We treat the sum , where α and γ are real with α positive. This sum was treated first by Hardy and Littlewood , and after them, by Behnke  and , Mordell , Wilton  and others. The reader will find its history in  and in the comments of the Collected Papers . Here we show that the sum can be expressed explicitly, together with an error term O(N1/2), using the regular continued fraction expansion of α. As the statements have complications we will divide them into two theorems. In the followings all letters except ϑ, i, σ, ζ, χ and those in 3° are real, N is a positive real, and always k, n, a, A, B, C, D and E denote integers. The author expresses his thanks to Professor Tikao Tatuzawa and Professor Tomio Kubota for their encouragements.