Question about Riemann Zeta Function

In summary, the conversation discusses the Reimann-Zeta function and its convergence in the complex plane. It is clarified that Euler's Identity is not related to the function, and that Euler's Formula may be what the person is referring to. It is explained that the value of the function can be assigned by calculating a series of terms and observing its convergence.
  • #1
willr12
17
2
I understand how to calculate values of positive values ζ(s), it's pretty straightforward convergence. But when you expand s into the complex plane, like ζ(δ+bi), how do you assign a value with i as an exponent? Take for example
ζ(1/2 + i)
This is the sequence
1/1^(1/2+i) + 1/2^(1/2+i) + 1/3^(1/2+i) ...
How do you assign a value to this? do you have to use euler's identity to calculate it? Or am I looking at it all wrong?
P. S. Try to dumb it down for me. I'm in algebra 2 right now...
 
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  • #2
First, Euler's Identity has nothing to do with the Reimann-Zeta function. Euler's Identity is just
##e^{i \pi} + 1 = 0##
Euler's Formula is probably what you're thinking of (which is where the above identity comes from):
##e^{ix} = \cos x + i \sin x##
But there's no exponential function (it works that way because the exponential function is re-written with a power series to get the above formula), and ##i## would have to be factored out of the exponent entirely to rewrite it that way (it's only in one of the terms).
But, in the case you're talkng about where ##s## is defined as ##\sigma + bi## for ##0 < \sigma < 1##, then it can be left as is, but the imaginary zeroes are plotted along the "critical line", ##\Re(s) = \frac{1}{2}##

http://mathworld.wolfram.com/RiemannZetaFunction.html
 
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  • #3
I'd say, calculate the first m terms, and the first m+1, m+2... and see what it converges to.
 

What is the Riemann Zeta Function?

The Riemann Zeta Function is a mathematical function that was introduced by the mathematician Bernhard Riemann in the mid-19th century. It is defined as the sum of the infinite series 1 + 1/2^s + 1/3^s + 1/4^s + ..., where s is a complex number with real part greater than 1.

What are the applications of the Riemann Zeta Function?

The Riemann Zeta Function has various applications in number theory, probability theory, and physics. It is used to study the distribution of prime numbers, the behavior of random variables, and the energy levels of quantum mechanical systems.

What is the Riemann Hypothesis?

The Riemann Hypothesis is one of the most famous unsolved problems in mathematics. It states that all non-trivial zeros of the Riemann Zeta Function lie on the line with real part equal to 1/2. It has important implications for the distribution of prime numbers and has been a subject of intense research for over 150 years.

How can the Riemann Hypothesis be proved?

As of now, there is no known proof for the Riemann Hypothesis. Many mathematicians have attempted to prove it, but none have been successful. The proof of this conjecture is considered to be one of the most difficult problems in mathematics.

Are there any consequences if the Riemann Hypothesis is proven?

Yes, there are many consequences if the Riemann Hypothesis is proven. It would have a significant impact on number theory, as it would provide a deeper understanding of the distribution of prime numbers. It would also have practical applications in cryptography and coding theory.

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