Odysseus vs Raskolnikov: Comparing the nature of Heroism in Dostoevsky vs Greek Epic

Man’s greatest desire is immortality and woman’s greatest desire is to find such an immortal man to love her. Most great stories explore this very endeavour. The last two books I read were Crime and Punishment by Fyodor Dostoevsky and the Odyssey of Homer (translated by Palmer in 1902). By virtue of the juxtaposition, I have been forced to compare the two men, Raskolnikov and Odysseus even if I hadn’t intended to, and have realized that the comparison is not so straightforward as one might assume. Both pursue heroism, and both achieve and fall short of it in different ways. What surprised me most were their similarities despite one man being a nearly destitute student and the other being a King. The most dramatic difference between them is the implication in the ending, the “Nostos”, which is in the heroic story, the home-coming. To what extent is the achievement of immortality the end goal of a hero-story? How do these characters achieve immortality and how is it meaningfully different from mere fame?