For the summatory function of 2^omega(n), the paper proves an unconditional decomposition with error O(x^{8/29+epsilon}) and a conditional decomposition involving zeros of zeta(2s).
Bourgain, Decoupling, exponential sums and the Riemann zeta function , J
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On the distribution of the strongly multiplicative function $2^{\omega(n)}$ on the set of natural numbers
For the summatory function of 2^omega(n), the paper proves an unconditional decomposition with error O(x^{8/29+epsilon}) and a conditional decomposition involving zeros of zeta(2s).