# asymptotic behavior of the real part of the Riemann zeta function for $0<\sigma<1$

consider the zeta function $\zeta(\sigma+it)$ for $\sigma>1$ : $$\zeta(\sigma+it)=\sum_{n=1}^{\infty}\frac{1}{n^{\sigma+it}}$$ And: $$\zeta(\sigma-it)=\sum_{n=1}^{\infty}\frac{1}{n^{\sigma-it}}$$ From the identity $\zeta\left(\overline{s} \right )=\overline{\zeta(s)}\;\;$, we have: $$\Re(\zeta(\sigma+it))=\sum_{n=1}^{\infty}\frac{\cos(t\ln)}{n^{\sigma}}\leq\zeta(\sigma)$$ Thus, for $\sigma>1$ the behavior of $\Re(\zeta(\sigma+it))$ is largely governed by its values along the real line. Now, using the functional equation of the zeta function, can we obtain similar results, on the asymptotic behavior/upper bound of $\Re(\zeta(\sigma+it))$ for $0<\sigma<1$!?

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No. Not the way You described it. The functional equation

$$\pi ^{-s/2} \Gamma \left( \frac{s}{2} \right) \zeta \left( s \right) = \pi ^{-(1-s)/2} \Gamma \left( \frac{1-s}{2} \right) \zeta \left( 1-s \right)$$

links region to the right of the critical line to the region to the left of the critical line. More precisely, it links behavior of zeta at $s$ to the behavior of zeta at $1-s$. Hence, the functional equation links region $\Re s >1$ to the region $\Re s <0$. This way, the functional equation does not link the behavior of zeta in $\Re s >0$ to the behavior of zeta in the critical strip.

However, some results on zeta in the critical strip can be obtained by other methods. For instance, a corollary of the Lindelöf's theorem proves that zeta is un-bounded on any line $\Re s = c$ with $c \leq 1$. Reference: Edwards, Riemann's zeta function, Dover Publications Inc.,2001,p. 184.

Also, zeta has the property of universality in the critical strip. Reference for universality: say, http://en.wikipedia.org/wiki/Zeta_function_universality.

Regards.

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