# Searching for numbers

In my last post, I mentioned finding the significance of the Niven constant $N=1+\sum_{k\geq 2}{(1-\zeta(k)^{-1})}=1.70521114\ldots$

using Google. In fact, I searched first for “0.70521”, then for “2.70521” (which is the value of real interest in my case), and then last for “1.70521”.

Yesterday, I had the apparently subtle insight that since the integral part of the constant was unclear, I should search for “.70521” instead. But it doesn’t work! Instead of showing prominently pages with the Niven constant, we get a lot of information on the 70521 ZIP code and other irrelevant information. Increasing the precision with 705211 does not help (one gets various types of objects with component number 705211, from light bulbs to UV filters). Going to 7052111 gives us a number of patents and transaction numbers, 70521114 is similar, and 705211140 gives…nothing. (Until this post appears in the Google index, of course — this effect has been noticed already by I. Lugo).

To get the Niven constant, one must therefore search correctly, and not too precisely. It starts working at 1.70521, which is the precision I had chosen (randomly) to see if my constant was already known. Amusingly, searching for 1.705211 finds a reference to an Advanced Problem in the American Math. Monthly in 1975, which asks precisely to prove the result of Niven going back to 1969 where the constant first occurred. (It should also find the MathWorld page on the Niven constant, where the value is given with this precision, but it is hidden in an image containing the formula and Google apparently does not index the ALT text in the HTML tag). But from 1.7052111 on, it is all terra incognita.

I wonder if this “optimal” precision is typical if we want to find other constants on the web?  For e and π, we can certainly go much further, but what about other less glamorous numbers?