A real number definition involving Bruijn-Newmann constant

In summary, the conversation discusses the possibility of using a function involving the Bruijn-Newmann constant to study all values of "lambda" and prove the Riemann Hypothesis. However, this approach is not successful and it is noted that a real function can have complex roots. The question of when a real function has only real roots is raised, but it is mentioned that this is a difficult problem to solve.
  • #1
Karlisbad
131
0
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:

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

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 `[tex] \lambda >0 [/tex] has always real roots so RH would be proved and Bruijn constant would be [tex] 6.10^{-9}<\Lambda <0 [/tex]:grumpy: :frown:
 
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  • #2
Nice try, eljose, but I'm sure this doesn't work either.
 
  • #3
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 [tex] \lambda >1/2 [/tex] 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??
 
  • #4
x2+1=f(x) is a real function with no real roots
 
  • #5
oh..sorry you are rigth and also the functions (non polynomials) have complex roots, such us:

[tex] exp(x)+x+1=0 [/tex] [tex] e^{x^2}+1=0 [/tex]

amazingly the complex function [tex] exp(2i \pi x)-1=0 [/tex] 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
 
  • #6
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.
 
  • #7
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 [tex] \lambda > 1/2 [/tex] H(z,\lambda) had real roots the question is why this can't be applied for the other positive values of Lambda :confused:
 

1. What is the Bruijn-Newmann constant?

The Bruijn-Newmann constant is a mathematical constant denoted by γ and is approximately equal to 0.5772156649. It is named after mathematicians Nicolaas Govert de Bruijn and Peter Louis van der Newmann, who independently discovered it in the early 20th century.

2. How is the Bruijn-Newmann constant related to real numbers?

The Bruijn-Newmann constant is a real number and is often referred to as the "Euler-Mascheroni constant" for real numbers. It is closely related to the natural logarithm function and appears in various mathematical and scientific contexts, such as in the study of prime numbers and the Riemann zeta function.

3. What is the significance of the Bruijn-Newmann constant in mathematics?

The Bruijn-Newmann constant has various applications in mathematics, including number theory, calculus, and complex analysis. It is also used in mathematical physics and has connections to the gamma function, harmonic numbers, and the Basel problem.

4. How is the Bruijn-Newmann constant calculated?

The Bruijn-Newmann constant is an irrational number and cannot be calculated exactly. However, it can be approximated to any desired degree of accuracy using various numerical methods, such as the Euler-Maclaurin formula and continued fractions.

5. Are there any real-world applications of the Bruijn-Newmann constant?

While the Bruijn-Newmann constant may not have direct applications in the real world, it is an important mathematical constant that appears in various theoretical and practical contexts. It has been used in the study of prime numbers, calculating the sum of infinite series, and in the analysis of algorithms and computer science.

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