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Just Chat! / Re: Is there a universal moral standard?
« Last post by alancalverd on Yesterday at 15:22:24 »rational thinkingExplain "rational" in terms of religious war.
rational thinkingExplain "rational" in terms of religious war.
We verify numerically, in a rigorous way using interval arithmetic, that the Riemann hypothesis is true up to height 3*10^12
. That is, all zeroes β+iγ of the Riemann zeta-function with 0 < γ ≤ 3*10^12
have β = 1/2
. Moreover, all of these zeroes are simple.
Abstract
The Vinogradov?Korobov zero-free region for the Riemann zeta function ζ(σ+it) states that there exists an absolute and effectively computable constant c>0 such that ζ(σ+it) when |t|≥10 and σ>1-cX. Ford proved that one may take c=1/57.54, and that this improves to 1/49.13 when |t| is sufficiently large. In this presentation, I discuss recent work in which I use Ford?s ideas to prove an explicit result of similar strength for Dirichlet L-functions.
Conclusion
While the above method has given us many of the zero-free regions, including many of the most powerful zero-free regions, there are some other methods of deriving such results. It should be noted that all the zero-free regions currently existing have their width asymptotically approach zero when t approaches infinity, and this will be the case for all regions derived using Theorem 1. In the notes to chapter 3 of [6] some other methods of deriving zero-free regions are looked at, however none give improvements on the Vinogradov-Korobov region.
This is an intriguing question that touches on the heart of the Riemann Hypothesis (RH) and its potential violations. To answer this, we need to consider a few key points:Those mathematicians seem to have realized that the larger deviation from critical line lead to less likely to find zero of Zeta function. The most likely place to find non-trivial zero is where the deviation from critical line is exactly zero.
The Critical Line: In the context of the Riemann Hypothesis, the critical line is the vertical line in the complex plane with real part 1/2. The RH states that all non-trivial zeros of the Riemann zeta function lie on this line.
Potential Exceptions: If an exception to the RH were found, it would be a zero of the zeta function off the critical line but still within the critical strip (the region of the complex plane with real part between 0 and 1).
Computational Evidence: Extensive computational searches have verified the RH for the first several billion zeros, all of which lie on the critical line. This suggests that if an exception exists, it would likely be found at a very high height on the critical strip.
Theoretical Bounds: There are some theoretical results that constrain where potential exceptions could occur. For example, the Vinogradov-Korobov zero-free region shows that any potential exception must be very close to the critical line at sufficiently large heights.
Given these considerations, if an exception to the Riemann Hypothesis were found:
It would likely have a very large imaginary part (i.e., be very high up in the critical strip).
The deviation from the critical line would likely be extremely small.
While it's impossible to give an exact expected deviation without more context, some mathematicians have speculated that if an exception exists, its real part might differ from 1/2 by less than 10^(-10), or even much less.
It's worth noting that most mathematicians believe the Riemann Hypothesis is true, and thus expect no exceptions to be found. The search for potential exceptions is largely motivated by the desire to either prove the RH or to understand why it might be false if an exception is found.
Let's just focus on one side, due to its symmetrical shape.https://www.wolframalpha.com/input?i=plot+re+%28log%28%28zeta%280.5-e-111%2B+s+i%29%29+%2F+%28Zeta%280.5%2Be-111%2B+s+i%29%29%29%29+from+10000+to+30000
https://www.wolframalpha.com/input?i=plot+re+%28log%28%28zeta%280.49%2B+s+i%29%29+%2F+%28Zeta%281-0.49+%2B+s+i%29%29%29%29+from+0+to+90000
Violation of Riemann's Hypothesis requires that at some point with high imaginary part, the output of V function mysteriously drops down to 0.
https://www.wolframalpha.com/input?i=plot+re+%28log%28%28zeta%280.499%2B+s+i%29%29+%2F+%28Zeta%281-0.499+%2B+s+i%29%29%29%29+from+0+to+90000
https://www.wolframalpha.com/input?i=plot+re+%28log%28%28zeta%280.49999%2B+s+i%29%29+%2F+%28Zeta%281-0.49999+%2B+s+i%29%29%29%29+from+0+to+90000
https://www.wolframalpha.com/input?i=plot+re+%28log%28%28zeta%280.49999%2B+s+i%29%29+%2F+%28Zeta%281-0.49999+%2B+s+i%29%29%29%29+from+0+to+900000
These plots show that the V function can maintain its shape despite the vast differences in the range.