On an asymptotic formula of Srinivasa Ramanujan
On an asymptotic formula of Srinivasa Ramanujan
复制标题
DOI：
10.4064/aa109-4-5
复制
发表时间：
2003
期刊：
影响因子：
0.7
通讯作者：
K. Ramachandra;A. Sankaranarayanan
中科院分区：
文献类型：
--
作者：
K. Ramachandra;A. Sankaranarayanan
他还记录了(无需证明)的结果：在黎曼假设的假设下，(1.1)中的误差项可以改进为O(N1/2+ε)。鉴于H.L.Montgomery和R.C.Vaughan(见[9])的方法，误差项很可能是O(N1/2)。我们提出这是一个猜想(另见[15]，[17])。无条件地，与D2(J)相关的误差项已知为O(N1/2+ε)，对于任何正常数ε(例如，参见[6]和[5]的公式(14.30))。A.Schinzel教授已经考虑了Ramanujan的一些问题(见[19])，即对于算术函数R2(N)，由于W.G.Nowak的一篇未发表的工作(另见[8]和[18])，他证明了相应的误差项是Ω(n3/8)，并且相应的误差项是O(n1/2(Logn)8/3(Loglogn)1/3)。让我们
Also he records (without proof) the result that on the assumption of the Riemann hypothesis, the error term in (1.1) can be improved to O(n1/2+ε). In view of a method due to H. L. Montgomery and R. C. Vaughan (see [9]), it is very likely that the error term is O(n1/2). We propose this as a conjecture (see also [15], [17]). Unconditionally, the error term related to d2(j) is known to be O(n1/2+ε) for any positive constant ε (see for example the equation (14.30) of [6] and also [5]). Professor A. Schinzel has already considered some of the problems of Ramanujan (see [19]), namely for the arithmetic function r2(n), and he has proved that the corresponding error term is Ω(n3/8) and also the corresponding error term is O(n1/2(logn)8/3(log logn)1/3) due to an unpublished work of W. G. Nowak (see also [8] and [18]). Let