New month, new password, but I've already forgotten! Help!
What is my phone's password this month?
As per usual, my password is pretty long (maybe around 40 digits?)
In March I took up an interest in analytical number theory
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Your password is
the first 40 digits, or perhaps decimal places, of the Euler-Mascheroni constant, sometimes denoted $\gamma$.
So it will begin
either 5772 or 05772 or 0.5772.
to 41 decimal places the constant's value is 0.57721566490153286060651209008240243104215, the last digit will (if you've used 40 decimal places rather than 40 digits including the leading zero) depend on how you've chosen to round it.
Explanation in case the above doesn't make it clear:
the harmonic series is $1+\frac12+\frac13+\frac14+\cdots$ and the sum of its first $n$ terms is $\log n+\gamma+\varepsilon(n)$ where $\varepsilon(n)\rightarrow0$ as $n\rightarrow\infty$. So the limiting difference between the harmonic series and the natural logarithm is $\gamma$.