Sommes de Gauss modulo \(p^\alpha\). I, II (Q1052380)

Let \(p\) be a prime number, \(\alpha\) an integer \(> 1\), \(\chi\) a primitive character modulo \(p^\alpha\). A Gaussian sum \(\tau(\chi)\bmod^\alpha\) is defined by \[ \tau(\chi)= \sum_{x\in (\mathbb Z/ p^\alpha \mathbb Z)^\times} \chi(x)\exp(2\pi ix/p^\alpha). \] In Part I of the two consecutive papers, the author demonstrates the evaluation of \(\tau(\chi)\) according to the parity of the exponent \(\alpha\) of the odd prime power \(p^\alpha\). But, in the final remark, the author himself adds that the same result has already been established by \textit{R. Odoni} [Bull. Lond. Math. Soc. 5, 325--327 (1973; Zbl 0269.10020)] with a completely different method of proof. In Part II, the author treats the case when \(p=2\) with appending a generalization of \(\tau(\chi)\) to the higher dimensional case.