Modular divisor functions and quadratic reciprocity

Steiner, R. (2010) Modular divisor functions and quadratic reciprocity. American Mathematical Monthly, 117(5), pp. 442-448. (doi: 10.4169/000298910X485969)

Abstract

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.

• Item Type: Articles
• Additional Information: Copyright the Mathematical Association of America 2010. All rights reserved.
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Steiner, Dr Richard
• Authors: Steiner, R.
• Subjects: Q Science > QA Mathematics
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: American Mathematical Monthly
• Publisher: Mathematical Association of America
• ISSN: 0002-9890
• ISSN (Online): 1930-0972
• Copyright Holders: Copyright © 2010 Mathematical Association of America
• First Published: First published in American Mathematical Monthly 117(5):442-448
• Publisher Policy: Reproduced with the permission of the publisher.