The question os fairly simple, though I couldn’t find its answer on the internet.
Is the norm of the complex output value of the Riemann Zeta Function limited? Or can I plug in input values to make it as large as I want?
Thanks in advance
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2What input values do you allow? In the most general input it seems the Maximum Modulus Principle tells us it will be unbounded. – hardmath Jul 02 '18 at 18:53
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8$\zeta$ has a simple pole at $1$, so it's unbounded. – Daniel Fischer Jul 02 '18 at 18:56
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@hardmath Any complex number different than 1. Is the Maximum Modulus Principle applicable since the function has a pole (as Daniel Fischer mentioned)? Or the principle does not relate to that? – leleo222 Jul 02 '18 at 19:27
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@DanielFischer This means that the function applied to numbers very close to 1 (in all directions) outputs large normed numbers? – leleo222 Jul 02 '18 at 19:31
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2Yes, if $\lvert s-1\rvert$ is small, then $\lvert \zeta(s)\rvert$ will be large. We have (near $1$) $$\zeta(s) = \frac{1}{s-1} + \gamma + O(s-1)$$ where $\gamma$ is (as usual) the Euler-Mascheroni constant. – Daniel Fischer Jul 02 '18 at 19:34
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@DanielFischer thanks, that solves my question:) – leleo222 Jul 02 '18 at 21:03
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A somewhat interesting way to ask about (un)boundedness of the Riemann zeta function $\zeta(s)$ for any "complex number different than $1$" is to look at the entire function $\zeta(s) - \frac{1}{s-1}$.
As a now deleted Comment on the Question mentioned, the power series expansion at $s=1$ of this function is given by the Stieltjes constants $\gamma_n$:
$$ \zeta(s) - \frac{1}{s-1} = \sum_{n=0}^\infty \frac{(-1)^n}{n!} \gamma_n (s-1)^n $$
Here $\gamma_0 = \gamma \approx 0.577\ldots$ is the Euler-Mascheroni constant. All of the other singularities of $\zeta(s)$ besides the simple pole at $s=1$ are removable, so the power series above converges in the entire complex plane.
By applying the Maximum Modulus Principle to this entire function, we find that it is unbounded in the complex plane once we confirm that it is not constant. It is enough to know that $\gamma_1 \approx −0.0728\ldots$ is nonzero. More information is given at the Wikipedia "Stieltjes constants" article linked above.
Indeed the Riemann zeta function has a curious universality property in the critical strip, where for any holomorphic function we can find a location that approximates it arbitrarily well. This is another way to see how the Riemann zeta function is unbounded away from its simple pole at $s=1$.
hardmath
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