In the relevant scholarly literature, it has actually been claimed that Luke used the writings of Josephus (specifically ‘Antiquities of the Jews’). Since Josephus wrote in 93 CE, this would date Acts no earlier than this time.
The following passages are typically claimed as examples of Luke’s dependence on Josephus.
Luke 3:1: Josephus and Luke record the census of Quirinius, but Luke’s differs from that of Josephus and cannot be verified independently; both Luke and Josephus refer to Lysanias the tetrarch of Abilene
Luke 13:1: Luke’s description of the murder of the Galileans is similar to Josephus’ description of an assault on Samaritans
Acts 5:36-37: Luke mentions Theudas and Judas the Galilean, but reverses the order in which Josephus listed them, dates Theudas 15 years before the date Josephus gives
Acts 11:28-9: Luke and Josephus both record famine during Claudius’ reign
Acts 12:21-3: Luke describes Agrippa I’s death in a manner similar to Josephus, but with certain differences
Acts 21:38: Luke describes ‘the Egyptian’ rebel leading sicarii into the wilderness but Josephus’s reference to sicarii in the wilderness is separate from his reference to ‘the Egyptian’
Acts 25:13, 23; 26:30: Like Josephus, Luke implies that Agrippa II and Berenice are married, or consorts
Acts 24:24-6: Like Josephus, Luke shows he is aware Drusilla (the wife of Felix), is a Jew
The claim is so insubstantial that most scholars consider it highly debatable at best, rejecting it on a range of grounds and arguing Luke and Josephus used common traditions and historical sources.
‘Arguments for the dependence of passages in Acts on Josephus (especially the reference to Theudas in Acts v. 37) are equally unconvincing. The fact is, as Schurer has said: “Either Luke had not read Josephus, or he had forgotten all about what he had read”‘ 
‘But it is hardly logical to hold that Luke depends on Josephus and yet be obliged to admit that Luke shows wide divergence from him in relating events that are supposedly the same.’ 
‘The argument that Luke used the historian, Josephus (ad 93), was never fully convincing (HJ Cadbury, BC 11, 357). Today it is seldom pressed.’ 
‘Sterling concludes that, while it is impossible to establish a literary dependence of Luke-Acts on the writings of Josephus, it is reasonable to affirm that both authors not only had access to similar historical traditions but also shared the same historiographical techniques and perspectives.’ 
‘After examining the texts myself, I must conclude with the majority of scholars that it is impossible to establish the dependence of Luke-Acts on the Antiquitates. What is clear is that Luke-Acts and Josephos shared some common traditions about the recent history of Palestine.’ 
‘It seems probable that Luke and Josephus wrote independently of one another; for each could certainly have had access to sources and information, which he then employed according to his own perspectives. A characteristic conglomerate of details, which in part agree, in part reflect great similarity, but also in part, appear dissimilar and to stem from different provenances, accords with this analysis.’ 
‘A. T. Robinson, Redating, p. 88, regards the Josephus line of approach as almost totally abandoned.’ 
‘From Krenkel’s remarks it can be seen that this proof can be offered only with very powerful mental contortions. See Hemer, Acts (n.37), 95: ‘the theory of Lukan dependence on Josephus has had in its day a certain vogue, and has been used as a major argument for the late dating of Luke-Acts’; cf. also Sterling, Historiography (n.37),365f. n.281.’ 
‘Nevertheless, direct literary dependence on Josephus by Luke is consistently dismissed for various reasons.’ 
‘The relationship between Luke and Josephus has produced an abundant literature, which has attempted to show the literary dependence of one on the other. I do not believe that any such dependence can be proved.’ 
‘Most scholars today deny any dependence one way or the other, and we think this judgment is correct.’ 
‘When we consider both the differences and the agreement in many details of the information in the two accounts, [of the death of Herod Agrippa I] it is surely better to suppose the existence of a common source on which Luke and Josephus independently drew.’ 
‘Some attempt to argue a literary dependence on Josephus, and date Luke-Acts after 93CE. But, without a doubt, Luke’s theology is of an earlier type than Justin.’ 
This consensus is even acknowledged by those who argue for Luke’s dependence on Josephus, or the other way around.
“Neither position has much of a following today, because of the significant differences between the two works in their accounts of the same events.”
 ‘This theory was maintained by F. C. Burkitt (The Gospel History and its Transmission, 1911, pp. 105–110), following the arguments of Krenkel’s Josephus und Lucas (1894).’, Guthrie, ‘New Testament Introduction’, p. 363 (4th rev. ed. 1996); Two recent examples are Richard Pervo’s ‘Dating Acts’ (2006), and ‘Acts: A Commentary’ in the series ‘Hermeneia: a Critical and Historical Commentary on the Bible’ (2008), and Steve Mason’s ‘Josephus and the New Testament’ (1992); Pervo’s is considered an academic argument worthy of response (though it has failed to convince most scholars), whereas Mason’s is rarely referred to in the relevant scholarly literature.
 ‘If Acts is dependent on Josephus for information, it cannot be earlier than 93. But such dependence is not proved and is highly unlikely.’, in Douglas & Tenney, ’New International Bible Dictionary’, p, 13 (1987).
 ‘A number of events to which allusion is possibly being made are discussed by J. Blinzler*, 32–37. These include: 1. the affair of the ensigns in Jos. Bel. 2:169–174; Ant. 18:55–59, but this took place in Caesarea in AD 26; 2. the tumults associated with the building of an aqueduct (Jos. Bel. 2:175–177; Ant. 18:60–62), but this incident involved the murder of Judaeans with cudgels outside the temple; 3. an attack on some Samaritans (Jos. Ant. 18:85–87), but this took place in AD 36; 4. the slaughter of about 3,000 Jews offering Passover sacrifices by Archelaus in 4 BC (Jos. Bel. 2:8–13; Ant. 17:213–218). This incident, however, took place some thirty years earlier and was committed by a different ruler; moreover, the murder of 3,000 men would not bear comparison with an accident to 18. It is wisest to conclude that the event is not attested from secular sources. This, however, is no argument against its historicity, since Josephus’ account of Pilate’s career is very incomplete (cf. Philo, Leg. 299-305). Pilate would have been in Jerusalem at Passover time, and the Galileans had a reputation for rebelliousness. The suggestion that Zealots were involved (O. Cullmann, The State in the NT, London, 1957, 14) lacks proof.’, Marshall, ‘The Gospel of Luke: A Commentary on the Greek Text’, New International Greek Testament Commentary, p. 553 (1978).
 ‘There are two problems: (1) Since Gamaliel was speaking well before AD 44 (the year in which Herod Agrippa I died, 12:20-23), a reference to the Theudas mentioned in Josephus would be anachronistic on his lips. (2). Gamaliel goes on to describe the rising of Judas after this; but the rising of Judas took place in AD 6 before the Theudas incident in Josephus. So, it is argued, Luke makes Gamaliel commit an anachronism and put the two stories in reverse chronological order. It has been argued that Luke was led to this error by misreading Josephus who goes on after the Theudas story to mention the sons of Judas and then to explain parenthetically who this Judas was and how he had led a revolt against Rome. But this supposition is highly unlikely, since Josephus’ works were not published till c. AD 93, and since Luke cannot possibly have got the details of his story (the 400 men) from him. No plausible explanation of Luke’s alleged error has been offered. There is, therefore, much to be said for the suggestions either that Josephus got his dating wrong or (more probably) that Gamaliel is referring to another, otherwise unknown Theudas. Since there were innumerable uprisings when Herod the Great died, and since ‘Josephus describes four men bearing the name of Simon within forty years and three that of Judas within ten years, all of whom were instigators of rebellion’ (cited by Knowling, p. 158), this suggestion should not be rejected out of hand.’, Marshall, ‘Acts: An Introduction And Commentary’, Tyndale New Testament Commentaries, volume 5, pp. 122-123 (1980).
 ‘Famines are mentioned in various parts of the empire during the time of Claudius. Josephus tells of a famine in Palestine during the governorship of Tiberius Alexander (46/48 C.E.):’, Conzelmann, Epp, & Matthews, ‘Acts of the Apostles: A Commentary on the Acts of the Apostles’, Hermeneia, p. 90 (1987).
 ‘The details of Herod’s death are recorded slightly differently by Josephus, but the accounts are complementary. …Luke’s description of Herod as being eaten by worms is probably directly related to the abdominal pains referred to in Josephus’ account.’, Carson, ‘New Bible Commentary: 21st Century Edition’ (4th ed. 1994).
 ‘According to Josephus (Bel. 2:261–263) there had been an Egyptian false prophet who had led 30,000 men to the Mount of Olives in order to take Jerusalem; he promised that they would see the walls of the city fall down. The governor, Felix, killed or captured his followers, while the prophet himself managed to escape. Clearly the tribune thought that this person had reappeared; the discrepancy between the number of his followers in Acts and in Josephus reflects the latter’s well-known tendency to exaggeration, and the tribune’s estimate will have been nearer the mark.’, Marshall, ‘Acts: An Introduction And Commentary’, Tyndale New Testament Commentaries, volume 5, p. 371 (1980).
 ‘There was gossip about the relationship between the brother and sister (Josephus Ant. 20.145; Juvenal Sat. 6.156–60). ‘,Conzelmann, Epp, & Matthews, ‘Acts of the Apostles: A Commentary on the Acts of the Apostles’, Hermeneia, p. 206 (1987).
 ‘The use of the LXX is not debatable, but the influence of Josephus and Paul has been and is subjected to considerable debate.’, Tyson, ‘Marcion and Luke-Acts: a defining struggle’, p. 14 (2006).
 Geldenhuys, ‘Commentary on the Gospel of Luke’, p. 31 (1950).
 Harrison, ‘Introduction to the New Testament’, p. 240 (1971).
 Ellis, ‘The Gospel of Luke’, p. 55 (1977).
 Verheyden, ‘The Unity of Luke-Acts’, p. 678 (1990).
 Sterling, ‘Historiography and Self-Definition: Josephos, Luke-Acts, and Apologetic Historiography’, Supplements to Novum Testamentum, pp. 365-366 (1992).
 Schreckenberg & Schubert, ‘Jewish Historiography and Iconography in Early and Medieval Christian Literature’, Compendia Rerum Iudicarum Ad Novum Testamentum, volume 2, p. 51 (1992).
 Guthrie, ‘New Testament Introduction’, p. 364 (4th rev. ed. 1996).
 Hengel & Schwemer, Paul between Damascus and Antioch: the unknown years’, p. 325 (1997).
 Denova, ‘The Things Accomplished Among Us: prophetic tradition in the structural pattern of Luke-Acts’, p. 207 (1997).
 Marguerat, ‘The First Christian Historian: writing the “Acts of the Apostles”‘, p. 79 (2002).
 Heyler, ‘Exploring Jewish literature of the Second Temple Period: A Guide for New Testament Students’, p. 362 (2002).
 Klauck & McNeil, ‘Magic and Paganism in Early Christianity: the world of the Acts of the Apostles’, p. 43 (2003)
 Hear, ‘Simon Magus: the first gnostic?’, p. 71 (2003).
 Mason, ‘Josephus and the New Testament’, p. 185 (1992).
For more, see https://lukeprimacy.com/