Hello P. Contreras, I am not a mathematian but I am very curious. So, about the Riemman z funtion there is no misteries by itself...in certain domain of the function, and it has specific applications, but the Riemman's hypotesis is related with an specific case of the solutions (zeros) into the "critical line". I think you will find very interesting the lecture of the next article thas explains with great detail the issue and some attempts to solve it:

https://mathworld.wolfram.com/RiemannHypothesis.html

Other people have given you some examples of how the zeta function can arise in

various fields of physics and engineering. More interestingly, sometimes relations involving the zeta function are found during the research that are novel and spark (or renew) interest in completely unrelated areas of mathematics.

For example, the following surprising relation was found in the 1970's during

the solution of a problem first posed by Ludwig Prandtl in 1933.

$$\sum^{\infty}_{n=1} \frac{L_{n}}{n^{2}} = \frac{7}{4}\zeta(3)$$

where

$$L_n = 1 + \frac{1}{3} + \frac{1}{5} + \dots + \frac{1}{2n-1}$$

It was later picked up by Jonathan Borwein and his colleagues in the 1990's during their investigation of Euler (double) sums. Other researchers then found connections to work by Ramanujan and other pure mathematicians.

To me it's quite remarkable that such a simple relation went undiscovered for so long, and then just popped out during the analysis of an (arguably!) simple boundary value problem in incompressible thin-wing theory.

The Riemann-Zeta function, Z(s), is the analytic continuation of the Dirichlet series, 1 + 1/s + 1/s^2 + 1/s^3 +... and is the bread and butter of analytic number theory. It is closely tied the the distribution of prime numbers. G. H. Hardy showed that the function Z(1/2 + it) has an infinite number of real zeros. The Riemann Hypothesis simply states that all the non trivial zeros of Z(s) have Re(s)=1/2.

Z(s) has close relationships with many other special functions in mathematics. See https://en.wikipedia.org/wiki/Riemann_zeta_function and https://mathworld.wolfram.com/RiemannZetaFunction.html

A great reference is the wonderful book by book "Complex Analysis" by Elias. Stein and Rami Shakarchi in the Princeton Lectures in Analysis series where there are two chapters devoted to the Zeta function.

The Zeta functions has close ties to all most areas of mathematics and in

http://library.msri.org/books/Book57/files/20kirsten.pdf

Kirsten looks at physical applications of a larger family of Zeta functions related to the Riemann Zeta function. Another reference to relationships between distributions of prime numbers and physics.

http://www.bristol.ac.uk/maths/research/highlights/riemann-hypothesis/

The Riemann hypothesis is considered one of the great unsolved problems in mathematics. It is Hilbert's 8th problem on the famous problem list of David Hilbert. It is number 1 on Smale's list of 18 of the greatest problems list. It is one of the 7 Millennium problems of the Clay Mathematics Institute. Only one of the 8 has been solved, Poincaré's Conjecture.

https://www.claymath.org/millennium-problems

Solve the Riemann hypothesis and you will be a million dollars richer.

The Zeta function is a very important function in mathematics. While it was not created by Riemann, it is named after him because he was able to prove an important relationship between its zeros and the distribution of the prime numbers. His result is critical to the proof of the prime number theorem.

For more details see

https://www.britannica.com/science/Riemann-zeta-function

Dear Ait Mansour El Houssain

The Riemann zeta function is defined by

ζ(x)=∑(1/nx) where n runs over natural numbers.

Euler's succeeded to write this sum in a product form

that depends on primes as follows

(1-2-x)ζ(x)=(1-2-x)∑(1/nx)

= ∑(1/nx) -∑(1/(2n)x) = 1 + ∑(1/nx)

where the new sum runs over numbers that are not divisible by 2

proceed the same up to prime P,

(1-2-x)(1-3-x).....(1- p-x)ζ(x) = 1 + ∑(1/nx)

where the new sum runs over numbers that are

not divisible by 2,3,...P.

As P→∞, we obtain ∏(1-p-x)ζ(x)=1, where the product runs over all prime numbers and then ζ(x) = ∑(1/nx) = ∏(1-p-x})⁻¹.

This new form is of great significance in the field of number theory and the distribution of prime numbers. Where the millennium problem is the Riemann hypothesis to show that the nontrivial singular values have a real part is 1/2.

Best regards

Of course, it is a basic object in Number Theory, but this fact tells nothing about its interest in other areas outside of Mathematics. Perhaps the main reason is that it allows to regularize divergent series, that is, to assign a definite value to expressions such as a1+a2+a3+... where an infinity of such terms, each one positive, are added up (for instance, the result of adding up all the natural numbers). There are many situations in Physics and Engineering where this kind of sums arise, mainly when determining the energy of a system (by counting normal modes as in string theory, or microstates, as in statistical physics). If you naively consider how to assign a definite number to the sum of infinite positive quantities, you very quickly will realize that you have to introduce some notion of "limit". The more sophisticate this "limit", the greater the number of sums you can regularize. The usual Cauchy-Weierstrass notion of limit, although quite intuitive, is not the most useful in Physics (it is unable of regularizing a lot of sums), and here is where Riemann's Zeta enters the stage: from a practical point of view, it allows for the regularization of a large number of infinite sums that arise in Physics. In addition to the works mentioned by P. Contreras you can take a look at

https://en.wikipedia.org/wiki/Zeta_function_regularization

and a very popular video in Numberphile:

https://www.youtube.com/watch?v=E-d9mgo8FGk

Thanks to José Antonio Vallejo , Issam Kaddoura , Maged Gumaan Bin-Saad , Truman Prevatt , P. Contreras et all for your responses. Is there any alternative formulation based on infinite sum of zeta function ?

It is not so much the Riemann Zeta function that is important to physics - particularly the "regulation" and "renormalization" it is techniques developed in the proof that the Dirichlet series, ds(s) 1+ 2^-s + 3^-s+... which only converges to a function when Re(s)>1 could be extended to a function, zeta(s), such that zeta(s) = ds(s) for Re(s)>1 but was analytic in the entire complex plane except a simple pole at s=-1.

Riemann used the Euler identity that relates ds(s) to a different expression and then the techniques being developed for the analysis of functions of a complex variable that led to the definition of the Zeta function and the functional relationships of this function. This technique Riemann used in the proof is important in application to physics - as it gives an approach for calculating answers for "ill-posed" conditions like non-convergence.

Riemann's original proof was quite elegant. http://people.math.harvard.edu/~elkies/M259.02/zeta1.pdf

But to physics it is the technique he develop in the proof supports the process of "regulation" Much of mechanics of the approach in general situations was developed by British mathematicians Hardy and Littlewood.

It's famous because the conjecture on its zeros on the critical strip is awfully hard to tackle. Many tried, few succeeded to obtain (some) progress. Plus, it's got connections, sometimes surprising, with various area of mathematics. Personally, I have not been touched by this "virus" - I have only two small contributions called "A recurrence formula for Riemann's zeta function" (1987) and "Contour integral representation of Riemann's zeta function" (1988), which Tom Apostol reviewed favourably, but surely won't solve the famous conjecture on its zeroes.