348 SCIENCE PROGRESS 



S(97) = 0(7/"+'), where a={k — i)lk, for every positive e, and 

 that this index a is the best possible one. They also obtain 

 an explicit series formula for N(?;) analogous to that given by 

 Voronoi for the lattice-points of a rectangular hyperbola. 

 Corresponding results in Problem B are also obtained ; some 

 of them are given in Ostrowski's paper. Allied problems are 

 also considered, namely the series 



/i(s) = S-:,/2(5) = 2^^^n . . . , where a«^{m^}, 

 n yr 



and the fzth term of the /)th series is essentially the ^th 

 Bernoullian function of a„ + \. Hecke {I.e.) had shown 

 that fi(s) is meromorphic over the whole of the complex 

 s-plane, having at most a double infinity of simple poles 

 at the points —2k ± 2m^'Tri, {k, m = o, i, 2 . . .), 7 being 

 a constant depending on ^ ; by their more elementary methods 

 Hardy and Littlewood show that fp{s) is regular if the real 

 part of s is greater than i — p/{i +X). These series are also 

 the subject of a subsequent paper by H. Behnke {Abhl. math. 

 Sem. Hamburg, 1, 1922, 252-67). 



R. D. Carmichael {Bull. Am. Math. Soc, 28, 1922, 109-10) 

 points out that the proof which he gave in 1907 of the theorem 

 that for a given number n the equation (f){x) = n has either no 

 solution or has at least two solutions, where cj}{x) is the number 

 of numbers less than x and prime to it, is not correct. The 

 theorem, however, is probably true ; the author shows at any 

 rate that a number x which is a unique solution must have 

 at least 38 digits. 



If ^ is a prime and q a factor of p — i, we can obtain an 

 equation of degree q with rational coefficients each of whose 

 roots is the sum of {p - i)/q of the primitive pth roots of 

 unity, no such ^th root being included in more than one of 

 the sums. The equation is an Abelian equation with a cyclical 

 group. The determination of a system of relations expressing 

 any rational function of the roots of this equation as a linear 

 function of the roots was called by Cayley the problem of 

 cyclotomic ^-section. He himself worked out the complete 

 theory for q=3, 4, and Burnside in 191 5 treated the case 

 q= S' O- Upadhaya {Tohoku Math. Journal, 19, 1921, 183-6 ; 

 21, 1922, 46-50) makes a beginning for ^ = 6 by examining 

 the cases in which the prime p is 13, 19, and 31. 



Other recent papers on the theory of numbers are : 



R. FuETER, " Kummers Kriterium zum letzten Theorem von Fermat," 



Math. Ann., 85, 1922, 11-20. 

 H. S. Vandiver, " Note on Some Results concerning Fermat's Last Theorem," 



Bull. Am. Math. Soc, 28, 1922, 255-60. 



