I don't know when Artaxerxes I began to rule. Experts in ancient chronology should have a good idea.
Monarchs reigned from the time the previous monarch died or otherwise stopped reigning until they died or stopped reigning. Elected or appointed officials could begin their rule on the first day of a year and end it on the last day of that or a later year. But monarchs tended to have reigns that began and ended in various parts of various years.
So how did ancient writers decided how long a monarch reigned? A monarch usually reigned and/or ruled for all of several years, and for parts of two different years, one at the beginning and one at the end of their reign.
Suppose that a king started reigning in year 681 and died in year 691 of the local calendar. He would have reigned in all of 8 years, plus parts of two others. So some persons might say that the king ruled for 8 years. Others might assume that the part of year 681 and the part of year 691 that he reigned in would average half a year each, and so say he reigned for 9 years. Others might say that since he ruled in parts of 2 years plus all of 8 years that he ruled for 10 years.
If someone added up the lengths of the reigns of the kings of a kingdom to get a total, how did they count the years when the previous king died and the new king began ruling? If in my example the years 681 and 691 were counted as full years in the reigns of both kings, approximately 10 years of time would seem like 12 years. So there had to be an rule, official or otherwise, for a historian to count the length of the reigns of the kings to avoid counting years twice.
According to the Old Testament, the united kingdom of Saul, David, and Solomon was divided in the reign of Solomon's son Rehoboam, when Jeroboam revolted and founded the Kingdom of israel, leaving Rehoboam with the small rump kingdom of Judah. The lengths of the reigns of all the kings of Israel and Judah are given. Some events are dated to the time since a king began his reign, which thus gives a total time since the two kingdoms split. And some events that involved both Israel and Judah are dated with the regnal yeas of both kings.
And I remember reading that when an event is dated in both the Israel and the Judah chronology, the time spans since the split between Israel and Judah are not the same. And it was said that the records of Israel and the records of Judah used in the old testament must have used different methods of counting how long a king reigned.
Obviously the best way to say how long a king ruled was to take the exact date when his reign began, and the exact date when his reign ended, and then calculated how many years, months, and days that was, when the exact dates of the beginning and end of the reign were recorded. But sometimes ancient historians wouldn't know the exact dates when a reign began and ended. And usually ancient historians just gave a round number of years as the length of a reign.
In ancient Rome, the civil year began on the day the consuls began their term of office, and ended the day before the next pair of consuls began their term of office, so it was usual to call a year the year of the consulship of X and Y.
My answer here:
Lists the various dates when the Consuls took office. Note that they began to take office on January 1, beginning in 153 BC. So the practice of beginning the new year on January 1 began in the Roman Calendar used in the Republic in 153 BC. and was copied by the Julian Calendar established in 44 BC.
But the ancient Roman January 1 was not on the same date as our January 1, because the Republican Calendar used leap months instead of leap days, so their January 1 was often weeks off what would be January 1 in the proleptic Julian or Gregorian Calendar.
A proleptic calendar is a calendar that is applied to dates before its introduction. Examples are:
So after centuries of using the Republican Calendar, the dates were moving around the seasons too much, and Caesar reformed the calendar, producing the Julian Calendar which used a leap day every fourth year. And then the persons in charge of the calendar managed to mess up the application of leap days, interpreting every fourth year as what we would call every third year. So when this was discovered Emperor Augustus had to intervene and stop adding leap days until the calendar was realigned with the proper dates. As I remember, the regular use of leap days every fourth year then began about our year AD 4.
So the new year has been on January 1 of the Julian calendar since about AD 4. And it is possible that about AD 4 was the first time ever in human history that the new year in any calendar was on January 1. And it did not remain on January 1 forever, not even in the Julian Calendar. Later Christian societies used the Julian Calendar, but they often choose to begin the year on a significant Christian date.
In 567 AD, the Council of Tours formally abolished 1 January as the beginning of the year. At various times and in various places throughout mediaeval Christian Europe, the new year was celebrated on 25 December in honour of the birth of Jesus; 1 March in the old Roman style; 25 March in honour of Lady Day and the Feast of the Annunciation; and on the movable feast of Easter. These days were also astronomically and astrologically significant since, at the time of the Julian reform, 25 March had been understood as the spring equinox and 25 December as the winter solstice. (The Julian calendar's small disagreement with the solar year, however, shifted these days earlier before the Council of Nicaea which formed the basis of the calculations used during the Gregorian reform of the calendar.) Mediaeval calendars nonetheless often continued to display the months running from January to December, despite their readers reckoning the transition from one year to the next on a different day.
European countries converted to making January 1 the start of the year during late medieval and modern times.
So obviously, since there are a lot of different calendars used in ancient times, the sources about the length of a king's reign might have different starting dates for years, which could mean that two different sources might give a particular king reign lengths that might differ by one year.
Fortunately, the reign of Artaxerxes I was during the era when a lot of clay tablets from Mesopotamia have been found. Many of them would give the exact date in the Babylonian calendar, and number the year from the beginning of the reign of Artaxerxes I. So the reign of Artaxeres I should be at least as long as the number of years, months, and days between the earliest and latest Babylonian dates in his reign.
And I think that the Babylonian Calendar would very well known by now, and that there should not be any problems converting Babylonian dates to proleptic Julian calendar dates. So experts on ancient chronology should probably have a good idea of when Artazwerxes I began to reign in Mesopotamia, which should be fairly close to when he began to reign in Persia.