346 SCIENCE PROGRESS 



been given by Landau {Jahresber. d. Math. Verein., 30, 1921, 



179-85). 



An attempt to extend the method so as to apply to algebraic 



numbers has been made by C. L. Siegel (Jahresber. d. Math. 



Verein., 31, 1922, 22-6 ; Math. Ann., 87, 1922, 1-35), the 



first paper being a general account ; the only general theorem 



so far known is one due to Hilbert (cf. Siegel, Math. Zs., 11, 



1 92 1, 246-75) on the decomposition of total positive numbers 



into the sum of four squares. 



Another extension of Waring's Problem is to the expression 



of any integer as the sum of a finite number of polynomials 



plus a finite number of units, the total number being less than 



N, where N is independent of the integer. Proofs have been 



given by Landau {Math. Zs., 12, 1922, 219-47) ^^d by E. 



Kamke {Math. Ann., 83, 1921, 85-112 ; Math. Zs., 12, 1922, 



323-8). 



(2) The Dirichlet divisor problem. 



This is concerned with the order of magnitude of T){x) = 

 Sd{n), where d{n) is the number of divisors of n. We have 



D{x) = X log X + (2C — \)x -\- A{x), 



C being Euler's constant, and it is a question of the order of A{x). 

 Dirichlet showed that A{x) = 0{x^), and Voronoi improved 

 this in 1903 to A{x) = 0{x^ log x) , while Hardy and Landau 

 independently (191 5) proved that A{x) 4= o{x^). In a Doctor's 

 dissertation at Gottingen (1922) Rogosinski has given another 

 proof of Voronoi's result, using the methods of Landau, involving 

 the " lattice-points " of a circle, and Landau {Gott. Nach., 1922, 

 8-16) discusses part of his work. J. G. van der Corput {Math. 

 Ann., 87, 1922, 39-65) shows that A{x) = 0{o(^), where 

 M<33/ioo. 



A generalisation of the problem was proposed by Piltz, 

 who discussed the sums 'D},{x)=^Xdu{n), where dk{n) is the 



number of decompositions of n into k factors {k = 2 gives the 

 Dirichlet problem). 



In this case we get a similar remainder term Ak{x). A 

 summary of the known results as to the order of Ai,{x) is given 

 by Hardy and Littlewood {Proc. L.M.S., 21, 1922, 39-74), 

 who by a study of the " approximate functional equation " in 

 the theory of the zeta-function obtain a new result. If a* 

 denote the least number such that Ai{x) = 0(jt:"i+^) for every 

 positive e, they prove that ak^{k- 2)/k, (^>3), it having 

 been known before that ak^{k- i)/{k-{- i). 



H. Cramer {Arkiv for Mat., 16, 1922, No. 21) obtains a 

 series for AJ^x) analogous to that obtained by Hardy and 

 Littlewood in the Dirichlet problem. 



