Riemann zeta function

Riemann zeta function
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A Indices of unstable zeros of the Riemann zeta function. A Unstable twins: pairs of consecutive numbers in A indices of unstable zeros of the Riemann zeta function.

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Researcher Academy Author Services Try out personalized alert features. Watch a Video Abstract on An improved upper bound for the argument of the Riemann zeta-function on the critical line II Read full article here. Return to Video Abstracts. Journal of Number Theory. Advertisers Media Information. These expressions can be used to prove the prime number theorem by means of the inverse Mellin transform.

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The Riemann zeta function can be given by a Mellin transform [23]. However, this integral only converges if the real part of s is greater than 1, but it can be regularized. This gives the following expression for the zeta function, which is well defined for all s except 0 and Abel—Plana formula.

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The Riemann zeta function or Euler–Riemann zeta function, ζ(s), is a function of a complex variable s that analytically continues the sum of the Dirichlet series. In mathematics, the Riemann zeta function is an important function in number theory. It is related to the distribution of prime numbers. It also has uses in other.

Another series development using the rising factorial valid for the entire complex plane is [ citation needed ]. On the basis of Weierstrass's factorization theorem , Hadamard gave the infinite product expansion.

A simpler infinite product expansion is. To ensure convergence in the latter formula, the product should be taken over "matching pairs" of zeros, i. Euler summation :. The series only appeared in an appendix to Hasse's paper, and did not become generally known until it was discussed by Jonathan Sondow in Peter Borwein has developed an algorithm that applies Chebyshev polynomials to the Dirichlet eta function to produce a very rapidly convergent series suitable for high precision numerical calculations.

Here p n is the primorial sequence and J k is Jordan's totient function. Hence this algorithm is essentially as fast as the Riemann-Siegel formula. Similar algorithms are possible for Dirichlet L-functions. The zeta function occurs in applied statistics see Zipf's law and Zipf—Mandelbrot law. Zeta function regularization is used as one possible means of regularization of divergent series and divergent integrals in quantum field theory.

In one notable example, the Riemann zeta-function shows up explicitly in one method of calculating the Casimir effect. The zeta function is also useful for the analysis of dynamical systems.

Riemann's Zeta Function

The zeta function evaluated at equidistant positive integers appears in infinite series representations of a number of constants. Other sums include. There are yet more formulas in the article Harmonic number. There are a number of related zeta functions that can be considered to be generalizations of the Riemann zeta function.

These include the Hurwitz zeta function. For other related functions see the articles zeta function and L -function. The Lerch transcendent is given by.

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The multiple zeta functions are defined by. One can analytically continue these functions to the n -dimensional complex space.

Journal of Number Theory

Share this article. Here, I is the Mathematica code for the imaginary unit i. Here, each subsequent multiplication by the th prime leaves only terms that are powers of. The pattern in figure 1 is reminiscent of the field lines of an electric dipole. The first group contains just one zero. Contour plots are a popular and efficient technique to visualize complex functions.

The special values taken by these functions at positive integer arguments are called multiple zeta values by number theorists and have been connected to many different branches in mathematics and physics. From Wikipedia, the free encyclopedia. Main article: Particular values of the Riemann zeta function. Proof of functional equation.

Main article: Riemann hypothesis.

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Some identities related to Riemann zeta-function

Unsourced material may be challenged and removed. December Learn how and when to remove this template message. Retrieved 4 January Bombieri, Enrico. Clay Mathematics Institute. Retrieved 8 August Cambridge University Press. A: Math.