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As a part of my math project, I am working on Infinite series and Riemann Zeta function. I am very new to this function, and am still studying how this series behaves. I am trying to work the problem out when $s= 1+i$ i.e. $\zeta(1+i)$.

Since Re(s) =1, so it should be outside critical strip. But I do not know more than this, and I looked up online as well on the previous queries related to complex exponentials as well as Riemann Zeta function, but found nothing to help me move forward. I do not know how to work with complex exponential in an infinite series. $\zeta(1+i)= \sum^\infty _i = 1+\frac{1}{2^{1+i} }+\frac{1}{3^{1+i} }+\frac{1}{4^{1+i} }+...$

So far I have done this using the exponential and logarithmic property, and the Euler representation. I would be super grateful if you have anything that will help me go to further steps after this:

$a^z=e^{z\cdot \ln(a)}=e^{(x+iy)\ln(a)}=e^{x \cdot \ln(a) } \cdot e^{i y \ln(a)}$

Now using this we get,

$2^{1+i}= e^{\ln(2)} \cdot e^{i \ln(2)} = 2e^{i \ln(2)}= 2(\cos{\ln(2)}+ i \sin{\ln(2)})$

And for $3^{1+i}= e^{\ln(3)} \cdot e^{i \ln(3)} = 2e^{i \ln(3)}= 3(\cos{\ln(3)}+ i \sin{\ln(3)})$

I am not sure how would I use this back in the series to examine anything.

optimistic_mathematician
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    What kind of answer are you looking for? Do you want a method on how to compute the actual value, or a different series which converges to $\zeta(1+i)$? This might be a good place to start... – RSpeciel Aug 04 '20 at 16:38
  • Your expansions of $2^{1+i}$ and $3^{1+i}$ are correct! If you're interested in the infinite series itself, would it suffice for you to understand $\zeta(2+i)$ instead of $\zeta(1+i)$? The convergence of $\zeta(1+i)$ is more delicate (it converges conditionally). – Greg Martin Aug 04 '20 at 16:57
  • @GregMartin The series $\sum_{n = 1}^{\infty} n^{-s}$ diverges for all $s$ with $\operatorname{Re} s \leqslant 1$. – Daniel Fischer Aug 04 '20 at 17:54
  • @GregMartin - yes, you are correct. I am interested in examining the infinite series, so $\zeta(2+i)$ would definitely suffice. Do you have any suggestions on how I can approach it? – optimistic_mathematician Aug 04 '20 at 20:14
  • @RomainS Yes, I am looking for a method that helps me to show how $\zeta(2+1)$ or any $\zeta(z)$ outside the critical strip would behave. I need to formally show the process for at least one of the values for the infinite series. – optimistic_mathematician Aug 04 '20 at 20:26
  • But what exactly do you mean by "behave"? – Greg Martin Aug 04 '20 at 20:43
  • @optimistic_mathematician If this is for a school math project, I would highly suggest looking at $\zeta(2)$ instead of this. Despite its apparent simplicity, it is actually a very interesting problem (hence a very well documented problem). See this introduction, as well as one of my all time favorite youtube videos here. Hope this helps. – RSpeciel Aug 04 '20 at 20:49
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    @RomainS and GregMartin I am sorry I am not sure how to clarify about this. For my math project, I am trying to work out the full process of showing Riemann Zeta function with a complex exponent. I have already looked at $\zeta(2)= \frac{\pi^2}{6}$ before posting here and was hoping to use similar approach. – optimistic_mathematician Aug 05 '20 at 22:00
  • You may be interested too by this answer with the result that $\zeta(1+i)$ (related links may be of interest too) may be rewritten in function of $\zeta(i)$ (cf $(6)$ and $(9)$ for the argument and modulus relations or $(5)$ directly). Studying $,\zeta(i),$ appears thus interesting too even if the most studied one will remain $,\zeta\left(\frac 12+i\right)$ since it produces a real function once multiplied by $e^{i\theta(t)}$ with $\theta$ defined here! – Raymond Manzoni Nov 12 '20 at 12:51

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