18th Jul, 2021

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Asked 14th Jun, 2021

Here we discuss about one of the famous unsolved problems in mathematics, the Riemann hypothesis. We construct a vision from a student about this hypothesis, we ask a questions maybe it will give a help for researchers and scientist.

- 129.89 KBRiemann zeta function Problem.pdf

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Mathematics is the queen of the sciences—and number theory is the queen of mathematics""

The distribution of prime numbers is a central point of study in number theory

In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem

It is used to define prime elements, a generalization of prime numbers to arbitrary commutative rings.

Several historical questions regarding prime numbers are still unsolved. These include Goldbach's conjecture, that every even integer greater than 2 can be expressed as the sum of two primes, and the twin prime conjecture, that there are infinitely many pairs of primes having just one even number between them.

Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers. analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers.

It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic progressions.

It is well known for its results on prime numbers (involving the Prime Number Theorem and Riemann zeta function) and additive number theory (such as the Goldbach conjecture and Waring's problem) In particular, the methods of zeta function regularization and Ramanujan summation assign the series a value of −+1/12, which is expressed by a famous formula,[2]

1 + 2 + 3 + 4 + ⋯ = − 1/12

where the left-hand side has to be interpreted as being the value obtained by using one of the aforementioned summation methods and not as the sum of an infinite series in its usual meaning. These methods have applications in other fields such as complex analysis, quantum field theory, and string theory. ..

Riemann hypothesis,...

following actual definition of prime number,... 1 is not one (Prime number) !!!!!! ""Prime, any positive integer greater than 1 that is divisible only by itself and 1—e.g., 2, 3, 5, 7, 11, 13, 17, 19, 23, ….""

Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative

The value s = 1 is therefore a singularity of the function.

In mathematics, a singularity is generally a point, a value or a case in which a certain mathematical object is not well defined or is undergoing a transition. i have the mathematics to describe such concept,...

Code ULTIMATE Prime Tools

This program can decompose any integer [+,-] into a prime addition term (termisation)

IF so

- Single specified one will be termisationed [S]

-Termisation of specified one [T] This program can also find all GOLDBACH number [pair,impair] until the specified one

Juan, the real part of z needs to be > 1 (by definition).

The easiest ways (in my opinion) to see that -2, -4, ... are trivial zeros:

(i) From the connection with the Bernoulli numbers, we have that:

(2n+1) ζ(-2n) = B_{2n+1} for natural n ≥ 1,

where the right hand side are the (2n+1)-Bernoulli numbers - which are equal to zero;

or

(ii) From Riemann's functional equation, we have:

ζ(-2n) = -2^{-2n} (2n)! π^{-2n-1} ζ(2n+1) sin (nπ) for natural n ≥ 1,

and the latter factor equals 0.

7 Recommendations

They say -2, -4, -6,....are trivial zeros of the Riemann Zeta function.

Just what is the proof?

Introducing -2, I seem to get

1+4+9+..very far from zero.

1/(2^z) with z=-2 is 1/(1/4) =4??

Where is the mistake?

1 Recommendation

Juan, the real part of z needs to be > 1 (by definition).

The easiest ways (in my opinion) to see that -2, -4, ... are trivial zeros:

(i) From the connection with the Bernoulli numbers, we have that:

(2n+1) ζ(-2n) = B_{2n+1} for natural n ≥ 1,

where the right hand side are the (2n+1)-Bernoulli numbers - which are equal to zero;

or

(ii) From Riemann's functional equation, we have:

ζ(-2n) = -2^{-2n} (2n)! π^{-2n-1} ζ(2n+1) sin (nπ) for natural n ≥ 1,

and the latter factor equals 0.

7 Recommendations

There is a lot of confused reasoning about the Riemann zeta function.

Consider it this way:

A. In the complex domain, zeta(s) is defined by the integral formulas

Integral_x=0..inf{x^(s-1)/(e^x-1)}/Integral_x=0..inf{x^(s-1)/e^x}.

B. This evaluates to 0 when s is a negative even integer.

C. For Re(s) > 1 this evaluates to the same value as the series Sum_n=1..inf{1/n^s}. Ok, nice to know.

But C has nothing to do with B.

The implication is from A to B and from A to C, not at all from C to B.

There exists thousands of examples of this kind.

For example, the Lorentzian function f(x) = 1/(1+x^2) is well defined for any real x. In the sub-domain of |x|<1 it evaluates to the same value as the series Sum_n=0..inf{(-x)^(2n)}. For |x|>=1, however, the series does not converge at all, while the function f(x) exists and is well behaved.

No mystery.

But name the Riemann zeta and many people think that there IS a mystery.

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has an intrinsic meaning as a complex number, independent of the requirements
of analytic continuation.
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