A question on the Riemann zeta function

Question

Yesterday, a certain very talented and passionate young student from Southern Africa asked me the following question about the Riemann zeta function $\zeta(s)$. He says he "thinks" he knows the answer, but he just wants to hear my views. However, I'm not a number theorist, hence I couldn't answer him. So below is the question:

Consider the Riemann zeta function $\zeta(s)$, and let $\alpha$ be the supremum of the real parts of its zeros. Let $\mu$ denote the Möbius function. Define $S(x)= \sum_{n\leq x} \frac{\mu(n)\log n}{n}$.

Note that

$$\Big(\frac{1}{\zeta(s+1)}\Big)' = -s \int_{1}^{\infty} S(x)x^{-s-1} \mathrm{d}x$$ for $\Re(s)> \alpha-1$, where the prime denotes differentiation. It is known that $S(x)=-1 + o(1)$, thus the above integral converges if and only if $\Re(s)>0$. The student's question is: what does this tell us, if anything, about the value of $\alpha$ ?

PS: Personally, i couldn't verify the above identity, neither could I verify the "known" result that $S(x) = -1 + o(1)$, hence I couldn't answer his question.

Answer 1

The identity $$\sum_{n=1}^\infty\frac{\mu(n)\log n}{n}=-1$$ was conjectured by Möbius (1832) and proved by Landau (1899). It is a consequence of the prime number theorem. Not surprisingly, the rate of convergence is determined by the (known) zero-free region of $\zeta(s)$. In particular, $$S(x)=-1+O_\epsilon(x^{\alpha-1+\epsilon})$$ holds for any $\epsilon>0$, and $\alpha$ in the exponent cannot be lowered.

Here is a sketch of the proof of the mentioned facts. By Perron's formula, we have (at least for $x\not\in\mathbb{N}$) $$S(x)=\frac{1}{2\pi i}\int_{1-i\infty}^{1+i\infty}\left(\frac{-1}{\zeta(s+1)}\right)' \frac{x^s}{s}\,ds.$$ The integration is meant over the vertical line with abscissa $1$. By truncating the integral at some height, and applying the residue theorem appropriately, we can move the line segment of integration to the left with the benefit of $x^s$ being much smaller there. This is the same technique by which the prime number theorem was originally proven. At $s=0$, the derivative inside the integral equals $-1$, while $x^s/s$ has a simple pole with residue $1$. Therefore, as we move the curve of integration to the left of $s=0$, we pick up the main term $-1$. The error term then depends on how far to the left we can move the curve of integration without encountering further poles, i.e. where the zeros of $\zeta(s+1)$ are located. The standard zero-free region already implies my first display. If $\alpha<1$, then we have a much wider zero-free region, and the second display follows. The fact that the exponent is optimal follows by reversing this logic, namely by examining the analytic continuation of the RHS of the OP's formula to the left of $s=0$.

I hope this helps your student, or perhaps this is exactly what she/he had in mind. It is standard material, but a good way to better understand the prime number theorem and its relation to the zeros of $\zeta(s)$.