1881.] Reciprocals of the Prime Numbers and of their Powers. 7 



This is the formula given by Mr. Grlaisher, and used both by him 

 and by myself in the calculation of 2«. 

 In the particular case of n—1, we have — 



2 1= log (log x) -0^31571 84520 73890 [*=«].* 



I have not been able to identify this constant 0"3157 .... with any 

 function of a known constant. It does not seem to have any imme- 

 diate connexion with the Eulerian constant 7 ; for 



log e7 = -0-5496, 7 2 = 0-33816. 



There appears no reason why it should be commensurable with any 

 simple function of 7. The following results may be useful for the 

 purpose of this or analogous comparisons : — 



6 y=1 -781072, 6-y=0 -5614595, 



6 -2 1 = l -371241,t 6^ = -7292647,t 



6 a+2 1 )=l -982347,f 6-<i+2x)=0 -5044525.f 



It is worth while to remark that the S series l + i + 3+i+ 



begins with unity, whereas the prime series | + 3 + i+ •••• begins 

 with J, and omits the unit. 



The last two tables are given to 15 decimal places, having been 

 calculated to 16 places. Still, the last figure is not reliable. The 

 tables are not continued to very high values of n, because, for such 

 values — 



S, 4 -l=log(S K ) = 2„ 

 =2(S^-1) =2 log (S«_ 1 )=22„_ 1 , 



true to 15 decimal places, or more. 



* There may be some doubt as to the correctness of this expression. This turns 

 on whether the assumption which it involves is justifiable, namely — 



log (S^)- 1 } -log (log*)=0 



when x = qc , and consequently when both terms on the first side of the equation re- 

 present divergent series. This, however, is not a question which much affects the 

 arithmetic. 



f Omitting the term log (log oc ) . 



