BIRKHOFF. — THE GENERALIZED RIEMANN PROBLEM. 555 



§ 17. Solution of the Problem of § 16. 



In order to treat the problem of § 16 we apply the preliminary 

 theorem. We shall take r = 1, and take Cy to be the axis of imagi- 

 naries in the complex plane unless | P(.v) | = at a point of that axis. 



The matrix Ai (.r) is taken equal to 



T (x) P (x) r-i (.r), T (x) = (x'^' (p.e-'^r x^Jdi^), 



except near to x = 0, where it is chosen in any way so as to satisfy the 

 restrictions of the preliminary theorem there (compare § 9). Since 

 the elements of T(x) are in general multiple-valued functions of x, 

 it is necessary to specify which branch of T(x) to select. We shall 

 choose a continuous branch of T(x) in the right half plane, and a 

 continuous branch in the left half plane in such a way that these 

 branches coincide along the upper half of the axis of imaginaries. 

 The first factor T{x) in the expression for Ai{x), will be identified 

 with the first of these branches, and the last factor T"^{x) will be the 

 inverse of the second of these branches. 



It is therefore clear that, along the upper half of the axis of imagina- 

 ries, the element in the ith. row and jib. column of Ai (x) is, for i 9^ j, 



g2.xy ^J~lXp.Xp^-x^^rrrj f^.^.(o) + . . . + ^..(M-i;g2.(M-i) V=ixj 



while the diagonal elements are the same as for P(x). But by defini- 

 tion of Xy, 



(49) 1 > U (\ij - ] (logp,- - logpO) ^ 0. 



V 27rV— 1 / 



Let us exclude at present the case of the equality sign; the element of 

 ^i(.r) in the ith row and ./th column (i 9^ j) will therefore vanish to 

 infinite order together with its derivatives as x goes to infinity along 

 the upper half of the axis of imaginaries. The diagonal elements 

 diminished by 1 have the same properties. 



Hence we have Ai(x) c^j / along the upper half of the axis of imagi- 

 naries, while all the derivative matrices of ^i(.r) tend to matrices of 

 zero elements as x becomes infinite. If Ai{x) has this character along 

 the lower half of the axis also, it is clear that this matrix satisfies all 

 the restrictions imposed in the theorem. 



Let us demonstrate that such is actually the case. The determina- 

 tion of T{x) on the left-hand side of the lower half of the axis of imagi- 



