Can Riemann zeta function be written as ##f(s)=u(s)+iv(s)##?

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Thread starter Adel Makram
Start date Jun 4, 2016
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Function Riemann Riemann zeta function Zeta function

In summary, the Riemann zeta function is a mathematical function defined by Bernhard Riemann in the 19th century. It has important applications in number theory and connections to other areas of mathematics. It can be written as u(s) + iv(s) and this decomposition allows for a better understanding of its behavior. There are known relationships between u(s) and v(s) including the functional equation and the Euler product formula.

Jun 4, 2016

Adel Makram

I don't recall that I have seen Riemann zeta function put in the form of ##f(s)=u(s)+iv(s)##.

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Jun 4, 2016

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Every complex-valued function can be written that way, but that doesn't mean it has to be useful.

What is the Riemann zeta function?

The Riemann zeta function is a mathematical function that was first defined by Bernhard Riemann in the 19th century. It is denoted by ζ(s) and is defined for all complex numbers s with real part greater than 1.

Why is the Riemann zeta function important?

The Riemann zeta function has important applications in number theory, particularly in the study of prime numbers. It also has connections to other areas of mathematics, such as complex analysis and physics.

Can the Riemann zeta function be written as a sum of real and imaginary parts?

Yes, the Riemann zeta function can be written as f(s) = u(s) + iv(s) where u(s) and v(s) are real-valued functions. This is known as the decomposition of the Riemann zeta function.

What is the significance of the decomposition of the Riemann zeta function?

The decomposition of the Riemann zeta function allows us to visualize and understand the behavior of the function in a more intuitive way. It also allows for the use of techniques from real analysis to study the function.

Are there any known relationships between u(s) and v(s)?

Yes, there are several known relationships between u(s) and v(s), including the functional equation ζ(s) = 2^sπ^(s-1)sin(πs/2)Γ(1-s)ζ(1-s) and the Euler product formula ζ(s) = ∏(p^(-s) / (1-p^(-s))) where the product is taken over all prime numbers p.