🚀 go-pugleaf

I have some questions about the growth of the Mahler coefficients of the
p-adic gamma function, which I'll denote by Gamma_p.  Let a_p(n) denote the
nth coefficient of the Mahler expansion (NOT the Taylor expansion) of
Gamma_p, and A_p(n) = ord_p( a_p(n)).  With some assistance I have produced
the following data:

For p = 2, 0 <= n <= 509, write n = 4k + r with 0 <= r <= 3.  Then
A_2(4k) = k
A_2(4k + 1) = k + 1
A_2(4k + 2) = k + 2
A_2(4k + 3) = k + 3 + ord_2(k + 1) = k + 3 + ord_2(k - 1/(1 - 2))

For p = 3, 0 <= n <= 1082, write n = 9k + r with 0 <= r <= 8.  Then 
A_3(9k) = 2k
A_3(9k + 1) = 2k
A_3(9k + 2) = 2k
A_3(9k + 3) = 2k + 2
A_3(9k + 4) = 2k + 1
A_3(9k + 5) = 2k + 3 + ord_3(2k + 1) = 2k + 3 + ord_3(k - 1/(1 - 3))
A_3(9k + 6) = 2k + 2
A_3(9k + 7) = 2k + 2
A_3(9k + 8) = 2k + 3

I have looked at A_p(n) as well for p = 5, 7, 11, 13, and 17 for n running
up to at least 400 (in the case of 5 and 7 for n up to 1400), and what
seems to be happening is 
      A_p(n) = (p - 1)[n/p^2] + r_n,   ( [ ] = greatest integer)
where r_n >= 0 and for "most" residue classes n(mod p^2) it turns out that
if m = n (mod p^2) then r_m = r_n.  For example, when p = 2 this "mod p^2"
periodicity of r_n" occurs for all classes mod 4 except for the class 3(mod
4).  When p = 3 there is one exception, namely the class of 5 mod 9.  When
p = 5 there seem to be (at least) four exceptions: the classes of 4, 7, 16,
and 22 mod 25.  When p = 7 I have so far found only one class mod 49 for
which  the n in that class don't have the same r_n value - it's 19 mod 49.
Due to the slower growth of A_p(n) as p increases, there are probably some
more exceptions to this when p = 7 which I have not located. Even in the
(seemingly) few exceptional classes mod p^2 not admitting a constant r_n
value, it seems that r_n = const. + s_n where the const. only  depends on
the residue class and s_n increases very slowly with n (see the data above
for p = 2 or 3 and the classes of 3 mod 4 or 5 mod 9).  

I'd like to hear from anyone who can explain this behavior of A_p(n),
ESPECIALLY the periodicity of the r_n's mod p^2.  The fact that A_p(n)
grows like (p - 1)n/p^2 seems to be due to Gamma_p(n) being analytic on
cosets of pZ_p but not on Z_p (well, for p > 2), although I haven't worked out
the details on this.

Responses can be given on this newsgroup, although I'd also appreciate it 
if they could be sent to me as well.  Thanks.

Keith Conrad
conrad@zariski.harvard.edu