A real number definition involving Bruijn-Newmann constant

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Discussion Overview

The discussion revolves around the definition of a "real" number in the context of the Bruijn-Newmann constant and its implications for the Riemann Hypothesis (RH). Participants explore mathematical functions, their roots, and the conditions under which real roots exist, particularly focusing on the Wiener-Hopf integral and related concepts.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant proposes a function involving the Bruijn-Newmann constant and suggests that studying the values of "lambda" could lead to proving that RH holds true.
  • Another participant expresses skepticism about the initial proposal, indicating that it may not work as intended.
  • A participant questions the reasoning behind Newmann's proof that for \lambda > 1/2, suggesting that a real function should have real roots, and seeks clarification on this point.
  • One participant provides a counterexample with the function x² + 1, which is a real function but has no real roots.
  • Another participant acknowledges the existence of complex roots in non-polynomial functions and discusses the criteria for determining whether a function has real roots.
  • A later reply emphasizes that a function has real zeroes if and only if the solutions to f(x) = 0 are real numbers, while critiquing the confusion around the topic.
  • One participant clarifies that their intention was not to offend mathematicians but to highlight the perceived lack of straightforward answers to seemingly simple questions regarding real roots.
  • There is a mention of Newmann's proof regarding H(z, λ) having real roots for λ > 1/2, raising questions about the applicability of this result for other positive values of λ.

Areas of Agreement / Disagreement

Participants express differing views on the validity of the initial proposal regarding the Bruijn-Newmann constant, with some skepticism about its effectiveness. There is also disagreement on the criteria for determining real roots, indicating a lack of consensus on these mathematical concepts.

Contextual Notes

Participants reference various mathematical functions and their properties, but the discussion contains unresolved questions about the conditions under which real roots exist and the implications of the Bruijn-Newmann constant for the Riemann Hypothesis.

Karlisbad
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A "real" number definition involving Bruijn-Newmann constant..

Ok, ths isn't anything new, but i would like to discuss this possibility, taking into account the function:

\xi(1/2+iz)=A(\lambda)^{-1/2}\int_{-\infty}^{\infty}dxe^{(-\lambda)^{-1} B (x-z)^{2}}H(\lambda, x)

then with the expression above we could study all the values of "lambda" so the Wiener-Hopf integral above involving a symmetryc tranlational Kernel has only real roots, or use it to prove that for `\lambda >0 has always real roots so RH would be proved and Bruijn constant would be 6.10^{-9}<\Lambda <0 :frown:
 
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Nice try, eljose, but I'm sure this doesn't work either.
 
arildno said:
Nice try, eljose, but I'm sure this doesn't work either.

It would be nice if someone could explain me the trick Newmann use to prove that \lambda >1/2 to take a look at it...:frown: :frown: I'm not a mathematician but i believe that a real function should always have real roots :confused: am i wrong??
 
x2+1=f(x) is a real function with no real roots
 
oh..sorry you are rigth and also the functions (non polynomials) have complex roots, such us:

exp(x)+x+1=0 e^{x^2}+1=0

amazingly the complex function exp(2i \pi x)-1=0 has only real roots.

the definition of a function f(x) is more than 300 years old, i don't know why mathematicians don't have some criteria to decide wether a real function has only real roots or complex appart from knowing that if f(x) and f(x*) (complex conjugate) are equal then there are pairs or complex roots changing only their imaginary part b to -b
 
We do have a simple criterion: f has real zeroes if and only if f(x)=0 implies x is in R. You're once more confusing several different issues, Jose. Testing when something satisfies some property is (always?) a hard problem except in toy examples as anyone can tell just by thinking about it for a few seconds, instead of offering yet another damning indictment of the stupidity of mathematicians.
 
I didn't want to offend mathematician :frown: i only questioned that such a "easy" (in appearance) question was not answered and that you could find some theorems for much more difficult questions,.. that's all, in fact Newmann could prove that for \lambda > 1/2 H(z,\lambda) had real roots the question is why this can't be applied for the other positive values of Lambda :confused:
 

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