Higurashi no Naku Koro ni/Analysis
Subversion Of The Messianic Archetype
Hanyuu is a subversion of the typical messianic archetype. Like a Christ-figure, she sacrificed her life to atone for the sins of humanity. However, this did not cleanse other people of their sins. Rather, they decided that they could atone for sin by sacrificing innocent scapegoats in their place. Hanyuu has thus by Massacre Chapter come to regret having sacrificed her life, and believes that one person cannot atone for the sins of another. People have to take responsibility for their own sins. This is exemplified by the scene in the final episode of Kai, where Hanyuu is set up for a Heroic Sacrifice but the moment is subverted when Takano's bullet misses. Hanyuu can't redeem Takano's sins by dying; Takano must atone by living.
Sin And Atonement
Sin and Atonement are major themes of the Higurashi series. Particularly, it pushes the idea that people must atone for their own sins. Rika could not break the cycle of tragedy because she was trying to take responsibility for people's sins from them. The cycle of tragedy is only broken in Atonement Chapter when Keiichi and Rena realize their own sins, and vow to atone for those sins. Likewise, in the backstory, Hanyuu died to atone for the sins of the inhabitants of Hinamizawa; but this resulted in a Religion Of Evil in which the innocent were sacrificed for the sins of the guilty. By Massacre Chapter, Hanyuu has come to acknowledge that one cannot die to redeem the sins of another; people must take responsibility for their own sins. It is the mark of villains of the series that they try to push their sins off on others, as Takano references using the metaphor of the game of "Old Maid": Nomura using Takano as a scapegoat for the Great Hinamizawa Disaster, and Takano trying to use Oyashiro as a scapegoat for the same. But Takano's final bullet misses, because in the Perfect World she cannot pass her sin off on others; she must bear the responsibility for her own actions.
Herostratic Fame
Herostratic Fame, or fame through infamy, is named after the Greek youth Herostratus, who burned down the Temple of Artemis and Ephesus, one of the Seven Wonders of the Ancient World. He proudly confessed to the crime, and said that he had committed it simply so that his name would be recorded in the history books and remembered forever. Takano's motivation is to become a God through Herostratic Fame. Takano noticed that people are fascinated by tragedy above all else. When they think of ships, they think of the Titanic; when they think of zeppelins, they think of the Hindenberg; when they think of Germany, they think of Hitler; and so on. Due to her horrible childhood, Takano came to believe that fear and tragedy are what truly control mankind. When her grandfather told her that he would "become a God" if his name was remembered forever, Takano made the mental leap that the easiest way to be remembered forever was to be responsible for an atrocity so horrific that it would never be forgotten—hence, her plan to instigate the Great Hinamizawa Disaster and kill the two thousand residents of Onigafuchi. (The first episode of Kai suggests that Takano has not in fact become famous, but this is probably due to Nomura instigating a cover-up rather than Takano's plan. Takano's actions, such as giving away her notebook before faking her death, suggest that she fully planned on becoming infamous).
On Sympathetic Murderers and the importance of Motive
Ryukishi07 asked this in the Staff Room portion of Meakashi-hen: "Do you believe that murder can be justified through the motive?" Each arc has a murder with a different culprit and motive, and the motives of each vary in justifiability. For example, Keiichi in Tatarigoroshi-hen kills Houjou Teppei. His motives are understandable: to protect Satoko from her uncle's abuse. This outlines one part of Ryukishi's answer to that question: degrading the victim's life in order to justify the murder.
In both Meakashi-hen and Tsumihoroboshi-hen, we're given a full view of the killer's thought processes, and it becomes up to the reader in order to discern whether the murderer has enough motive to justify their actions. The fact is that both perpetrators are highly unstable, but manage to earn a degree of sympathy from the audience since their thoughts are in full view of the reader. This is the second part of Ryukishi's answer: that each killer will earn a degree of sympathy which will vary from person to person.
In the end, the answer that Ryukishi gives is that sympathy varies from person to person, and that the motive is only a tool to help evoke that sympathy. However, the author also believes that murder is murder, no matter what the justification is.