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We compute the Adler function on the lattice from vacuum polarization data with twisted boundary conditions using numerical derivatives. The study is based on CLS ensembles with two flavours of $O(a)$ improved Wilson fermions. We extrapolate the lattice data for the Adler function to the continuum limit and to the physical pion mass and analyze its dependence on the momentum transfer. We discuss the application of this method to the extraction of the $u,d$ contribution to $a_\mu^{\mathrm{HLO}}$.

A Study of the anomalous magnetic moment of the muon computed from the Adler function