Error term of the mean value theorem for binary Egyptian fractions

In this article, the error term of the mean value theorem for binary Egyptian fractions is studied. An error term of prime number theorem type is obtained unconditionally. Under Riemann hypothesis, a power saving can be obtained. The mean value in short interval is also considered.

Egyptian Fractions Re-Revisited

Ancient Egyptians represented each fraction as a sum of unit fractions, i.e., fractions of the type 1/n. In our previous papers, we explained that this representation makes perfect sense: e.g., it leads to an efficient way of dividing loaves of bread between people. However, one thing remained unclear: why, when representing fractions of the type 2/(2k+1), Egyptians did not use a natural representation 1/(2k+1)+1/(2k+1), but used a much more complicated representation instead. In this paper, we show that the need for such a complicated representation can be explained if we take into account that instead of cutting a rectangular-shaped loaf in one direction – as we considered earlier – we can simultaneously cut it in two orthogonal directions. For example, to cut a loaf into 6 pieces, we can cut in 2 pieces in one direction and in 3 pieces in another direction. Together, these cuts will divide the original loaf into 2 * 3 = 6 pieces. It is known that Egyptian fractions are an exciting topics for kids, helping them better understand fractions. In view of this fact, we plan to use our new explanation to further enhance this understanding.