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We consider previously derived upper and lower bounds on the numberof operators in a window of scaling dimensions [\Delta - \delta,\Delta + \delta] [ Δ − δ , Δ + δ ] at asymptotically large \Delta Δ in 2d unitary modular invariant CFTs. These bounds depend on a choice offunctions that majorize and minorize the characteristic function of theinterval [\Delta - \delta,\Delta + \delta] [ Δ − δ , Δ + δ ] and have Fourier transforms of finite support. The optimization of thebounds over this choice turns out to be exactly the Beurling-Selbergextremization problem, widely known in analytic number theory. We reviewsolutions of this problem and present the corresponding bounds on thenumber of operators for any \delta \geq 0 δ ≥ 0 .When 2\delta \in \mathbb Z_{\geq 0} 2 δ ∈ ℤ ≥ 0 the bounds are saturated by known partition functions withinteger-spaced spectra. Similar results apply to operators of fixed spinand Virasoro primaries in c>1 c > 1 theories.