1881.] Reciprocals of the Prime Numbers and of their Powers. 5 



which presents itself, in the series of simple reciprocals of primes, as 

 the difference between the sum of the series and the double loga- 

 rithmic infinity to the Napierian base e. 



The summation of these series was shown by Euler to depend upon 

 the Napierian logarithms of the sums of the reciprocals of the powers 

 of the natural numbers. Euler's reason for taking even powers only 

 was doubtless that these latter summations could be expressed in 

 terms of Bernoulli's numbers, and of the powers of w, and could 

 therefore be computed directly, without any actual summation. A 

 table given by Legendre,* and reprinted by De Morgan, f however, 

 contains the whole series of the sums of the reciprocals of the powers 

 of the natural numbers, even as well as odd, and I have made use 

 of this table as the basis of my own work, being satisfied with the 

 verifications of Legendre. 



The first step was to find the Napierian logarithms of all the 

 numbers given in Legendre's table. These were then combined, so 

 as to give the summations of primes, by means of a theorem of M6bius£ 

 used by Mr. J. W. L. Glaisher,§ in calculating a corrected table of 

 Euler's values. Instead of giving my own results only, I have 

 thought it would be convenient to bring together the complete set jj 

 of— 



1. Legendre's table of the sums of the reciprocals of the powers of 

 the natural numbers, to 16 figures. 



2. The Napierian logarithms of the numbers in that table, to 

 15 figures. 



3. The whole series of the sums of the reciprocals of the powers of 

 primes, to 15 figures. 



I think it will be convenient also to give a very short note of the 

 means of obtaining the results, as the best explanation which they can 

 receive, and as saving some troublesome references to books not 

 accessible to everybody. 



Since the reciprocal of 1—- is the geometrical progression — 



i+uvv ■■■■> 



X X 4 X 6 



* "Traite des Fonctions Elliptiques," vol. ii, p. 432. 

 f See his " Diff. and Int. Calc," p. 554, 

 J Crelle's " Journal," vol. ix, p. 105. 



§ See the " Transactions" of the " Association Francaise pour 1'Avancenient des 

 Sciences," Havre, 1877. 



|! See Mr. J. W. L. Grlaisher's paper in the " London Mathematical Society's 

 Proceedings," vol. iv (for 1872), pp. 48-56, " On the Constants that occur in certain 

 Summations by Bernoulli's Series," for the summations of some allied series, 

 namely — 



l - " — 2~ n + S~ n — 4r n + . . . (natural numbers) , 

 and l _n + 3~" + 5 _n + 7~"" + . . . (odd numbers). 



