Science Record

SCIENCE RECORD New Ser. Vol. III, No. 1, 1959

MATHEMATICS ON DIRICHLET’S DIVISOR PROBLEM*

Yin WEN-LIN ($t3c3® ) (Peking University)

Let d(#) denote the number of divisors of » and D(x) =2id (n). It is

well known that as x > ©, Dirichlet proved that A(s) =DG@) — «loge — Qy — Ie = O(x?),

where y is the Euler constant. Geometrically D(«) is the number of lattice points under the hyperbola UV =. in the first quadrant of the UV-plane, containing those on the curve but not those on the coordinate axes.

Let 6 be the lower bound of the values of @ satisfying A(x) = O(2*). With regard to 9, on the one hand, Hardy™ proved in 1915 that 9> = and on the other hand, adopting Professor $, H. Min’s”! method, T. T. Chin‘! proved in 1950 that 0< =. The same result was published in 1953 by Richert!. In conclusion, the improvements of the upper bound of @ are

pee Se 3 100 82 46 These results belong respectively to Voronoi®!, van der Corput'*!, van der Corput™ and T. T. Chin.

It is well known that the divisor problem is harder than, but similar to, the circle problem. And the result on the last problem was improved to = in 1942 by Professor L. K. Hua'*!, Now, under the supervision of Professor

S. H. Min, the author has proved that

je 40

just like the circle problem.

*Received Nov, 15, 1958,