The origin, reliability, and dispersion of the Ep,i – Eiso and other spectral energy correlations is a highly debated topic in GRB astrophysics.\ud GRB080916C, with its enormous radiated energy (Eiso ∼ 10^55 erg in the 1 keV−10 GeV cosmological rest-frame energy band) and its intense GeV emission measured by Fermi, provides a unique opportunity to investigate this issue. In our analysis, we also study another extremely energetic event, GRB090323, more recently detected and localized by Fermi/LAT, whose radiated energy is comparable to that of GRB080916C in the 1 keV−10 MeV energy range. Based on Konus/WIND and Fermi spectral measurements,\ud we find that both events are fully consistent with the Ep,i – Eiso correlation (updated to include 95 GRBs with the data available as of April 2009), thus further confirming and extending it, and providing evidence against a possible flattening or increased dispersion at very high energies. This also suggests that the physics behind the emission of peculiarly bright and hard GRBs is the same as for medium-bright and soft-weak long events (XRFs), which all follow the correlation.\ud In addition, we find that the normalization of the correlation obtained by considering these two GRBs and the other long ones for which Ep,i was measured to high accuracy by the Fermi/GBM are fully consistent with those obtained by other instruments (e.g., BeppoSAX, Swift, Konus/WIND), thus indicating that the correlation is not affected significantly by “data truncation” because of detector thresholds and limited energy bands.\ud A Fermi/GBM accurate estimate of the peak energy of a very bright and hard short GRB with a measured redshift, GRB090510, provides robust evidence that short GRBs do not follow the Ep,i – Eiso correlation and that the Ep,i – Eiso plane can be used to discriminate between, and understand, the two classes of events. Prompted by the extension of the spectrum of GRB080916C to several GeV (in the cosmological rest-frame) without any excess or cut-off, we also investigated whether the evaluation of Eiso in the commonly adopted 1 keV−10 MeV energy band may bias the Ep,i – Eiso correlation and/or contribute to its scatter. By computing Eiso from 1 keV to 10 GeV, the slope of the correlation becomes slightly flatter, while its dispersion does not change significantly. Finally, we find that GRB080916C is also consistent with most of the other spectral energy correlations derived from it, with the possible exception of the\ud Ep,i – Eiso – tb correlation
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