riemann integral

In astronomy, the integral curve was a proper term. In the field of mathematics, curve integral was a type of integral. The value of the integral function followed a specific curve (called the integral path). When there was such a concept related to the integral path, it could also be understood as the concept of an integral curve. The curve integral included the first type of curve integral and the second type of curve integral. When the integral path was a closed curve, it was called loop integral or curve integral. In addition, there was also the integral curve defined by the differential equation. The answer could be found from the equation itself. At this time, the properties determined by the equation were qualitative.

The non-trivial zero of the Riemann zeta function was an extremely complicated mathematical problem. There was no accurate calculation method at present. Generally, numerical calculation methods could be used to approximate the calculation. For example, numerical approaches, iterations, optimization algorithms, and so on. In addition, there were some algorithms specifically for calculating the non-trivial zero of the Riemann zeta function, such as the Riemann-Siegel formula and the Gram Schmidt orthonormalization method. However, these methods required a certain mathematical background and programming skills to implement. The Extraordinary Ordinary Life novel is equally exciting. Everyone is welcome to click and read it!

Riemann discovered that the distribution of prime numbers was hidden in the zero point distribution of the Riemann zeta function. Riemann attributed the distribution of prime numbers to the problem of functions. He believed that there was a special function (Riemann zeta function), and a series of special points (non-trivial zeros of Riemann zeta function) that made it zero determined the detailed distribution of prime numbers. The real part of the non-trivial zeros of the Riemann zeta function (s is not a value of-2,-4,-6···, etc., these are all trivial zeros) is 1/2, which means that these non-trivial zeros are distributed on the line of the complex plane, Ri (z) = 1/2. By studying the Riemann zero point, one could gain insight into the secrets of the distribution of prime numbers. For example, with the help of these methods, the number of prime numbers less than x could be determined. The distribution of prime numbers was a normal distribution centered on the Riemann formula and the upper limit of the Gauss formula (this was an experimental formula that could only be proved to be a theorem). The Extraordinary Ordinary Life novel is equally exciting. Everyone is welcome to click and read it!

Riemann's hypothesis proposed that all non-trivial zeros were on a line with the real part equal to 1/2 (critical line). Since Riemann's hypothesis was proposed, the study of its non-trivial zeros continued to advance. In 1896, Jacques Hadamard and Farebusai were the first to independently prove that there were no zeros on a straight line. In 1903, Gran proved that the first 15 zeros were true for Riemann's hypothesis, which became the earliest result of the research of the hypothesis. In 1986, the computer was able to calculate the first 1.5 billion non-trivial zeros of the Zeta function that satisfied Riemann's hypothesis. On May 31, 2024, Fields Medal winner James Maynard and mathematics breakthrough award winner MIT mathematician Larry Gus published a paper that made substantial progress on the road to proving Riemann's hypothesis. However, it was still far from completely solving Riemann's hypothesis and determining all non-trivial zeros. The Extraordinary Ordinary Life novel is equally exciting. Everyone is welcome to click and read it!

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