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I'd like to introduce a function, which I call Y function, just because that's how its plot look like in complex plane. It can also called upsilon function, just to sound more geeky. It's defined as follow:Y(s)=ln{ζ(s) - ζ(1-s*)}
When m =1, the low slope part isn't visible.
I played around with this Y function, and found something interesting when 1 in the formula is replaced with other numbers. The plot looks simpler when s has high imaginary part.
The logarithm of a complex number z=log(r*e^(iθ)) is condensed as log(z)=log(r)+iθ, where r is the absolute value of z and θ is the argument of z. It is represented in polar coordinates.
when imaginary part of s is close to the smallest non-trivial zero of Riemann's zeta functionhttps://www.wolframalpha.com/input?i=plot+%28ln%28%28zeta%28s%2B14.134725+i%29%29+-+%28Zeta%281-s%2B14.134725+i%29%29%29+%29from+-22+to+23Any larger value for the imaginary part of s don't change shape of the real part of Y function, only the imaginary part gets longer period.https://www.wolframalpha.com/input?i=plot+%28ln%28%28zeta%28s%2B114.134725+i%29%29+-+%28Zeta%281-s%2B114.134725+i%29%29%29+%29from+-22+to+23
The Riemann Hypothesis is the most notorious unsolved problem in all of mathematics. Ever since it was first proposed by Bernhard Riemann in 1859, the conjecture has maintained the status of the "Holy Grail" of mathematics. In fact, the person who solves it will win a $1 million prize from the Clay Institute of Mathematics. So, what is the Riemann hypothesis? Why is it so important? What can it tell us about the chaotic universe of prime numbers? And why is its proof so elusive? Alex Kontorovich, professor of mathematics at Rutgers University, breaks it all down in this comprehensive explainer. 00:00 A glimpse into the mystery of the Riemann Hypothesis 01:42 The world of prime numbers02:30 Carl Friedrich Gauss looks for primes, Prime Counting Function03:30 Logarithm Function and Gauss's Conjecture04:39 Leonard Euler and infinite series06:30 Euler and the Zeta Function07:30 Bernhard Riemann enters the prime number picture08:18 Imaginary and complex numbers09:40 Complex Analysis and the Zeta Function10:25 Analytic Continuation: two functions at work at once11:14 Zeta Zeros and the critical strip12:20 The critical line12:51 Why the Riemann's Hypothesis has a profound consequence to number theory13:04 Riemann's Hypothesis shows the distribution of prime numbers can be predicted14:59 The search for a proof of the Riemann Hypothesis
Why is the Riemann Hypothesis hard? 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.https://www.johndcook.com/blog/2019/11/29/near-zeros-of-zeta/
Any idiot can build a bridge that stands, but it takes an engineer to build a bridge that barely stands.
when ζ(1-s)=0 → ζ(s)=0, ζ(s*)=0, ζ(1-s*)=0The sum and difference among those terms must also equal 0
Let's focus on the imaginary part of the addition equation.https://www.wolframalpha.com/input?i=plot+im%28zeta%280.5%2Bis%29+%2B+zeta%280.5-is%29%29+from+0+to+30And the real part of the subtraction equation.https://www.wolframalpha.com/input?i=plot+re%28zeta%280.5%2Bis%29+-+zeta%280.5-is%29%29+from+0+to+30They are always zero.
Compare them to the value of s where the real part is slightly different from 0.5.https://www.wolframalpha.com/input?i=plot+im%28zeta%280.5001%2Bis%29+%2B+zeta%280.4999-is%29%29+from+0+to+30https://www.wolframalpha.com/input?i=plot+re%28zeta%280.5001%2Bis%29+-+zeta%280.4999-is%29%29+from+0+to+30
Another result I got from searching. QuoteWhy is the Riemann Hypothesis hard? 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.https://www.johndcook.com/blog/2019/11/29/near-zeros-of-zeta/
https://www.wolframalpha.com/input?i=plot+%7Czeta%281%2F2+%2B+s.+i%29%7C+from+0+to+30https://www.wolframalpha.com/input?i=plot+%7Czeta%281%2F2+%2B+s.+i%29%7C+from+1000+to+1030The plots above show that at higher imaginary part of s, the density of Zeta zero is higher. It reduces the space for anomalous zero. Thus the higher the imaginary part, the less likely we can find anomalous zero of Zeta function. It provides a stronger indication that Riemann's hypothesis is true.
The Riemann zeta function ζ(s) is a function whose argument s may be any complex number other than 1, and whose values are also complex. It has zeros at the negative even integers; that is, ζ(s) = 0 when s is one of −2, −4, −6, .... These are called its trivial zeros. The zeta function is also zero for other values of s, which are called nontrivial zeros. The Riemann hypothesis is concerned with the locations of these nontrivial zeros, and states that:The real part of every nontrivial zero of the Riemann zeta function is 1/2Thus, if the hypothesis is correct, all the nontrivial zeros lie on the critical line consisting of the complex numbers 1/2 + i t, where t is a real number and i is the imaginary unit.https://en.wikipedia.org/wiki/Riemann_hypothesis
Corrections to an estimate of the prime-counting function using zeros of the zeta function. The magnitude of the correction term is determined by the real part of the zero being added in the correction.https://en.m.wikipedia.org/wiki/Riemann_hypothesis