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I posted this on /co/ because I assumed you guys wouldn't take me seriously or not have a full working knowledge of the DC cast of characters, but here goes:
There's no way this punk could ever outsmart Batman. Should a crossover between the two ever happen, Batman would nearly instantly realize that the murders are supernatural in origin. From there he would contact Zatanna or Dr. Fate about the situation, and either of them would know exactly what was going on and know how to trace the Death Note by some spell or something. That or transport Batman to the Shinigami realm, where he could interogate some of the Shinigami about anything weird. There he would figure out where Ryuk went. Maybe he'd get Deadman to help him search for Ryuk. And if he found Ryuk, he'd find Light.
There's no way this punk could ever outsmart Batman. Should a crossover between the two ever happen, Batman would nearly instantly realize that the murders are supernatural in origin. From there he would contact Zatanna or Dr. Fate about the situation, and either of them would know exactly what was going on and know how to trace the Death Note by some spell or something. That or transport Batman to the Shinigami realm, where he could interogate some of the Shinigami about anything weird. There he would figure out where Ryuk went. Maybe he'd get Deadman to help him search for Ryuk. And if he found Ryuk, he'd find Light.