TITLE: Interesting result on the Euler-Maschroni constant - what is the background?
QUESTION [11 upvotes]: Today I entered the following expression in maple:
$$a_i = H_{10^i} - ln(10^i) - \gamma$$
Here $H_j$ equals $\sum_{k=1}^{j} 1/k$ and $\gamma$ is the Euler-Mascheroni constant. 
When I computed $a_n$ for $i = 0$ to $10$ I obtained the following results:

 $i = 0$;    4.227843350984671393934879099175975689578406640600764011942327651151323 * $10^{-1}$

 $i=1$;   4.9167496072675423629464709201487329610707429399557393414873118115813 * $10^{-2}$

 $i=2$;   4.991666749996032162622676207122311664609813510982102304110919767206 * $10^{-3}$

 $i=3;$    4.999166666749999960317501984051226762153678825611388678758121701133 * $10^{-4}$

 $i=4$;     4.99991666666674999999960317460734126976551226762154503821179264423 * $10^{-5}$

 $i=5$;     4.9999916666666667499999999960317460321626984126226551226762154523 * $10^{-6}$

 $i=6$;     4.999999166666666666749999999999960317460317501984126984051226523 * $10^{-7}$

 $i=7$;     4.99999991666666666666674999999999999960317460317460734126984123 * $10^{-8}$

 $i=8$;                                                                            
4.9999999916666666666666667499999999999999960317460317460321623 * $10^{-9}$

 $i=9$;                                                                            
4.999999999166666666666666666749999999999999999960317460317423 * $10^{-10}$  

 $i=10$;
4.99999999991666666666666666666674999999999999999999960317423 * $10^{-11}$

So we see that the periodic strips of ...99999..., of ...66666... and ...99999... an many other periods increase for even larger $i$. The question is now: Is there any rule behind it that the remainder term $a_i$ behaves that way?

REPLY [2 votes]: Adding an example to @quid's answer:      
Using Pari/GP the harmonic numbers minus Euler-gamma can be obtained using the psi-function.
If we subtract 1 from the $\small a_i $-values the composition of the result by i-digits long blocks of decimal expansion of the bernoulli-numbers becomes then immediately visible. With $\small i \gt 20 $ or so it becomes even more impressive:
fmt(200,60)   \\ user function: internal prec 200, display prec 60 digits
i=6
psi(10^i)-1 - i*log(10)
 %681 = -1.00000050000008333333333332500000000000396825396824980158730
(psi(10^i)-1 - i*log(10))*6
 %683 = -6.00000300000049999999999995000000000002380952380949880952381
(psi(10^i)-1 - i*log(10))*42
 %684 = -42.0000210000034999999999996500000000001666666666664916666667
(psi(10^i)-1 - i*log(10))*42*30
 %685 = -1260.00063000010499999999998950000000000499999999999475000000

etc...