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New month, new password, but I've already forgotten! Help!

What is my phone's password this month?

enter image description here

Hint #1:

As per usual, my password is pretty long (maybe around 40 digits?)

Hint #2:

In March I took up an interest in analytical number theory

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  • 2
    $\begingroup$ Really March? Really 2018? $\endgroup$ – Gareth McCaughan Apr 3 at 15:26
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    $\begingroup$ If you look at my other puzzles you’ll see I’m starting January of 2018 and going month by month to the present day. Sorry for confusion! $\endgroup$ – Tyler22Alex Apr 3 at 15:59
  • $\begingroup$ Ah, OK. Fair enough. $\endgroup$ – Gareth McCaughan Apr 3 at 16:44
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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.

Since

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 $\gamma$.

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    $\begingroup$ That was fast! Nice job! $\endgroup$ – Tyler22Alex Apr 3 at 16:00
  • $\begingroup$ The second hint made it pretty hard not to get :-). $\endgroup$ – Gareth McCaughan Apr 3 at 16:44
  • $\begingroup$ (Though I think I'd probably have got it quickly even without.) $\endgroup$ – Gareth McCaughan Apr 3 at 16:44

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