What did kautilya's arthashastra say about the duty of the king?

Kauṭilya, answered from the texts and cited to the page.

Kauṭilya answers

The king's first duty is wakefulness, and the reason is plain: if he is energetic, his subjects will be energetic; if he is reckless, they will be reckless too, eat into his works, and leave him an easy prey for his enemies.¹ So I divide the day and the night into eight parts each and assign every part its business. In the day: the watchmen posted and the accounts of receipts and expenditure; the affairs of citizens and country people; bathing, dining and study; the receipt of revenue in gold and the appointment of superintendents; correspondence with the ministers and the secret information of the spies; then his amusements or self-deliberation; the review of elephants, horses, chariots and infantry; and last the plans of military operations with the commander-in-chief.²

The night is ordered likewise, beginning with secret emissaries and ending, before he goes into his court, with the benedictions of the priests, the sight of his physician, cook and astrologer, and the salutation of a cow with her calf.³ If his capacity requires, he may alter the time-table, but the duties remain.⁴ In court he shall never make petitioners wait at the door.

A king who makes himself inaccessible and leaves his work to his immediate officers breeds confusion in business and disaffection among the people, and makes himself a prey to his enemies.⁵ He attends in person to the business of the gods, of Brāhmaṇas learned in the Vedas, of cattle, of sacred places, of minors, the aged, the afflicted and the helpless, and of women, in that order or according to urgency.⁶

Urgent calls he hears at once and does not postpone them, for what is put off becomes hard or impossible to accomplish.⁷ I gather his office into this: the king's religious vow is readiness to action, the satisfactory discharge of his duties is his sacrifice, and equal attention to all is his offering of fees and his ablution towards consecration.⁸

His happiness lies in the happiness of his subjects and his welfare in theirs. Whatever pleases himself he shall not count as good; whatever pleases his subjects he shall.⁹ The root of wealth is activity, and of evil its reverse; without activity, acquisitions present and to come perish.¹⁰ As to justice, the king's duty is to protect his subjects with justice.

By observing it he reaches heaven, but he who does not protect his people or upsets the social order wields his sceptre in vain.¹¹ The rod must be exercised with impartiality and in proportion to guilt, over his son or his enemy alike; that alone maintains this world and the next.¹² He judges by sacred law, evidence, history and the edicts of kings, and where a text of sacred law conflicts with reasoned law, reason prevails, for in that case the original text on which the sacred law rests is not available.¹³

Sources

If a king is energetic, his subjects will be equally energetic. If he is reckless, they will not only be reckless likewise, but also eat into his works.
Arthasastra
during the eighth part, he shall consider various plans of military operations with his commander-in chief.
Arthasastra
during the first one-eighth part of the night, he shall receive secret emissaries
Arthasastra
Or in conformity to his capacity, he may alter the time-table and attend to his duties.
Arthasastra
When in the court, he shall never cause his petitioners to wait at the door, for when a king makes himself inaccessible to his people and entrusts his work to his immediate officers, he may be sure to engender confusion in business
Arthasastra
He shall, therefore, personally attend to the business of gods, of heretics, of Brahmans learned in the Vedas, of cattle, of sacred places, of minors, the aged, the afflicted, and the helpless, and of women
Arthasastra
All urgent calls he shall hear at once, but never put off; for when postponed, they will prove too hard or impossible to accomplish.
Arthasastra
Of a king, the religious vow is his readiness to action; satisfactory discharge of duties is his performance of sacrifice; equal attention to all is the offer of fees and ablution towards consecration.
Arthasastra
In the happiness of his subjects lies his happiness; in their welfare his welfare; whatever pleases himself he shall not consider as good, but whatever pleases his subjects he shall consider as good.
Arthasastra
the root of the wealth is activity, and of evil its reverse. In the absence of activity acquisitions present and to come will perish
Arthasastra
As the duty of a king consists in protecting his subjects with justice, its observance leads him to heaven. He who does not protect his people or upsets the social order wields his royal sceptre (danda) in vain.
Arthasastra
It is power and power (danda) alone which, only when exercised by the king with impartiality and in proportion to guilt, either over his son or his enemy, maintains both this world and the next.
Arthasastra
But whenever sacred law (sastra) is in conflict with rational law (dharmanyaya— king's law), then reason shall be held authoritative; for there the original text (on which the sacred law has been based) is not available.
Arthasastra