Functiones et Approximatio 48.1 (2013), 91–111 doi: 10.7169/facm/2013.48.1.7 VARIATIONS OF THE RAMANUJAN POLYNOMIALS AND REMARKS ON ζ (2j + 1)/π 2j+1 Matilde N. Lalín, Mathew D. Rogers Abstract: We observe that five polynomial families have all of their roots on the unit circle. We prove the statements explicitly for four of the polynomial families. The polynomials have coefficients which involve Bernoulli numbers, Euler numbers, and the odd values of the Riemann zeta function. These polynomials are closely related to the Ramanujan polynomials, which were recently introduced by Murty, Smyth and Wang [MSW]. Our proofs rely upon theorems of Schinzel [S], and Lakatos and Losonczi [LL] and some generalizations. Keywords: Ramanujan polynomials, Riemann zeta function values, reciprocal polynomials, roots on the unit circle, Bernoulli numbers, Euler numbers. 1. Introduction In a recent paper, Murty, Smyth and Wang considered the Ramanujan polynomials [MSW]. They were defined by Gun, Murty and Rath [GMR] using R 2k+1 (z) := k+1 ∑ j=0 B 2j B 2k+2-2j (2j )!(2k +2 − 2j )! z 2j , (1.1) where B j is the j th Bernoulli number. Among other fascinating results, Murty, Smyth and Wang showed that R 2k+1 (z) has all of its non-real roots on the unit circle. The purpose of this paper is to study some variants of R 2k+1 (z), which also have many roots on the unit circle. Conjecture 1.1. Let B j denote the Bernoulli numbers, and let E j denote the Euler numbers. Suppose that k  2. The following polynomials have all of their The first author is supported by NSERC Discovery Grant 355412-2008 and a start-up grant from Université de Montréal. The second author is supported by NSF award DMS-0803107. 2010 Mathematics Subject Classification: primary: 26C10; secondary: 11B68