26o 



NATURE 



{July 12, 1888 



Let us first, without any reference to convergence, consider 

 the product obtained by the usual mode of multiplication of the 

 infinite series 



S = 1 - 

 by the product 



1 I + 



I -_ 



+ *'- \ +h~ 



ad inf. 



I + 



ad inf. 





<h 



'/■1 



It is clear that the effect of the multiplication of S by the 

 numerator of the above product will be to deprive the series S 

 of all its negative terms. Then the effect of dividing by the 

 denominator of the product, with the exception of the factor 

 I - \, will be to restore all the obliterated terms, but with the 

 sign + instead of - . Lastly, the effect of multiplying by the 

 reciprocal of ( 1 - \) will be to supply the even numbers that 

 were wanting in the denominators of the terms of S, and we 

 shall thus get the indefinite series 



1 + 4 + \ + I + • 



Call now 



ad inf. 



Qs = 



1 + 



?N., 



i - i 1 1 - 1 - — 



Q N , which is finite when N is finite, may be expanded into 

 an infinite aggregate of positive terms, found by multiplying 

 together the series 



- + -, + r% + 



+ — + 



'/■i 



fk" 



•i-r 



Let 



222 



1 + + '■>+-' 



y-K.q '^n.q ?N.C 



■■•** 



then from what has been said it is obvious that we may write 

 Q N S N - t +: i + | + { + . . . . + -L + V-R, 



where V and R may be constructed according to the following 

 rule : Let the denominator of any term in the aggregate Q N 

 be called t, and let 6 be the smallest odd number which, multi- 

 plied by t, makes id greater than N ; then if 6 is of the form 

 4« + 1 it will contribute to V a portion represented by the 

 product of the term by some portion of" the series S N of the form 



--___+__ 



e e + 2 e + 4 



and if 6 is of the form 4« + 3 it will contribute to - R a portion 

 equal to the term multiplied by a series of the form 



- I + - - -- + • • • • 



e e + 2 e + 4. 



Hence R is made up of the sum of products of portions of the 

 a ggregate Q x multiplied respectively by the series 



1 — 1 4. 1 1-Ll 1_1_ 



1 — 1 4- 1 _ 1 _. 



tV - tV + • ■ • ■ 

 of which the greatest is obviously the first, whose value is 1 - S N . 



Consequently R must be less than the total aggregate Q N 

 multiplied by 1 - S N . 



Therefore 



+ - > log N, 



Qn s n + Qn( j - s K )> 1 + _.+ ! + * +._. 



i.e. Q N > log N, 



from which it follows that when N increases indefinitely the 

 number of factors in Q N also increases indefinitely, and there 

 must therefore be an infinite number of primes of the form 

 4» + 3- 



Denoting by M N the quantity 



- '■)(' " -' 



we obtain the inequality 



'/n 



N../ 



and taking the logarithms of both sides 



_! - A2 2 + i_ 3 - . . . . > 4 log log N + 1 log M N - I log 2, 



where in general 2, denotes the sum of the z'th powers of the 

 reciprocals of all prime numbers of the form 472 + 3 not 

 surpassing N. 



Hence it follows that 2 X > \ log log N. 



If we could determine the ultimate ratio of the sum of those 

 terms of Q N whose denominators are greater than N to the total 

 a gg re g ate > an d should find that ft, the limiting value of this ratio, 

 is not unity, then the method employed to find an inferior limit 

 would enable us also to find a superior limit to Q N ; for we 

 should have V < /xQ N added to the sum of portions of what 

 remains of the aggregate when /*Q N is taken from it multiplied 

 respectively by the several series 



i - 7 + I ~ i't + T V " tV + • • • • adinf 

 \ - t r + -h ~ T-s + • • • • ad inf. 

 Ts ~ t a + ■ ■ ■ ■ ad inf. 

 the total value of the sum of which products would evidently be 

 less than 



(1 -m)(S- i + DQ^ 



Hence the total value of V would be less than 



i.e. less than 



M Q N s + (i - M )Q N .(s - |y, 



Q N S - |(i - M )Q N> 

 and consequently we should have 



f(i - M )Q N < log N, 



l - e - Q v < 3 log N. 



N 2(1 - M 8 



From which we may draw the important conclusion that if fi is 



less than I, i.e. if when N is infinite the portion of the aggregate 



S N Q N comprising the terms whose denominators exceed N does 



not become infinitely greater than the remaining portion, the 



sum of the reciprocals of all the prime numbers of the form 



4« + 3 not exceeding N would differ by a limited quantity from 



half the second logarithm of N. 



A precisely similar treatment may be applied to prime numbers 



of the form 6« + 5. We begin with making 



S N = 1 

 We write 



1 

 Qn = 



1 4- 1 — 



I + _ 



I + _ 



We make 



Q N S N = i + h + h + i+ ■ ■ ■ 

 We prove as before that 



R < (1 - S)Q N , 



and thus obtain 



and then putting 

 M VI = 1 ] 



Q v > log N, 



and finally noticing that 



we obtain 



1 + - 



1 



N.r / 



I+I) 31 



''0/ 



> 4M M log N. 



