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

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 \(6 \pmod{7}\), and for every prime divisor \(p (\neq 7)\) of \(v\) such that the order \(w\) of \(2\) mod \(p\) satisfies \(2|\frac {\phi(p)}{\omega}\); 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 \(p (\neq 13)\) of \(v\) such that the order \(w\) of \(3\) mod \(p\) satisfies \(2|\frac {\phi(p)}{\omega}\). These results distinctly improve McFarland’s corresponding results and Turyn’s result.