ON THE THEORY OP NUMBERS. 337 



n— 1 



in which D = 3, mod 4, so that f—\=( — l)~^^^, ;J=4a, A still denoting 



( 2Tmr .N 

 ^ e *^ J is assigned by the formulae 



m— 1 



ofGanss; it is 2i' ( — 1) "^ (-)v^A, or zero, according as m is, or is 



not, prime to 4a*. We thus find 



the summation extending to all values of m prime to ^A and less than it. In 



* If ^ be any prime divisor of A, an uneven number admitting of no square divisor, and if, for 



brevity, P = - ; we have, by Gauss's formula, 

 P 



,_ , 2kmVn . 



:}r ©•^■=e)©-^"-"^-'=» « 



according as m is or is not prime to jo. If we multiply together the equations of this type, 

 corresponding to every prime divisor of A, and observe (1) that 6 = 2 . AP2 represents a system 



of residives prime to A, (2) that (|) = (^\ (l-\...= (^\ [h\ . . . , (3) that 

 n /-) j^(/'-i)''=(_l)Si(;',-i)(/'2-i),-2i(p-i)2==j^[2(7)-i)]2^ii(A-i)2, ^e find 



according as m is or is not prime to A. We have already met with this equation in art, 96, 

 ix. If in the equations (1) we write 4P for P, and join to them the equation 



At— 1 2km ^TT . m — 1 A — 1 



2(— 1) 6 =2z( — 1) , (m uneven), or =0 (m even), 



in which I: is either term of a system of residues prime to 4, we obtain after multipli- 

 cation the equation which is employed in the text. And similarly may the function 



f\6 * / be evaluated, whatever be the form of D. 



The formula; (A) and (A') of art. 20 are only particular cases of the general result 

 obtained by Gauss in the ' Summatio Serierum, &c.' The general formula, including (A), is 



2 r* =/-ji'' Vn, A denoting any number prime to n. When n is even, the 



formula (A') of art. 20 is similarly included in the following, 



/-A-l\2 



2r»^= = 0, or =g) t ^^ ^ (1+i) 'Jn, 



according as n is unevenly or evenly even. 

 When n is uneven and not divisible by any square, the two sums 



'T~r''*%nd2(^V** 



arc identical, as appears from a comparison of (2) with the generalization of (A), and as 

 has been already observed in the case when « is a prime (art. 21). 

 1861. z 



