Read Ezra 3-4 at Bible Gateway.
Then the people of the land tried to discourage the people of Judah. They troubled them in building, and hired counselors against them to frustrate their purpose, all the days of Cyrus king of Persia, even until the reign of Darius king of Persia. In the reign of Ahasuerus, in the beginning of his reign, they wrote an accusation against the inhabitants of Judah and Jerusalem. In the days of Artaxerxes also, Bishlam, Mithredath, Tabel, and the rest of their companions wrote to Artaxerxes king of Persia; and the letter was written in Aramaic script, and translated into the Aramaic language. Ezr 4:4-7
The Scripture mentions four kings of Persia: Cyrus, Darius, Ahasuerus, and Artaxerxes. Daniel mentions Cyrus and Darius the Mede. Esther mentions Ahasuerus, the Ahasuerus who reigned over one hundred and twenty- seven provinces from India to Ethiopia (Est 1:1).
Cyrus was the king of Persia and Media, who defeated the Babylonians on the night of Belshazzar’s feast (Dan 5), and inherited the Babylonian empire, becoming the first king of the Persian empire. His uncle was Darius the Mede (Dan 5:30), whom Cyrus made king of Babylonia under him. This Darius the Mede is not the same as the Darius king of Persia mentioned in Ezra.
The grammar of Ezr 4:4-7 reads as if Judah’s enemies troubled them, beginning in Cyrus’ reign, continuing all the way through, until the reign of Darius. Additional kings who reigned were named Ahasuerus, and Artaxerxes.
Who are these kings? Greek history, the history by which we have any knowledge of the Persian kings, tells us that Cyrus’ sons Cambyses and Smerdis ruled after his death; then a usurper known as False Smerdis, then Darius the Great:
Long before the Temple was finished, Cyrus died (529 BC), and the Persian Empire was divided between his two sons, Cam-by´ses, and Smer´dis. Now the Egyptians, whom Cyrus had made to submit to his authority, rebelled upon hearing of his death; thus Cambyses took an army thither to reassert Persian authority. But he seems to have gone insane while he was there, and perpetrated atrocities on Egyptian and Persian alike; and even sent to Persia to have his brother Smerdis killed, so that he might be sole ruler.
This order was carried out, but in secret; and a usurper took the throne. His name was also Smerdis, and his appearance was so like that of Cyrus’ son, that the court nobles and the people did not know that their rightful king had been replaced with a counterfeit (522 BC).
It may be that this false Smerdis was on the throne of Persia, when the Samaritans sent a letter to the king, accusing the Jews who had returned to Jerusalem of crimes against the state. Thus the king commanded that the work on the Temple be halted, and so it was.
In the Bible, we are told that the name of this king was Artaxerxes (ar-tuh-zurk´sees, Ezr 4:7), but you must not imagine that this was the personal name of the king. “Artaxerxes” was a title borne by many Persian kings, which simply means “high king,” just as the title “Pharaoh” was borne by Egyptian kings, or “Abimelech” was borne by Philistine kings.
In due course, however, the usurper was discovered, and killed; the mad Cambyses also meeting the same fate. Therefore the seven greatest nobles of Persia, who had formed the king’s council from of old, chose a new king from among their number, who was related to the royal house. His name was Darius Hys-tas´pes, who is known in history as Darius the Great (521 BC).
— The Story of the Ancient World, pgs 247-248
It is probable, that since not even in Persia was the true identity known of the man who was sitting on the Persian throne at this juncture, that the king here in Ezra was referred to by his title “Artaxerxes” rather than by his personal name.
As far as the identity of Ahasuerus, this was also a title borne by Persian kings, rather than a personal name, and it meant “lion king.” Many Bible scholars before the modern era of Scriptural skepticism, believed that the Ahasuerus of Esther, was none other than the Darius king of Persia of Ezra, or Darius the Great. Esther’s king is identified as “the Ahasuerus who reigned over one hundred and twenty- seven provinces …” lending credence to the theory that there had been more than one Ahasuerus, Esther’s Ahasuerus needing to be distinguished from others by the detail of the one hundred and twenty- seven provinces — something history ascribes to Darius the Great.
It is probable that the Ahasuerus of Ezra and the Ahasuerus of Esther are two different kings, Ezra’s Ahasuerus being one of the sons of Cyrus or the usurper, and Esther’s Ahasuerus being Darius the Great.
For further reading, especially on the identity of the Persian kings:
Esther, Historical and Chronological Comments I (Biblical Chronology Newsletter)
Esther, Historical and Chronological Comments II (Biblical Chronology Newsletter)
Esther, Historical and Chronological Comments III (Biblical Chronology Newsletter)