In South Africa, a hadith is quoted that says that if a person recites the durood Allaahumma sallee 'ala Muhammadin Nabee al-'ummiy waslamon a Friday after the 'Asr prayer 80 times, then so many years of their sins will be forgiven. Is this hadith authentic?
All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah and that Muhammad is His slave and Messenger.
Perhaps you are referring to the hadeeth cited by Al-Ghazaali and others; its wording is similar to what you have mentioned in the question. Al-Ghazaali wrote:
"It is recommended to invoke the blessings of Allaah upon the Messenger of Allaah on that day (Friday), as he said, 'Whoever invokes blessing upon me on Friday eighty times, Allaah forgives him the sins of eighty years.' He was asked, 'O Messenger of Allaah, how do we invoke blessings upon you?' He said, 'You say, 'Allaahumma salli ʻala Muhammadin ʻabdika wa Nabiyyika wa Rasoolika, An-Nabiyy Al-Ummiyy,' (O Allaah, confer Your blessings upon Muhammad, Your slave, prophet, and messenger, the illiterate prophet) and you count it as one.'" [Ihyaa’ ʻUloom Al-Deen]
Al-Haafith Al-ʻIraaqi commented in his revision of the book Ihyaa’ ʻUloom Al-Deen, "The hadeeth that reads, 'Whoever invokes blessings upon me on Friday eighty times...' was narrated by Ad-Daaraqutni on the authority of Ibn Al-Musayyib. He said, 'I think it was reported by Abu Hurayrah, and it is a ghareeb hadeeth (i.e. a hadeeth or version reported by one reliable or unreliable narrator which differs in context with another hadeeth or version reported by a group of reliable narrators). Ibn An-Nuʻmaan declared it a hasan (good) hadeeth.'"
Al-Munaawi said in his book Faydh Al-Qadeer that Ibn Hajar declared it inauthentic.
Hence, Hadeeth scholars held different opinions regarding the authenticity of this hadeeth; some declared it dha'eef (weak) and others declared it hasan.
Even if we assumed that the hadeeth is inauthentic, it is acceptable to act upon the inauthentic hadeeth provided that certain conditions are fulfilled. As-Sakhaawi wrote:
"There are three conditions for the permissibility of acting upon an inauthentic hadeeth:
1) It should not be very weak; this excludes the reports transmitted by narrators declared as liars, of those who are accused of lying, or of those whose mistakes in transmission are serious. This condition is agreed upon by scholars.
2) It should have a basis in the sharee'ah; this excludes the fabricated reports with no ground or foundation in the sharee'ah.
3) When acting upon it, one should not believe that the hadeeth is authentic..." [Al-Qawl Al-Badeeʻ fi As-Salah ʻala Al-Habeeb Ash-Shafeeʻ]
We would like to highlight that invoking blessings and peace upon the Prophet is a very rewardable thikr (remembrance) that the Muslim should keenly recite more often, especially on Fridays. The Prophet said, "Among the most excellent of your days is Friday, so invoke more blessings on me on that day, for your invocations will be presented to me." The people asked, "O Messenger of Allaah, how will our invocations be presented to you while your body will have decayed?" He replied, "Allaah, the Exalted, has prohibited the earth from consuming the bodies of Prophets." [Abu Daawood and others]
Allaah knows best.
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