This morning I am trying to think more about the function $$f(x):=-\sum_{n=1}^\infty\frac{\mu(n)}{n}x^{n-1}\tag{1}$$ defined over the interval $[0,1]$, being $\mu(n)$ the Möbius function (see the definition of this arithmetic function, for example, from this MathWorld).
I think that this function $(1)$ is convex. You can see a plot of the function (a toy model for our function $(1)$)
plot sum -mu(n)/n x^(n-1), from n=1 to 1000, for 0<x<1 using Wolfram Alpha online calculator. I know that a sum of convex functions is convex, and this property holds for series, but I don't know how provide us a rigurous proof that $f(x)$ is convex over the unit interval.
Claim. On assumption that $f(x)$ is convex, then invoking Dragomir's theorem (see frist pages of [1] or [2]) one gets $$0.61\approx\frac{1}{\zeta(2)}\leq \sum_{n=1}^\infty\frac{\mu(n)}{n}\sum_{k=0}^{n-1}\frac{\binom{n-1}{k}}{k+1}t^k\left(\frac{1-t}{2}\right)^{n-1-k}\leq \sum_{n=1}^\infty\frac{\mu(n)}{n}2^{-n+1}\approx 0.66.\tag{2}$$
I would like to know if my Claim is numerically right or if can be improved (calculate better upper bounds for the series in the middle of $(1)$).
Question. A) Can you find numerically (I say with your computer) a good approximation (preferably a new upper and a new lower bound to do a comparison with mine) of $$ \sum_{n=1}^\infty\frac{\mu(n)}{n}\sum_{k=0}^{n-1}\frac{\binom{n-1}{k}}{k+1}t^k\left(\frac{1-t}{2}\right)^{n-1-k}\tag{3}$$ when $t\in [0,1]$? I am asking to do a comparison with my calculations, thus I think that I am asking about the infimum and supremum of our function in $(3)$ when $t$ runs over the unit interval.
B) On assumption that our function $(1)$ is convex can be improved the lower and upper bounds of $(2)$? (I believe that it is very difficult, even on assumption that our $f(x)$ is convex).
C) (Optional) I would like to know if can be potentitally interesting to study if our function $f(x)$ is a Wright-convex function (see first pages of [1] or [3]). Thus if you think that our function $f(x)$ can be Wright-convex function, or if it can be interesting answer this optional question with a yes or not ( I fantasized about creating functions like $f(x)$ enconding some valuable fact realted to unsolved problems concerning the Riemann's Zeta function). Many thanks.
The article [1] is free-access from the archive of the journal OCTOGON MATHEMATICAL MAGAZINE.
References:
[1] Ciobotariu-Boer, Hermite-Hadamard and Fej´er Inequalities for Wright-Convex Functions, OCTOGON MATHEMATICAL MAGAZINE Vol. 17, No.1, (April 2009).
[2] Dragomir, Two mappings in connection to Hadamard’s inequalities, J. Math. Anal. Appl., 167 (1992).
[3] Klee, Solution of a problem of E.M. Wright on convex functions, Amer. Math. Monthly, 63 (1956).
Maybe this becomes helpful:
$f$ is represented in power series so f is smooth (even analytic ) in its convergent radios which is $(-1 , 1)$, so to show $f$ is convex on $(0 , 1)$ you need to show that $f'' \ge 0$ on (0 , 1 ).
$$f''(x):=-\sum_{n=3}^\infty\frac{(n-1)(n-2)\mu(n)}{n}x^{n-3} = \frac{1}{3} -\sum_{n=5}^\infty\frac{(n-1)(n-2)\mu(n)}{n}x^{n-3} $$
Plotting point shows even $f'' \ge \frac{1}{3}$. So you need to show that $\mu (n)$is often $-1 $ rather than $1$. which is really a number theory stuff rather than analysis (so it is out of my skill!)
Note that first terms of $\mu$ after $5$ are $$ -1, 1,-1,0 ,0,1, -1,0 $$