Yesterday, a very talented and passionate young student from South Africa asked me the following question about Riemann's zeta function. $ zeta (s) $. He says he "thinks" he knows the answer, but he just wants to hear my views. However, I am not a number theorist, therefore I was unable to answer you. So below is the question:

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

Note that

$$ Big ( frac {1} { zeta (s + 1)} Big) & # 39; = -s int_ {1} ^ { infty} S (x) x ^ {- s-1 } mathrm {d} x $$ for $ Re (s)> alpha-1 $, where the cousin denotes differentiation. It's known that $ S (x) = – 1 + or (1) $, therefore, the previous integral converges if and only if $ Re (s)> 0 $. *What does this tell us, in any case, about the value of $ alpha $ ?*

**P.S:** I could not verify the previous identity, nor could I verify the "known" result that $ S (x) = -1 + or (1) $.