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We present detailed calculations of the prompt spectrum of gamma-ray bursts(GRBs) predicted within the fireball model framework, where emission is due tointernal shocks in an expanding relativistic wind. Our time dependent numericalmodel describes cyclo-synchrotron emission and absorption, inverse and directCompton scattering, and e^\pm pair production and annihilation (including theevolution of high energy electro-magnetic cascades). It allows, in particular,a self-consistent calculation of the energy distribution of e^\pm pairsproduced by photon annihilation, and hence a calculation of the spectraresulting when the scattering optical depth due to pairs, \tau_\pm, is high. Weshow that emission peaks at ~1MeV for moderate to large \tau_\pm, reaching\tau_\pm ~ 10^2. In this regime of large compactness we find that (i) A largefraction of shock energy can escape as radiation even for large \tau_\pm; (ii)The spectrum depends only weakly on the magnetic field energy fraction; (iii)The spectrum is hard, epsilon^2 dN/depsilon \propto epsilon^\alpha with0.5<\alpha<1, between the self absorption (epsilon_{ssa}= 10^{0.5\pm0.5} keV)and peak (epsilon_{peak}= 10^{0.5\pm0.5} MeV) photon energy, (iv) and shows asharp cutoff at ~10 MeV; (v) Thermal Comptonization leads to emission peakingat epsilon_peak>=30 MeV, and can not therefore account for observed GRBspectra. For small compactness, spectra extend to >10 GeV with flux detectableby GLAST, and the spectrum at low energy depends on the magnetic field energyfraction. Comparison of the flux at ~1 GeV and ~ 100 keV may therefore allow todetermine the magnetic field strength. For both small and large compactness,the spectra depend only weakly on the spectral index of the energy distributionof accelerated electrons.Comment: Accepted for publication in ApJ. Improved analytic analysis of  numerical result

Prompt Gamma‐Ray Burst Spectra: Detailed Calculations and the Effect of Pair Production