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GW190521 is a merger of two black holes (BHs), wherein at least one BH lieswithin the pair-instability (PI) mass gap, and it is difficult to form becauseof the effects of PI supernovae (PISNe) and pulsational PI (PPI). In thisstudy, we examined the formation of GW190521-like BH-BHs under Population (Pop)III environments by binary population synthesis calculations. We reveal thatconvective overshooting in stellar evolution strongly affects the formation ofGW190521-like BH-BHs. A model with a small overshoot parameter (similar toGENEC) can form GW190521-like BH-BHs. The derived merger rate is $4 \times10^{-2}$ ${\rm yr}^{-1}$ ${\rm Gpc}^{-3}$ at a redshift of $\sim 0.82$, whichis comparable to the merger rate of GW190521-like BH-BHs inferred bygravitational wave (GW) observations. In this model, a $\sim 90~M_\odot$ starcollapses to form a $\sim 90~M_\odot$ BH by avoiding PPI and PISN even if it isa member of a binary star. This is because it expands up to $10^2~R_\odot$, andlose only little mass through binary evolution. However, a model with a largeovershoot parameter (similar to Stern) cannot form GW190521-like BH-BHs at all.Thus, we cannot conclude that a Pop~III binary system is the origin of GW190521because determination of the overshoot parameter involves highly uncertain. Ifa Pop~III binary system is the origin of GW190521, the merger rate of BH-BHsincluding a $100-135~M_\odot$ BH is substantially smaller than that ofGW190521-like BH-BHs. This will be assessed by GW observations in the nearfuture.

Population III Binary Black Holes: Effects of Convective Overshooting on Formation of GW190521