204 Prof. H. J. S. Smith on Quadratic Forms ^c. [Dec. 5^ 



product (— Ij^'I^xIgX .. XI2V-1, and the summation extends to all 

 uneven values of in, which are prime to D, from 1 to oo . The sum of 

 this infinite series can in every case be obtained in a finite form by the 

 methods employed by Dirichlet (in the 21st volume of Crelle's Journal) 

 and by Cauchy (in the 17th volume of the Memoires de 1' Academic des 

 Sciences, p. 679). As the result of the summation does not seem to have 

 been given, we shall present it here in one of many various forms which it 

 may assume. Let represent the quotient obtained by dividing D by 

 its greatest square divisor ; let q be any uneven prime dividing D, but 



JGO /J) \ J 



not D,, and let Y= — 2( — - ) — , the siD;n of summ.ation extendino- to all 

 values of in prime to 2D^ ; we then have the equation 



To obtain the value of V, let A represent the positive value of D^, so that 

 A=D3 when v is even, and A=— when v is uneven. Also let 



^--(-)=S-^-"-^ + n.2.^n.2 ^--""^ 



n.2cr-4.n.4'^ . 2 . 11 .2(r-2^'"-' ^ 



where /3p /Sg, . . . are the fractions of Bernoulli, so that V^{x) is the function 



s=x—l 



which, when x is an integral number, is equivalent to the sum 2 s^. 



Then, if 1)^^''+^'', or ( — 1)-"+^-*, according as v is even or uneven, 



the value of eV is 



(1) when Di =1, mod 4, 



(2) in every other case, 



A 



the summation S extending to every integral value of s inferior to A and 



4A 



prime to A, the summation 2 extending to every integral value of 5 inferior 



1 



to 4A, and prime to 4A. The formula (1) is inapplicable when A — D^=] ; 



