This monograph explores several classical issues of modern mathematics, and discusses both the historical and research aspects. The brief historical part is focused on Riemann's zeta function and few related problems. The research part starts from direct formulation of simple proofs of both the prime number theorem and Riemann's hypothesis, two intriguing problems of modern mathematics, which applies the concept of Mertens's function and is based on Apostol's and Littlewood's criterions of equivalence.