Weighted partial sums of a random multiplicative function and their positivity

In this paper, we study the probability that some weighted partial sums of a random multiplicative function f are positive. Applying the characteristic decomposition, we obtain that if S is a non-empty subset of the multiplicative residue class group (Z/mZ) ^ with m being a fixed positive integer and A=\a+mn n=0, 1, 2, 3, \ with a S, then there exists a positive number δ independent of x, such that \ P (₀[₁, ₗ) f (n) nδ\ unless the coefficients of the real characters in the expansion of the characteristic function of S according to the characters of (Z/mZ) ^ are all non-negative, and the coefficients of the complex characters are all zero, in which case we have \ P (₀[₁, ₗ) f (n) n<0) =O ( (- (xC₂x) ) ) \ for a positive constant C. This includes as a special case a result of Angelo and Xu. We also extend the result to the cyclotomic field K₍=Q (ζ₍) with ζ₍=e^2πi/n and study the probability that these generalized weighted sums are positive. In addition, we deal with the positivity problem of certain partial sums related to the celebrated Ramanujan tau function τ (n) and the Ramanujan modular form Δ (q), and obtain an upper bound for the probability that these partial sums are negative in a more general situation.