Fascinating prime numbers.

Is there some non-random sequence in prime number series? 

Prime numbers are divisible by only one and by itself. That thing makes prime numbers the prime tools for ASCII (American Standard Code for Information Interchange) encryption. The ASCII means the numeric code of keyboard symbols. In that process, computers count ASCII numbers using prime numbers. That allows the encryption computer to hide messages from outsiders. And that makes the prime numbers so important. Prime numbers have also fascinated mathematicians throughout history. 

Famous mathematician Carl Friedrich Gauss (1777-1855) spent 15 minutes per every day in his career. Calculating prime numbers. Another mathematician, Bernhard Riemann (1826-1866), created the "Riemann's Zeta function" for calculating prime numbers. The Zeta Function zero points should not include any other than prime numbers. That's when researchers use normal numbers. But there are zeros in complex numbers. 

"The real part (red) and imaginary part (blue) of the Riemann zeta function along the critical line Re(s) = 1/2. The first non-trivial zeros can be seen at Im(s) = ±14.135, ±21.022 and ±25.011." (Wikipedia, Riemann zeta function)

"This image shows a plot of the Riemann zeta function along the critical line for real values of t running from 0 to 34. The first five zeros in the critical strip are clearly visible as the place where the spirals pass through the origin." (Wikipedia, Riemann zeta function)

Numbers from 1 to 49 placed in spiral order. (Wikipedia, Ulam Spiral)

and then marking the prime numbers:(Wikipedia, Ulam Spiral)

There can be a couple of nonprime numbers in Riemann's Zeta Function zero-point series in the case that the zeros. Riemann's hypothesis is one of the biggest unsolved mathematical problems. The problem is this: is there some kind of non-random order of prime numbers? Is there some kind of non-random sequence in the prime number series? If that series exists and somebody finds it, that thing can revolutionize number theory. 

But then we can think about another way to find the prime numbers than calculating. Mathematician Stan Ulam created the Ulam Spiral. If we want to calculate prime numbers using the Riemann Zeta function that takes time. The Riemann Zeta function is the tool whose weakness is that it forms prime numbers in reading lines. Faster computers must only calculate that series that slower computers used. And that breaks the code immediately. The Ulam spiral is the tool that should respond to the need to select certain prime numbers without linear models. 

"The Ulam spiral or prime spiral is a graphical depiction of the set of prime numbers, devised by mathematician Stanisław Ulam in 1963 and popularized in Martin Gardner's Mathematical Games column in Scientific American a short time later. It is constructed by writing the positive integers in a square spiral and specially marking the prime numbers." (Wikipedia, Ulam Spiral) 

"Ulam and Gardner emphasized the striking appearance in the spiral of prominent diagonal, horizontal, and vertical lines containing large numbers of primes. Both Ulam and Gardner noted that the existence of such prominent lines is not unexpected, as lines in the spiral correspond to quadratic polynomials, and certain such polynomials, such as Euler's prime-generating polynomial x2 − x + 41, are believed to produce a high density of prime numbers. "(Wikipedia, Ulam Spiral) 

"Nevertheless, the Ulam spiral is connected with major unsolved problems in number theory such as Landau's problems. In particular, no quadratic polynomial has ever been proved to generate infinitely many primes, much less to have a high asymptotic density of them, although there is a well-supported conjecture as to what that asymptotic density should be."(Wikipedia, Ulam Spiral) 

"In 1932, 31 years prior to Ulam's discovery, the herpetologist Laurence Klauber constructed a triangular, non-spiral array containing vertical and diagonal lines exhibiting a similar concentration of prime numbers. Like Ulam, Klauber noted the connection with prime-generating polynomials, such as Euler's" (Wikipedia, Ulam Spiral) 

https://www.3blue1brown.com/lessons/zeta

https://www.quantamagazine.org/mathematicians-uncover-a-new-way-to-count-prime-numbers-20241211/

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

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

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

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

The Gemini AI tells Riemann's hypothesis in this way.

What is the Riemann zeta function?

The Riemann zeta function is a special mathematical function that is closely related to the distribution of prime numbers. It is defined over the complex numbers and has many interesting properties. The zeta function can be used to study the distribution of prime numbers among positive integers.

Zeros

The zeros of the zeta function are the complex numbers for which the function's value is exactly zero. These zeros are distributed throughout the complex plane, and their location has deep connections to number theory.

Trivial zeros: The zeta function has infinitely many so-called trivial zeros at negative even integers (-2, -4, -6, ...).

Non-trivial zeros: These are more interesting. The Riemann hypothesis states that all non-trivial zeros lie on the so-called critical line, where the real part is exactly 1/2.

Riemann Hypothesis

The Riemann hypothesis is one of the most famous unsolved problems in mathematics. It states that all non-trivial zeros of the 1  zeta function lie on the critical line.   

Why is the hypothesis important? If the hypothesis were proven to be true, it would be a huge breakthrough in number theory and would have significant applications in many other areas of mathematics.

Why has the hypothesis not been proven yet? The Riemann hypothesis is a very difficult problem, and proving it would likely require entirely new mathematical tools.

Summary

The Riemann zeta function is a complex and fascinating mathematical object whose zeros are closely related to the distribution of prime numbers.