### Mathematics by Experiment, I & II: Plausible Reasoning in the 21st Century

, 2003

"... Abstract. In the first of these two lectures I shall talk generally about experimental mathematics. In Part II, I shall present some more detailed and sophisticated examples. The emergence of powerful mathematical computing environments, the growing availability of correspondingly powerful (multi-pr ..."

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Abstract. In the first of these two lectures I shall talk generally about experimental mathematics. In Part II, I shall present some more detailed and sophisticated examples. The emergence of powerful mathematical computing environments, the growing availability of correspondingly powerful (multi-processor) computers and the pervasive presence of the internet allow for research mathematicians, students and teachers, to proceed heuristically and `quasi-inductively'. We may increasingly use symbolic and numeric computation visualization tools, simulation and data mining. Many of the benefits of computation are accessible through low-end `electronic blackboard ' versions of experimental mathematics [1, 8]. This also permits livelier classes, more realistic examples, and more collaborative learning. Moreover, the distinction between computing (HPC) and communicating (HPN) is increasingly moot. 2 The unique features of our discipline make this both more problematic and more challenging. For example, there is still no truly satisfactory way of displaying mathematical notation on the web; and we care more about the reliability of our literature than does any other science. The traditional role of proof in mathematics is arguably under siege. Limned by examples, I intend to pose questions ([9]) such as: ffl What constitutes secure mathematical knowledge? ffl When is computation convincing? Are humans less fallible? ffl What tools are available? What methodologies? 3 ffl What about the `law of the small numbers '? ffl How is mathematics actually done? How should it be? ffl Who cares for certainty? What is the role of proof? And I shall offer some personal conclusions.

### Experimental Mathematics: Apéry-Like Identities for ζ(n)

, 2005

"... We wish to consider one of the most fascinating and glamorous functions of analysis, the Riemann zeta function. (R. Bellman) Siegel found several pages of... numerical calculations with... zeroes of the zeta function calculated to several decimal places each. As Andrew Granville has observed “So muc ..."

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We wish to consider one of the most fascinating and glamorous functions of analysis, the Riemann zeta function. (R. Bellman) Siegel found several pages of... numerical calculations with... zeroes of the zeta function calculated to several decimal places each. As Andrew Granville has observed “So much for pure thought alone. ” (JB & DHB) www.cs.dal.ca/ddrive AK Peters 2004 Talk Revised: 03–29–05Apéry-Like Identities for ζ(n) The final lecture comprises a research level case study of generating functions for zeta functions. This lecture is based on past research with David Bradley and current research with David Bailey. One example is Z(x): = 3 k=1 n=1

### Centre for Discrete Mathematics and Theoretical Computer ScienceSearching for Spanning k-Caterpillars and k-Trees

, 2008

"... We consider the problems of finding spanning k-caterpillars and k-trees in graphs. We first show that the problem of whether a graph has a spanning k-caterpillar is NP-complete, for all k ≥ 1. Then we give a linear time algorithm for finding a spanning 1-caterpillar in a graph with treewidth k. Also ..."

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We consider the problems of finding spanning k-caterpillars and k-trees in graphs. We first show that the problem of whether a graph has a spanning k-caterpillar is NP-complete, for all k ≥ 1. Then we give a linear time algorithm for finding a spanning 1-caterpillar in a graph with treewidth k. Also, as a generalized versions of the depth-first search and the breadth-first search algorithms, we introduce the k-tree search (KTS) algorithm and we use it in a heuristic algorithm for finding a large k-caterpillar in a graph. 1

### Dirichlet Series of Squares of Sums of Squares

"... Hardy & Wright records elegant forms for the generating functions of the divisor functions k (n) = P djn d k and 2 k (n): 1 X n=1 k (n) n s = (s)(s k) (1) 1 X n=1 2 k (n) n s = (s)(s k) 2 (s 2k) (2s 2k) : (2) We have extended this elegant pair to: Theorem 1 For ..."

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Hardy & Wright records elegant forms for the generating functions of the divisor functions k (n) = P djn d k and 2 k (n): 1 X n=1 k (n) n s = (s)(s k) (1) 1 X n=1 2 k (n) n s = (s)(s k) 2 (s 2k) (2s 2k) : (2) We have extended this elegant pair to: Theorem 1 For completely multiplicative f 1 ; f 2 and g 1 ; g 2 , 1 X n=1 (f 1 g 1 )(n) (f 2 g 2 )(n) n s = (3) L f 1 f 2 (s)L g 1 g 2 (s)L f 1 g 2 (s)L g 1 f 2 (s) L f 1 f 2 g 1 g 2 (2s) where f g(n) := P djn f(d)g(n=d) and L f (s) := P 1 n=1 f(n)n s is a Dirichlet series. 2 Let r N (n) be the number of solutions of x 2 1 + +x 2 N = n and let r 2;P (n) be the number of solutions of x 2 + Py 2 = n. One application of Theorem 1 is to obtain closed forms, in terms of (s) and Dirichlet L- functions, for the generating functions of functions such as r N (n); r 2 N (n); r 2;P (n) and r 2;P (n) 2 for certain P and (even) N = 2; 4; 6; 8. We also use these generating functions to obtain asymptotics for the average values of each function for which we obtain a Dirichlet series. We nish by discussing the more vexing case N = 3, and related matters. This is joint work with Stephen Choi (SFU). These transparencies are at www.cecm.sfu.ca/personal/jborwein/talks.html The CECM preprint 01:167 is at: www.cecm.sfu.ca/preprints/2001pp.html 3 OUTLINE 1. Motivation and Background. 2. Theorem on L-series of Squares of Arithmetic Functions. 3. Why Squares not Cubes? 4. Applications to r 4 ; r 6 and r 8 . 5. Applications to Quadratic Forms. 6. Three, Twelve and Twenty Four Squares. 4 1. MOTIVATION and BACKGROUND Evaluating residues of the corresponding g.f.s at their largest real poles and Cauchy's integral theorem leads to X nx r 2...