Modular divisor functions and quadratic reciprocity

It is a well-known result by B. Riemann that the terms of a conditionally convergent series of real numbers can be rearranged in a permutation such that the resulting series converges to any prescribed sum s: add p1 consecutive positive terms until their sum is greater than s; then subtract q1 consecutive negative terms until the sum drops below s, and so on. For the alternating harmonic series, with the aid of a computer program, it can be noticed that there are some fascinating patterns in the sequences pn and qn. For example, if s = log 2 + (1/2) log (38/5) the sequence pn is 5, 7, 8, 7, 8, 7, 8, 8, 7, 8, 7, 8, . . . in which we notice the repetition of the pattern 8, 7, 8, 7, 8, while if s = log 2+ (1/2) log (37/5) the sequence pn is 5, 7, 7, 7, 8, 7, 8, 7, 7, 8, 7, 8, . . . in which the pattern is 7, 7, 8, 7, 8. Where do these patterns come from? Let us observe that 38/5 = 7 + 3/5 and 37/5 = 7 + 2/5. The length of the repeating pattern is the denominator 5, the values of pn, at least from some n on, are 7 and 8, and the number 8 appears 3 times in the pattern of the first example, and 2 times in that of the second one. These are not coincidences: we explain them in this paper