Let $$H_n=1+\frac{1}{2}+\cdots+\frac{1}{n},$$ the nth harmonic number and $$\sigma(n)=\sum_{d\mid n}d,$$ the sum of divisor function, for example $\sigma(6)=12$.
I believe that this could be a nice problem in analysis.
Question. For which integer $M\geq 3$ can you claim easily, without assuming Riemann Hypothesis, that previous arithmetical functions satisfies $$\sigma(n)\leq H_n+M^{H_n}\cdot\frac{\log H_n}{\log M},$$ for all $n\geq 1$?
In a phrase, what is the minimum integer $M\geq e$ that satisfies a modified Lagarias' statement, and it is easily to prove, unconditionally, with analysis?
Thanks in advance.