The paper explores the nucleosynthetic signature of the first generation of stars, known as Population III, which were likely massive ($\sim 100 - 300 \, \text{M}_\odot$). The authors investigate the nucleosynthesis of helium cores in the mass range $M_{\text{He}} = 64$ to $133 \, \text{M}_\odot$, corresponding to main-sequence star masses of approximately $140$ to $260 \, \text{M}_\odot$. For helium core masses above $133 \, \text{M}_\odot$, without rotation and using current reaction rates, a black hole is formed and no nucleosynthesis is ejected. For lighter helium core masses, $\sim 40$ to $63 \, \text{M}_\odot$, violent pulsations occur, induced by the pair instability, accompanied by supernova-like mass ejection, but the star eventually produces a large iron core in hydrostatic equilibrium. This core, too, is likely to collapse into a black hole. The authors find that pair-instability supernovae produce a roughly solar distribution of even-charged nuclei but are deficient in odd-charged elements. The Fe/Si ratio is sensitive to whether the initial mass function (IMF) is truncated at $\sim 200 \, \text{M}_\odot$ or between $140$ and $260 \, \text{M}_\od-m$. The distinctive nucleosynthetic signature of Population III stars is discussed, and possible observational tests are suggested.The paper explores the nucleosynthetic signature of the first generation of stars, known as Population III, which were likely massive ($\sim 100 - 300 \, \text{M}_\odot$). The authors investigate the nucleosynthesis of helium cores in the mass range $M_{\text{He}} = 64$ to $133 \, \text{M}_\odot$, corresponding to main-sequence star masses of approximately $140$ to $260 \, \text{M}_\odot$. For helium core masses above $133 \, \text{M}_\odot$, without rotation and using current reaction rates, a black hole is formed and no nucleosynthesis is ejected. For lighter helium core masses, $\sim 40$ to $63 \, \text{M}_\odot$, violent pulsations occur, induced by the pair instability, accompanied by supernova-like mass ejection, but the star eventually produces a large iron core in hydrostatic equilibrium. This core, too, is likely to collapse into a black hole. The authors find that pair-instability supernovae produce a roughly solar distribution of even-charged nuclei but are deficient in odd-charged elements. The Fe/Si ratio is sensitive to whether the initial mass function (IMF) is truncated at $\sim 200 \, \text{M}_\odot$ or between $140$ and $260 \, \text{M}_\od-m$. The distinctive nucleosynthetic signature of Population III stars is discussed, and possible observational tests are suggested.