Analytic Number theory in TCS
Codes/Lattices are certain combinatorial objects that are commonly used within TCS. A basic question for both of them is finding "short" codewords/lattice points, known as the Minimum Distance problem/Shortest Vector problem (MDP/SVP).

Both have been known to be NP-hard under randomized reductions for 20+ years. Roughly 10 years ago, the NP hardness proof for MDP was derandomized using some Analytic Number Theory (Weil's Character Sum bound).
SVP has still not been derandomized, but can be derandomized if the following result in analytic number theory is proven:


  For any $\epsilon &gt; 0$, there exists $c\in\mathbb{N}$ such that for all sufficiently large $n$, the interval $[n, n + n^\epsilon]$ contains a square-free integer with no prime factors larger than $\log^c(n)$


Details of both of the above are described in Micciancio's paper Locally Dense Codes.

It's worth mentioning that this "existence of smooth numbers in short intervals" question is known within analytic number theory. For example, section 1.2 of Smooth numbers: computational number theory and beyond mentions that a slight relaxation of the problem (no prime factors larger than $n^\epsilon$) implies Vinogradov's conjecture, a problem in analytic number theory with no progress in 40 years.
There are definitely recent publications on this problem (say this), but I am not an expert, and can't say what the limits of current techniques are.