Unless I made a mistake, the expected value of the largest exponent in the prime factorization of random positive integer (defined in the appropriate way) is $$\eta := \sum\_{n=1}^\infty \Big(1-\zeta(n)^{-1}\Big)$$ with the convention $\zeta(1)^{-1} = 0$ (for aesthetics). I was just wondering whether this constant $\eta$ has a name, whether it's been studied, etc..



Has this number-theoretic constant been studied?
If you calculate it to a few decimals, you find
$$
1.705211140105\ldots
$$
which is enough to locate it in the OEIS.


It's Niven's constant: [MathWorld](https://mathworld.wolfram.com/NivensConstant.html), [Wikipedia](https://en.wikipedia.org/wiki/Niven%27s_constant), [OEIS](https://oeis.org/A033150).


As [mentioned](https://mathoverflow.net/questions/422356/has-this-number-theoretic-constant-been-studied#comment1085332_422366) by GH from MO in the comments, it was in fact proven by Niven in 1969 that the average largest exponent tends to $\eta$.


Since the question is essentially about finding literature related to a given numerical constant, I should probably mention [this answer](https://math.stackexchange.com/questions/4263890/finding-a-closed-form-for-constant/4265349#4265349) by myself on the other site, exhibiting some other methods (Steven Finch's book *Mathematical Constants* & how to google the decimals effectively).


