In his third lecture, entitled 'Levantine Epigraphy and Samaria, Judaea and Idumaea during the Achaemenid Period', Prof. Lemaire showed the vital importance of epigraphic material for the political and social history of the southern Levant in the Persian period, particularly in the 4th century BCE. My notes are less ordered, as the presentation was less structured within each discussed region.

PART I: SAMARIA
The documents and bullae from Wadi Daliyeh – widely referred to as Samaria documents – were recently re-edited by Jan Dušek, a very fine epigrapher (Les manuscrits araméens du Wadi Daliyeh et la Samarie vers 450-332 av. J.-C. [Culture and History of the Ancient Near East 30; Brill: Leiden; Boston], 2007; for the coins see Meshorer and Qedar). These documents shed light mainly on the last decades of Persian control of the area:
The names in the papyri attest to a mixed society (North-Arabic, Hebrew, Phoenician and Aramaic).
While names do not map religion 1:1 it is likely significant that 57% of names have a yahwistic theophoric element, with other deities attested only between 1-3% each.
The order of governors of Samaria in the Persian period is as follows:

• Sanballat I (from before 445 until about 410/407 BCE)
• Delayah (, son of Sanballat; ca. 410/407 - ca. 370 BCE)
• Shelemyah [Dušek does not agree with this]
• Ḥananyah / 'Ananya (ca. 354 BCE)

This fits neatly with the depiction of Ezra's mission according to Neh 12, dated to 398 BCE

According to the available evidence, Samaria extended as far north in the Galilee as Yoqne'ām [on the basis of one Aramaic ostracon, unlike the Phoenician ostraca found further north. I find this particular argument less convincing. If the ostracon had been in Samarian Hebrew, or at least written in Palaeo-Hebrew letters, like the bullae and coins, I could understand it, but otherwise, the fact that the ostracon was written in Aramaic solely indicates that it was written in the Persian empire.
The Eastern border ws probably the Jordan

The Southwestern border of Samaria unclear. After the revolt of Sidon lands formerly controlled by Sidon may have been given to Samaria but this remains unclear.

Part II: YEHUD
Yehud stamps indicate that the Northern border of Yehud was North of Mizpah and Bethel.
A problem for historians of the period is that to this date, not a single papyrus from this period was unearthed in Yehud, but there are a few ostraca, such as the Ketef Yeriho ostracon, found together with an Alexander coin, therefore dating both to the last third of the 4th century BCE. A few other ostraca were found in Jerusalem, Heshbon, En-Gedi
In general, several low-denomination coins have been found dated to the 4th century onwards

Lemaire speculated that Ezra mission and promulgation of the law may well have been contemporary with the fall of the Elephantine since the relation between Jerusalem and Elephantine seems to have been relatively good at the end of the 5th century BCE, and the Judean community in Elephantine seems to have been well informed who the important people were in Jerusalem.

We can identify the following governors of Yehud:

• Bagavahya (Bagohi), ca. 407 
• Yehizqiyah

it is not impossible that these two were the only governors in the Persian 4th century

As high-priest we have a Yehohanan mentioned in the Elephantine letters, and a Yohanan in a coin which, according to Fried, should be dated to ca .370 BCE. Possibly, this Yohanan should be identified also with the Yohanan in Neh 12. This fits well with the presentation of governors of Samaria

The Hebrew Bible seems to know of two sets of boarders for Yehud, a small and a large one. The small one is attested, e.g. in Neh 3 and the larger Yehud mainly in Neh 11 (there extending south to Beer Sheva) .

Lemaire's second Schweich Lecture was a tour de force of epigraphic evidence for Judeans in the Near East in the Persian period. He started his lecture by referring to the cuneiform evidence that directly links to the Hebrew Bible and the exiles, such as the cuneiform tablet which shows that Nabû-šarussu-ukin (the Bible's Nebo-sarsekim) did exist and was a Babylonian Eunuch, the Weidner lists, the Murašû texts an the so-called āl-Yāhūdū texts. He gave a good summary of the currently available cuneiform evidence.
He then moved on to the Aramaic and – according to him at least in one case (one name is written as 'X ben Y' instead of 'X bar Y') – Hebrew dockets on cuneiform tablets.
According to him, in Babylonia, Jews demonstrably participated in three distinct cultures: Hebrew, Aramaic and Akkadian. 
to me, this statement was the most problematic that Prof. Lemaire made during the entire presentations. None of the scribes writing the cuneiform evidence was Judean (according to their name), and one Hebrew name does not make fully 'Hebrew culture' (whatever that is). What we have evidence for is for the use of Aramaic to express either partly content or one of the people involved in the dockets. The distinctions may appear minor, but they are rather important, I believe.

The second, and by far the longest part of the lecture, focussed on Elephantine. Not, as is so often the case only on the papyri from there, but mostly, in fact, on the ostraca of the Collection Claremont-Ganneau, recently published by Hélène Lozachmeur (2006). He argued that they are far closer to the everyday life than the papyri, and therefore are more representative of the Judean community's everyday life. He mentioned the use of the ostraca in indicating that the Aramaic documents in Ezra-Nehemiah are historically possible (the historian in me would ask, whether historical possibility is all that is needed for historical likelihood, but that is another question). He then went on to analyse the ostraca in particular with regard to the image of the religion of everyday Judean life in Elephantine and came to the result that the community's life was very much focussed around Yahu, and that other deities only appear very rarely indeed. Indeed, it seemed to Lemaire that the Judean community from Elephantine had a kind of 7th century Judean religiosity.
In an aside he then pointed to other sites of Judean activity in Egypt during the Persian period: Thebes, Abydos, Memphis, Saqqara, Daphne and Edfu. 

The third and final part of the presentation concerned evidence for Judeans in Aramaic texts from elsewhere in the Persian empire, in particular two funerary inscriptions from Didaskaleion (which he thought were both unlikely), and then 5 funerary inscriptions from Cyprus which mention individuals with Judean names. 

All in all a very interesting and useful summary of the currently available evidence for Judeans in Aramaic epigraphy in the Persian empire. Few know the texts as well as Lemaire.