What is the ruling on fasting the entire month of Rajab, which month is it recommended to fast frequently in, and is the hadith: "Rajab is the month of Allah, Sha'ban is my month, and Ramadan is the month of my Ummah" authentic from the Prophet, may Allah bless him and grant him peace?

The months in which it is recommended to increase fasting are summarized as follows: Sha'ban, the sacred months, in addition to six days of Shawwal for those who fasted Ramadan (definition). This is indicated by several prophetic hadiths, among them: "The best fasting after Ramadan is Allah's month, Muharram," and the hadith (definition) of Aisha, may Allah be pleased with her, regarding the Prophet's frequent fasting in Sha'ban, and the hadith of Usama, may Allah be pleased with him, in which the Prophet explained the virtue of Sha'ban as a month in which deeds are raised. As for the hadith in question (fasting in Rajab), scholars have deemed it weak.