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Let $\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$. We have $\frac{\zeta'}{\zeta}(s) = \sum_{n=1}^\infty \frac{\Lambda(n)}{n^s}$ for $s>1$, where $\Lambda$ stands for the von Mangoldt function ( It is easy to prove that there exists a constant $A$ such that $-\frac{\zeta'}{\zeta}(s) \leq \frac{1}{s-1} + A$ for every $s>1$. By plotting $-\frac{\zeta'}{\zeta}(s)-\frac{1}{s-1}$ or $-\frac{\zeta'}{\zeta}(s)\cdot(s-1)$ on any computation program, it seems fair to conjecture that we can choose $A=0$. Unfortunately, I am unable to prove this. Is it a well-known inequality? Does anyone know how to prove this, if true?

I tried to use the formula $-\frac{\zeta'}{\zeta}(s) = \frac{1}{s-1} + 1 + s \int_1^\infty \frac{\psi(t)-t}{t^{s+1}} \mathrm{d}t$, where $\psi(x) = \sum_{n \leq x} \Lambda(n)$, but the erractic behavior of $\psi$ makes it useless (and any Chebyshev-type inequality does give $A>0$).

Thanks in advance.

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In Davenport (Multiplicative Number Theory) you find (Chapter 12, page 81) that $$ \lim_{s\rightarrow} \left(\frac{\zeta'}{\zeta}(s)+\frac{1}{s-1}\right)=\gamma\;, $$ where $\gamma = 0.57721566...$ is the Euler-Mascheroni constant. So for $s$ values "close" to 1 seems true.... – PITTALUGA Oct 11 '13 at 14:54
Maybe that is a good clue. I tried to understand how he found that $\lim_{s \to 1} \left(\frac{\zeta'}{\zeta}(s)+\frac{1}{s-1}\right) = 1-I(1)$ (where $I(s) = \int_1^\infty \frac{t-[t]}{t^{s-1}}$), and I guess he just differentiated the formula $\zeta(s) = 1-sI(s)$ and then simplified at his best the expression we want. If I am not mistaken, one has $\frac{\zeta'}{\zeta}(s)+\frac{1}{s-1} = \frac{1-(2s-1)I(s)-s(s-1)I'(s)}{\textrm{something positive}}$, I hope the sign is now easy to evaluate... – Bernikov Oct 14 '13 at 0:59
up vote 1 down vote accepted

Ok, I succeeded in proving this. From the equality $\frac{\zeta'}{\zeta}(s)+\frac{1}{s-1} =\frac{1-(2s-1)I(s)-s(s-1)I'(s)}{\textrm{something positive}}$ (see my comment below my question), since $I'(s)<0$, we only have to prove thaht $1-(2s-1)I(s)>0$. Since $I(s)$ can be expressed in terms of $\zeta$, it is equivalent to an inequality only involving $\zeta$, that is, $\zeta(s) > \frac{s^2}{(s-1)(2s-1)}$. This is easy to prove, because we have refined lower bounds for $\zeta$ thanks to the comparaison between series and integrals.

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