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There are numerous equivalent formulations of Riemann Zeta function (that are valid in the critical strip). One of them I came across is: "The real and imaginary parts of the Riemann Zeta Function can be formulated as:

$$R(s) = \sum_{n=1}^\infty \frac{(−1)^n} {n^\sigma} \cos(t \log n), \qquad I(s) = \sum_{n=1}^\infty \frac{(−1)^n} {n^\sigma} \sin(t \log n)$$

where $s=t+i\sigma$ lies in the critical strip ($0 < \sigma < 1$)."

I have a 2 part Q.

Q1: Is the above formulation correct in the critical strip and if yes is there an integral representation of the above (valid in the critical strip)

Q2: How do we calculate (and I am talking from a Computer Science perspective here) the value of Zeta Function for the Real and Imaginary parts separately. (The simplest known formula/algorithm when applied gives the value of Zeta for any given complex input)? I have heard about the zeroes of Zeta but I am unaware how to calculate the value of Zeta for any Input in general?

I am a newbie with CS background here so please take that into consideration.

Semiclassical
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stack.tarandeep
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  • I don't think this representation is valid within the critical strip. If one takes $s=1/2$ (so $t=0,\sigma=1/2$) this would yield $\zeta(1/2)$ as $$R(1/2)+i I(1/2)=\sum_{n=1}^\infty \frac{(-1)^n}{n^{1/2}}.$$ But Mathematica yields this as instead $(\sqrt{2}-1)\zeta(1/2)$. 2) Where did you source this from?
    • – Semiclassical Jul 21 '24 at 21:35
  • In any case can this be represented as an integral? and the second Q is what is more important to me. how to calculate zeta values ? – stack.tarandeep Jul 21 '24 at 22:43
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    It looks like you're referring to the equation at the bottom of page 97. But that's explicitly a representation of the eta function, not the zeta function: $$\eta(s)=\sum_{n=1}^\infty (-1)^{n-1}\frac{\cos(t\log n)}{n^\sigma}-\sum_{n=1}^\infty (-1)^{n-1}\frac{\sin(t\log n)}{n^\sigma}$$ The identity $\zeta(s)=\eta(s)/(1-2^{1-s})$ means that this representation is useful for finding zeta zeros with positive real part, but it's still a representation of $\eta(s)$ not $\zeta(s)$. – Semiclassical Jul 21 '24 at 22:45
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    As to computing $\zeta(s)$ within the critical strip, see https://math.stackexchange.com/questions/1082139/how-are-zeta-function-values-calculated-from-within-the-critical-strip – Semiclassical Jul 21 '24 at 22:46
  • aah ok. So zeta will be 0 iff eta will be 0? Now is there an equivalent integral representation of the eta function? – stack.tarandeep Jul 21 '24 at 22:52
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    It's a little more subtle than that, owing to the $1-2^{1-s}$ factor, but the zeros of that factor lie along the line $\sigma=1$ and thus don't fall within the critical strip. So pretty much yes. – Semiclassical Jul 21 '24 at 23:01