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We discuss analytic continuation as a tool to extract the cumulants of thequark number fluctuations in the strongly interacting medium from lattice QCDsimulations at imaginary chemical potentials. The method is applied to $N_f =2+1$ QCD, discretized with stout improved staggered fermions, physical quarkmasses and the tree level Symanzik gauge action, exploring temperatures rangingfrom 135 up to 350 MeV and adopting mostly lattices with $N_t = 8$ sites in thetemporal direction. The method is based on a global fit of various cumulants asa function of the imaginary chemical potentials. We show that it isparticularly convenient to consider cumulants up to order two, and that below$T_c$ the method can be advantageous, with respect to a direct Montecarlosampling at $\mu = 0$, for the determination of generalized susceptibilities oforder four or higher, and especially for mixed susceptibilities, for which thegain is well above one order of magnitude. We provide cumulants up to ordereight, which are then used to discuss the radius of convergence of the Taylorexpansion and the possible location of the second order critical point at real$\mu$: no evidence for such a point is found in the explored range of $T$ andfor chemical potentials within present determinations of the pseudo-criticalline.

Higher order quark number fluctuations via imaginary chemical potentials in $N_f=2+1$ QCD