Fred wrote: << At 03:43 PM 6/7/2007, Schroeppel, Richard wrote: < I don't know the history: I don't think the extended notations are used in Hardy & Wright, except maybe for (mod 1) to mean fractional part, or for stuff with algebraic integers.
Actually Hardy & Wright do use this notation, and in exactly the same context as the Monthly article that Dan cited. Theorem 116 says "If p>3, and 1/i is the associate of i (mod p^2), then 1 + 1/2 + 1/3 + ... + 1/(p-1) == 0 (mod p^2)." (By "associate" they mean "multiplicative inverse".)
Wait a sec -- the quote from Hardy & Wright looks as though it has no immediate connection with harmonic numbers H_n. Rather, it seems to be saying that if, in the ring Z/(p^2), we add the multiplicative inverses* of 1,2,...,p-1, then we get 0. (Otherwise, what does "i" mean here?) Maybe there is some clever way to connect this with harmonic numbers. But my first impression is that Hardy & Wright are not using the "mod" notation to refer to anything but integers (or integers mod N). --Dan __________________________________________________________ * Of course for p prime, not all members of Z/(p^2) have multiplicative inverses, but 1,2,...,p-1 do.