<p> Supervisors Project Submission Form</p><p>Supervisor: M D Coleman</p><p>1 Project Title Bernoulli numbers, polynomials and the Euler-Maclaurin Formula 2 Category Pure</p><p>3 Level 3/4 4 Semesters 2 (length of project) 5 Description You may wish to consider the question, which came first, the Bernoulli numbers or the Bernoulli polynomials? Following discussion of the many definitions of the Bernoulli numbers you may wish to discuss their occurrence in formula for the sum of the k-th powers of the first n integers, then onto their recurrence relations. The Bernoulli numbers are rational numbers and Clausen-von Staudt's theorem says something about the factorization of the denominators. Some proofs of this, and perhaps its most natural setting, use p-adic numbers. Whichever came first you would have to study Bernoulli polynomials. In particular the simple facts about the behaviour of the Bernoulli polynomials on [0, 1] should be derived, as should the expansions of the Bernoulli polynomials on [0, 1] as Fourier series. An appearance of Bernoulli numbers is made in the Euler- Maclaurin summation formula. This can be derived and applied to the investigation of a number of finite and infinite series (e.g. harmonic numbers, Stirling's approximation and its improvement, connection to the Riemann zeta- function and resulting asymptotic results and the “acceleration” of zeta function series such as used by Aprey in his proof that ζ(3) is irrational). 6 References 1. Hardy, G.H., Wright, E.M. An Introduction to the Theory of Numbers, 6th edition, Oxford, [Chapter 7] 2. Ireland, K., Rosen, M. A Classical Introduction to Modern Number Theory, Graduate Texts in Mathematics 84, Springer-Verlag, 1982 [Chapter 15] 3. K. Knopp, Theory and Application of Infinite Series, R. C. Young, translator. Hafner, New York, 1951. 4. Montgomery, H, Vaughan, R, Multiplicative Number Theory, 1 Classical Theory, Cambridge Studies in advanced mathematics 97, CUP 2007. [Appendix B] 7 Prerequisite courses</p><p>8 Additional notes</p>