ON DIRICHLET'S DIVISOR PROBLEM

Let d(n) denote tho number of divisors of n and . It is well-known that as x≡θ, Dirichlet proved thatΔ(x)≡D(x) -xlogx- (2r-l)x= O(x1/2)where r is Euler’s constant. Actually Δ (x) =O(xz), for some a loss than1/2. Let θ be tho greatest lowor bound of α, tho best result hither-to obtained is that θ≤15/46 which belongs to T. T. Chin and was obtained and published in 1953 by Richert independently. Tho main object of this paper is to prove thatθ≤13/40.