A Method of Studying the Multiplier Conjecture and Some Partial Solutions for It

Abstract

This paper sketches the method of studying the Multiplier Conjecture that we presented in [1], and adds one lemma. Applying this method, we obtain some partial solutions for it: in the case $v=2n_1$, the Second Multiplier Theorem holds without the assumption ”$n_1>\lambda$”, except for one case that is yet undecided where $n_1$ is odd and $7\mid \mid v$ and $t\equiv 3,5,$ or $6(\mathrm{mod}7)$, and for every prime divisor $p(\ne 7)$ of $v$ such that the order $w$ of $2$ mod $p$ satisfies $2|\frac{\varphi (p)}{\omega }$; in the case $n=3n_1$ and $(v,3.11)=1$, then the Second Multiplier Theorem holds without the assumption “$n_1>\lambda$” except for one case that is yet undecided where $n_1$ cannot divide by $3$ and $13\mid \mid v$ and the order of $t$ mod $13$ is $12,4,$ or $6,2$, and for every prime divisor $p(\ne 13)$ of $v$ such that the order $w$ of $3$ mod $p$ satisfies $2|\frac{\varphi (p)}{\omega }$. These results distinctly improve McFarland’s corresponding results and Turyn’s result.