论文标题
在Ramanujan型算术功能的扩展
On a Ramanujan type expansion of arithmetical functions
论文作者
论文摘要
Srinivasa Ramanujan根据$ c_r（n）= \ sum \ limits _ {\ ordack {{m = 1} \\（m，r）= 1}}}^e^{r} e^$ frac {ravimn} $ nrac {ramanujan提供了某些算术函数的系列扩展。剑桥菲洛斯。 Soc，22（13）：259-276，1918]。在这里，我们根据E. Cohen定义的Cohen-Ramanujan总和给出了类似类型的扩展类型（h，r^s）_s = 1}}}^{r^s} e^{\ frac {2πin h} {r^s}} $。我们还提供了一些必要和充分的条件，以使这种扩展存在。
Srinivasa Ramanujan provided series expansions of certain arithmetical functions in terms of the exponential sums defined by $c_r(n) = \sum\limits_{\substack{{m=1}\\ (m,r)=1}}^{r} e^{\frac{2 πimn}{r}}$ in [Trans. Cambridge Phillos. Soc, 22(13):259-276,1918]. Here we give similar type of expansions in terms of the Cohen-Ramanujan sum defined by E. Cohen in [Duke Mathematical Journal, 16(85-90):2, 1949] as $c_r^s(n)=\sum\limits_{\substack{h=1\\ (h,r^s)_s=1}}^{r^s}e^{\frac{2πi n h}{r^s}}$. We also provide some necessary and sufficient conditions for such expansions to exist.
