# A lower bound for the Riemann zeta function

‎In our recent researches, we obtain the following lower bound for the (real) Riemann zeta function:‎ ‎ ‎‎\begin{align*}‎‎ ‎\zeta‎(s)‎\geq ‎\frac{5 - 2s}{s - 1}‎;\;\;\;\; 1<s<‎\frac{5}{2}. ‎\end{align*}‎‎‎ (a) Is it a known result?‎‎

(b) What about the case ‎$‎s>‎\frac{5}{2}$‎?‎‎

(c) Do you know any useful references?‎

‎ Note. We know that ‎$‎‎\zeta‎(s)‎\geq‎‎\frac{1}{s -1}‎$‎ ‎for ‎‎$‎s>1‎$ ( ‎‎because, ‎we have $‎‎\frac{1}{k^s}\geq‎\frac{1}{x^s}‎$ ‎for ‎every ‎‎$‎k\in‎\mathbb{N}‎$ ‎and ‎‎$‎x\geq k‎$‎, where ‎$‎s‎$ ‎‎ is ‎constant. So, we ‎get ‎the ‎result).‎ ‎ Therefore, ‎the ‎lower ‎bound ‎‎$‎‎\frac{5 - 2s}{s - 1}‎$ ‎is ‎stronger ‎than ‎‎$‎‎\frac{1}{s - 1}‎$‎ ‎if ‎‎$‎1<s<2‎$‎. ‎‎

• $\zeta(s)(s-1)$ converges to $1$ as $s\to 1$ so any inequality cannot violate this. You have $\zeta(s)(s-1)\geq 5-2 = 3$ as $s\to 1$. For a more visual disproof just plot it – Winther Jun 19 '18 at 15:23
• thank you for your guide – koohyar eslami Jun 19 '18 at 16:15

This cannot be a known result because it is wrong: For $s=3/2$ you have

$$\zeta(s) \approx 2.6123753486854883, \quad \frac{5-2s}{s-1}=4$$