QUESTION: The Gemara says that since Koresh (Cyrus II) was a virtuous king, the years of his reign were counted from the month of Nisan, like the years of a Jewish king. The verses which the Gemara cites clearly refer to Daryavesh (Darius II), the king who ruled over the Persian Empire after Achashverosh. The Gemara explains that the names Koresh, Daryavesh, and Artachshasta refer to the same person (he was named Koresh because he was a "kosher" king, Artachshasta was the name of his kingdom, and Daryavesh was his name). The Gemara implies that had he not been a righteous king, his reign would have been counted from Tishrei since he was not Jewish.
The Gemara later (4a) also implies that Daryavesh was not Jewish. The Gemara there explains that when he eventually became corrupt, the years of his reign were counted from Tishrei. The proof that he became corrupt is based on the verses which describe how he made considerable donations to the Beis ha'Mikdash on condition that the Kohanim should pray for him. While it is admirable for a Jew to donate money with such a condition, for a Nochri it is a mark of wickedness.
TOSFOS (3b, DH Shnas) calls this king "Darius the son of Esther," based on the Midrash (Vayikra Rabah 13:5) which says he was born from the union of Esther and Achashverosh.
Why does the Gemara imply that the son of Esther, a Jewess, was not Jewish?
(a) Even if Daryavesh had the legal status as a Jew, the years of his reign could be counted from Nisan only if his kingdom was a Jewish one (or an upright kingdom that was kind to the Jews). Since his kingdom was not Jewish, the years of his reign were counted from Tishrei.
Why, though, was his act of giving Tzedakah for the sake of personal benefit considered a sin, if an act with such intent is considered a sin only for a Nochri? TOSFOS (4a, DH Bishvil) explains that a Nochri gives only in order to receive something in return. In contrast, a Jew gives Tzedakah out of the goodness of his heart, and he merely appends a prayer for himself as an aside. If the Jew's prayer is not answered favorably, he does not regret having given Tzedakah. Daryavesh, who was raised in an environment which did not espouse Jewish values, had the traits of any other Nochri in this regard and he gave Tzedakah for impure motives.
(b) RASHI (4a, DH Kalbesa) explicitly refers to Daryavesh as a "Ben Noach." Rashi apparently understands that the Gemara here follows the opinion which argues with the Midrash and maintains that Daryavesh was not the son of Esther (see Esther Rabah 8:3).
(c) The Gemara in a number of places records a dispute with regard to whether a child born from a union between a Nochri and a Jewish woman is considered a Mamzer or is of pure lineage. According to some Rishonim, the opinion that the child is considered to be of pure lineage maintains that the child is a proper Nochri and not a Jew at all. If he converts, he will not be considered a Mamzer (RASHI to Kidushin 68b, DH Leima; TOSFOS to Yevamos 23a, DH Kasavar). (This is not the Halachic opinion.) According to those Rishonim, the Gemara may mean that Daryavesh was a legitimate Nochri. (See also BEIS YITZCHAK, Even ha'Ezer 1:29:8, cited in Insights to Bechoros 47:1.)