Is it obligatory to fast for three days as an expiation for someone who swore by Allah not to browse the Internet for three days, then mistakenly started browsing before the period ended due to forgetfulness?

If you broke your oath by accessing the Internet before the end of the three days you swore upon, then it is more prudent to expiate your oath to avoid disagreement among scholars. The expiation for an oath is to feed ten needy persons, or clothe them, or free a slave. If you cannot find these, then you must fast for three days, as Allah Almighty says: "Its expiation is the feeding of ten needy people from the average of that which you feed your [own] families or clothing them or freeing a slave. But whoever finds not [the means to do so] - then a fast of three days" (Al-Ma'idah: 89). As for forgetting, the Shafi'i school and one narration from the Hanbali school do not require expiation, while the Hanafi and Maliki schools consider it obligatory.