicates that there is a beautiful idea one is missing, so my interpretation is something wonderful is out there waiting to be discovered.
In a different context, Weil discovered that when they hung numbers of points of varieties over finite fields on the Zeta function that an analogy to the Riemann Hypothesis appeared to hold. In the 70's beautiful results of Groethendieck, Deligne, and many others came together to prove this. The ideas linked topology, number theory, and algebraic geometry together and proved incredibly fruitful in many other areas. Groethendieck had hoped these methods would get the Riemann Hypothesis and also the Hodge Conjectures, but the program proved extremely difficult.
Written Dec 29 •
As we know what's going on with zeroes outside the critical strip his result is an error term of size n. The Riemann Hypothesis is the best possible error term, yielding n^.5. Analytic number theory is sensitive enough to such computations that one can squeeze a lot of results out of a tighter error bound, but I am not competent to describe them.
I can describe some compelling evidence for why the problem is interesting: the best results I know of for zero free regions are due to Selberg. He proved there is a zero free region that looks roughly like a Gaussian pushing into the critical strip, but the boundary of this region tends to the line real part = 1 as the imaginary part gets very large! So even our best results after years of trying hard aren't enough to improve the error term in Riemann's original proof! Often times when one discovers that the best methods one has get tantalizingly close to an improvement it indicates that there is a beautiful idea one is missing, so my interpretation is something wonderful is out there waiting to be discovered.
In a different context, Weil discovered that when they hung numbers of points of varieties over finite fields on the Zeta function that an analogy to the Riemann Hypothesis appeared to hold. In the 70's beautiful results of Groethendieck, Deligne, and many others came together to prove this. The ideas linked topology, number theory, and algebraic geometry together and proved incredibly fruitful in many other areas. Groethendieck had hoped these methods would get the Riemann Hypothesis and also the Hodge Conjectures, but the program proved extremely difficult.
Written Dec 29 •
