Zeta Partial Sums

The Riemann zeta function $\zeta(s)$ is closely tied to the distribution of primes. Writing $\operatorname{Re}(s)$ for the real part of a complex number $s$, define

When $\operatorname{Re}(s) > 1$, one has $\zeta(s) = \lim_{k \to +\infty} \psi_k(s)$, so the partial sums $\psi_k(s)$ matter for studying $\zeta(s)$.

1. For every positive integer $n$ there exist a unique odd integer $a_n$ and a unique natural number $b_n$ with $n = a_n \cdot 2^{b_n}$. Find $\displaystyle\sum_{n=1}^{10} (a_n + b_n)$. 2. Does there exist a positive integer $k$ such that $\psi_k(1) = 2024$? Prove your conclusion. 3. Prove that $\psi_k\!\left(\dfrac{3}{2}\right) < 3$.