We provide examples of multiplicative functions f supported on the kfree integers such that at primes f (p) = ±1 and such that the partial sums of f up to x are o(x 1/k ). Further, if we assume the Generalized Riemann Hypothesis, then we can improve the exponent 1/k: There are examples such that the partial sums up to x are o(x 1/(k+ 1 2 )+ ), for all > 0.

Let T > 0 and consider the random Dirichlet polynomial S T (t) = Re n≤T X n n −1/2−it , where (X n ) n are i.i.d. Gaussian random variables with mean 0 and variance 1. We prove that the expected number of roots of S T (t) in the dyadic interval [T, 2T ], say EN (T ), is approximately 2/ √ 3 times the number of zeros of the Riemann ζ function in the critical strip up to height T . Moreover, we also compute the expected number of zeros in the same dyadic interval of the k-th derivative of S T (t). Our proof requires the best upper bounds for the Riemann ζ function known up to date, and also estimates for the L 2 averages of certain Dirichlet polynomials.