A trio of Bernoulli relations, their implications for the Ramanujan polynomials and the special values of the Riemann zeta function

Abstract

We study the interplay between recurrences for zeta related functions at integer values, 'Minor Corner Lattice' Toeplitz determinants and integer composition based sums. Our investigations touch on functional identities due to Ramanujan and Grosswald, the transcendence of the zeta function at odd integer values, the Li Criterion for the Riemann Hypothesis and pseudocharacteristic polynomials for zeta related functions. We begin with a recent result for ζ(2s) and some seemingly new Bernoulli relations, which we use to obtain a generalised Ramanujan polynomial and properties thereof.I would like to thank Professor M N Huxley for all his invaluable support and guidance in this problem and Professor M W Coffey for his perceptive appraisal.

Bernoulli identities, zeta relations, determinant expressions, Mellin transforms, and representation of the Hurwitz numbers

“…On pp. 22 and 23 of [20], the following typographical errors occur. The upper index for the multinomial coefficient for the summations for η(2s), φ(2s), and θ(2s) in Lemma 3.3 should be t. In the display equation for the proof of Theorem 1.3, an "=" should be inserted after x 2s−2 .…”

“…2 On p. 17 of [20], (−1) k should be (−1) s−k in the summand on the right side of the second display equation, and vice versa for the summand of the right side of the third display equation. At the bottom of p. 8, Ψ should read Ψ n (twice).…”

“…In fact, the minor corner layered determinants of [19,20] are very special cases of lower Hessenberg determinants. Following the proof of Theorem 3, we recall a much more general determinantal result.…”

“…The minor corner layered determinants of [19,20] with superdiagonal of all 1's and comprised of at most 3 vectors are obviously special cases. Any time that a recurrence can be made to take the form just above, a determinant expression may then be written.…”

“…Remarks. We note that part (c) of the Theorem suffices to write the set of identities implied by (2.3) of [20], wherein [14] F…”