In this informal talk, I will present some research problems that are currently open, emanating largely from my earlier research work on Artin's primitive root conjecture and related questions.

Bio: M. Ram Murty obtained his PhD from MIT (1980) and then held post-doctoral fellowships at IAS (Princeton) and TIFR (Mumbai). He was professor at McGill University (1982-1996), and now is Queen's Research Chair and the A. V. Douglas Distinguished University Professor at Queen's University. He was awarded the Coxeter-James Prize (1988), Balaguer Prize (1996), Jeffery-Williams Prize (2003), Queen's Research Prize (2003) and elected Fellow of the Royal Society of Canada (1990), Fields Institute (2003), Indian National Science Academy (2008), American Mathematical Society (2012) and Canadian Mathematical Society (2018). He received the E.W.R. Steacie Fellowship (1991), Killam Research Fellowship (1998) and the Simons Fellowship (2013). In 2018, he was awarded the Excellence in Graduate Supervision Prize and in 2024, the CRMFields-PIMS Prize.

He is also adjunct professor at IIT (Mumbai), TIFR (Mumbai), HRI (Prayag), IISER (Bhopal), IMSC (Chennai) and Chennai Mathematical Institute. He has written more than 275 research papers, a dozen books, and supervised more than 20 doctoral students, 40 masters students and 45 post-doctoral fellows. He is cross-appointed to the Department of Philosophy at Queen's University where he gives regular courses on mathematical logic and Indian philosophy. His introductory book on Indian philosophy has been published by Broadview Press.