Alexandru Gica:Cumulative sum of the Liouville function related to Gauss's problem of the tenth field, p.47-53

Abstract:

We present in this paper a connection between Gauss's problem of the tenth field and some cumulative sums of the Liouville function. Namely, we compute $L(x)=\displaystyle\sum_{n\leq x}\lambda (n)$ for $x=\frac{p-3}{4},x=\frac{p-1}{2},x=p-1$, where $p$ is a prime number, $p\equiv 19\pmod{24}$ such that $\mathbb{Z} [\frac{1+i\sqrt {p}}{2} ]$ is a principal ring. We use this result for finding another approach for the problem of the tenth field. We present also a short survey about some previous achievements related with this subject.

Key Words: Gauss's theorem of the tenth field, length, Liouville function.

2000 Mathematics Subject Classification: Primary: 11R29,
Secondary: 11N37.

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