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This could be quite long. Apologies. I am a physics student trying to understand the Zeta function and the Riemann hypothesis. Its not on my coursework, but I am interested in pure mathematics. I have a few questions. Perhaps you can help me out. Thank you.

My questions are concerned with the Wikipedia page on the Riemann Hypothesis.

(Q1) In the formula for the Zeta function i.e. [tex] \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} [/tex] is the [tex] s [/tex] a complex number like [tex] s = a + bi [/tex] Can [tex] a [/tex] and [tex] b [/tex] take any value?

(Q2)

On the Wikipedia page for the Riemann Hypothesis, there is a diagram in the top right hand corner. They state that the diagram is a plot of [tex] s = \frac{1}{2} + i \ x [/tex] . Does that mean that I would have to compute the sum of the Zeta function for each value of x, in order to plot that diagram?

(Q3) So how do I calculate the sum of an infinite series if that is the case? For instance, how would I calculate [tex] \zeta(\frac{1}{2} + i \ 14.135) [/tex]. I ask this because I want to see how they arrive at the zero of the Zeta function in that Wikipedia diagram.

(Q4) Does the value of [tex]a[/tex] and [tex]b[/tex] determine whether I can calculate the sum of the infinite series of the zeta function?

(Q5) Could you recommend a good introductory book on the Zeta function. As you can see I am pretty much flying solo on figuring this out.

Thanks for your help.

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# Zeta Function for dummies.

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