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It is shown that with a precise determination of a few derivatives of thehadronic vacuum polarization (HVP) self-energy function $\Pi(Q^2)$ at $Q^2=0$,from lattice QCD (LQCD) or from a dedicated low-energy experiment, one canobtain an evaluation of the lowest order HVP contribution to the anomalousmagnetic moment of the muon $a_{\mu}^{\rm HVP}$ with an accuracy comparable tothe one reached using the $e^+ e^-$ annihilation cross-section into hadrons.The technique of Mellin-Barnes approximants (MBa) that we propose isillustrated in detail with the example of the two loop vacuum polarizationfunction in QED. We then apply it to the first few moments of the hadronicspectral function obtained from experiment and show that the resulting MBaevaluations of $a_{\mu}^{\rm HVP}$ converge very quickly to the fullexperimental determination.

The Mellin-Barnes Approach to Hadronic Vacuum Polarization and $g_{\mu}-2$