# Has this number-theoretic constant been studied?

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..

• What do you mean by “random positive integer”? May 12 at 9:35
• @AlessandroDellaCorte I guess my last comment provided the definition for "largest exponent in prime factorization of a random positive integer" rather than for "a random positive integer" which is undefined. May 12 at 10:01
• 👍(and since some more characters are needed, I add these ones) May 12 at 10:59
• Don't you mean to set $\zeta(1)^{-1}=-12$? (Joke, joke! Put down the pitchforks...) May 12 at 16:31
• @StevenStadnicki You're thinking of $\zeta(-1)^{-1}=-12$ (down, pitchforks, down). I may have just accidentally explained the joke.
– J.G.
May 13 at 8:16

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, Wikipedia, OEIS.

As mentioned 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 by myself on the other site, exhibiting some other methods (Steven Finch's book Mathematical Constants & how to google the decimals effectively).

• The OP's $\eta$ is Niven's constant. The OP's term corresponding to $n=1$ equals $1$. More importantly, it seems that the OP rediscovered Niven's theorem from 1969 (doi.org/10.2307%2F2037055). May 12 at 13:53
• Thanks! You just added a new tool to my toolbox: I've been using inverse symbolic calculators instead of inputting the digits as a sequence to the OEIS! Also, as GH said, $1$ must be added. May 12 at 15:01
• Glad if it helps! BTW you can input it into OEIS either as a sequence of digits ("7,0,5,2,1,1,1") or more conveniently as the decimal (".7052111"). I didn't know the latter one works until I tried it today. May 12 at 15:26
• GH, thanks, somehow I misread the first term as zero. Will fix that in my answer. May 12 at 15:27
• @JukkaKohonen It would be weird if the average of the largest exponent were less than $1$ :) May 12 at 17:15