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On a θ-Weyl sum
- Source :
- Nagoya Mathematical Journal. 52:163-172
- Publication Year :
- 1973
- Publisher :
- Cambridge University Press (CUP), 1973.
-
Abstract
- We treat the sum , where α and γ are real with α positive. This sum was treated first by Hardy and Littlewood [4], and after them, by Behnke [1] and [2], Mordell [9], Wilton [11] and others. The reader will find its history in [7] and in the comments of the Collected Papers [4]. 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.
Details
- ISSN :
- 21526842 and 00277630
- Volume :
- 52
- Database :
- OpenAIRE
- Journal :
- Nagoya Mathematical Journal
- Accession number :
- edsair.doi...........528547c8ab8a276022389f1505a7446e