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

M. C. Lettington (2013)

Acta Arithmetica

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 pseudo-characteristic polynomials for zeta related functions. We begin with a recent result for ζ(2s) and some seemingly new Bernoulli relations,...

An Alternative Form of the Functional Equation for Riemann’s Zeta Function, II

Andrea Ossicini (2014)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

This paper treats about one of the most remarkable achievements by Riemann, that is the symmetric form of the functional equation for ζ ( s ) . We present here, after showing the first proof of Riemann, a new, simple and direct proof of the symmetric form of the functional equation for both the Eulerian Zeta function and the alternating Zeta function, connected with odd numbers. A proof that Euler himself could have arranged with a little step at the end of his paper “Remarques sur un beau rapport entre...

Arithmetic theory of harmonic numbers (II)

Zhi-Wei Sun, Li-Lu Zhao (2013)

Colloquium Mathematicae

For k = 1,2,... let H k denote the harmonic number ∑ j = 1 k 1 / j . In this paper we establish some new congruences involving harmonic numbers. For example, we show that for any prime p > 3 we have ∑ k = 1 p - 1 ( H k ) / ( k 2 k ) ≡ 7 / 24 p B p - 3 ( m o d p ² ) , ∑ k = 1 p - 1 ( H k , 2 ) / ( k 2 k ) ≡ - 3 / 8 B p - 3 ( m o d p ) , and ∑ k = 1 p - 1 ( H ² k , 2 n ) / ( k 2 n ) ≡ ( 6 n + 1 2 n - 1 + n ) / ( 6 n + 1 ) p B p - 1 - 6 n ( m o d p ² ) for any positive integer n < (p-1)/6, where B₀,B₁,B₂,... are Bernoulli numbers, and H k , m : = ∑ j = 1 k 1 / ( j m ) .

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