Below is what we wrote on the topic of Daryavash in our Insights to Rosh Hashanah:

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1) DARIUS, SON OF ESTHER

QUESTION: The Gemara says that Koresh was an upright king, and therefore the years of his reign were counted like the years of a Jewish king, from the month of Nisan. From the verses that the Gemara cites it is clear that the Gemara is discussing Darius II, the king who ruled over the Persian empire after Achashverosh. The Gemara implies that had he not been an upright king, his years would have been counted from Tishrei since he is not Jewish.

In addition, the Gemara later (4a) implies that he was not Jewish. The Gemara concludes that he became corrupt, and therefore the years of his reign after that point were counted from Tishrei. The Gemara proves that he became corrupt from the verses which describe how he made considerable donations to the Beis ha'Mikdash, but he told the Kohanim to pray for him in return for the donations. The Gemara says that giving Tzedakah with such a condition is a mark of wickedness when a Nochri does it. Only for a Jew is it an admirable trait.

TOSFOS (3b, DH Shnas) calls this king "Darius the son of Esther," based on the Midrash (Vayikra Rabah 13:5) that says that he was born from the union of Esther and Achashverosh. If his mother, Esther, was a Jew, then he should also be considered a Jew! Why, then, does the Gemara say that he was not Jewish?

ANSWERS:

(a) Even if Koresh had a legal status as a Jew, the years of his reign can only be counted from Nisan if his kingdom is a Jewish kingdom (or an upright kingdom that was kind to the Jews). Since his kingdom was not Jewish, even though he was himself Jewish the years of his reign were counted from Tishrei.

As far as giving Tzedakah for personal benefit, TOSFOS (4a, DH Bishvil) explains that the reason why giving Tzedakah with intention to personally benefit is considered a wrongdoing when a Nochri does so is because the Nochri gives only in order to receive something in return. In contrast, a Jew gives Tzedakah in order to give, and he only appends a prayer for himself as an aside. If not given his wish, he will not regret the Tzedakah that he gave. Since Darius was raised in an environment that did not espouse Jewish values, he had the traits of any other Nochri in this regard.

(b) RASHI (4a, DH Kalbesa) explicitly calls Darius a "ben Noach." It is possible that Rashi learns that our Sugya follows another opinion which argues with the Midrash in Vayikra Raba and holds that Darius was not the son of Esther (see Esther Raba 8:3).

(c) There is a Machlokes in the Gemara in a number of places whether a child born from a union between a Nochri and a Jewish woman is considered a Mamzer or is Kasher (of pure lineage). Some Rishonim explain that the opinion which says that the child is Kasher hold that the child is a proper Nochri and not a Jew at all, and that when he converts he will not be considered a Mamzer (RASHI, Kidushin 68b, DH Leima; TOSFOS, Yevamos 23a, DH Kasavar -- this is not the Halachic opinion). According to those Rishonim, our Gemara might be following those Tana'im who say that the child is Kasher, which means that he is a Kasher Nochri . (See also BEIS YITZCHAK, Even ha'Ezer 1:29:8.)