Open Access

Integral representations of equally positive integer-indexed harmonic sums at infinity

Research in Number Theory20173:10

https://doi.org/10.1007/s40993-017-0074-x

Received: 14 November 2016

Accepted: 25 January 2017

Published: 10 May 2017

Abstract

We identify a partition-theoretic generalization of Riemann zeta function and the equally positive integer-indexed harmonic sums at infinity, to obtain the generating function and the integral representations of the latter. The special cases coincide with zeta values at positive integer arguments.

Keywords

Harmonic sum Integral representation Zeta value

1 Background

The harmonic sum of indices \(a_{1},\ldots ,a_{k}\in \mathbb {R}\backslash \left\{ 0\right\} \) is defined as (see [1, eq. 4, pp. 1])
$$\begin{aligned} S_{a_{1},\ldots ,a_{k}}\left( N\right) =\sum _{N\ge n_{1}\ge \cdots \ge n_{k}\ge 1}\frac{\text {sign}\left( a_{1}\right) ^{n_{1}}}{n_{1}^{\left| a_{1}\right| }}\times \cdots \times \frac{\text {sign}\left( a_{k}\right) ^{n_{k}}}{n_{k}^{\left| a_{k}\right| }}, \end{aligned}$$
which is naturally connected to the Riemann zeta function, by noting that \(N=\infty \), \(k=1\) and \(a_{1}>0\) gives \(S_{a_{1}}\left( \infty \right) =\zeta \left( a_{1}\right) \). A variety of the study can be found in the literature. For instance, Hoffman [4] established the connection between harmonic sums and multiple zeta values. We especially focus on the equally positively indexed harmonic sums, given by the case \(a_{1}=\cdots =a_{k}=a>0\)
$$\begin{aligned} S_{\varvec{a}_{k}}\left( N\right) :=S_{\underset{k}{\underbrace{a,\ldots ,a}}}\left( N\right) =\sum _{N\ge n_{1}\ge \cdots \ge n_{k}\ge 1}\frac{1}{\left( n_{1}\cdots n_{k}\right) ^{a}}, \end{aligned}$$
(1.1)
and also the equally positive integer-indexed harmonic sums (EPIIHS), namely \(a=m\in \mathbb {Z}_{>0}\). If \(N=\infty \), we additionally assume \(m\in \mathbb {Z}_{>1}\) for convergence.
Recently, Schneider [6] studied the generalized q-Pochhammer symbol and obtained [6, pp. 3]
$$\begin{aligned} \prod _{n\in X}\frac{1}{1-f\left( n\right) q^{n}}=\sum _{\lambda \in \mathcal {P}_{X}}q^{\left| \lambda \right| }\prod _{\lambda _{i}\in \lambda }f\left( \lambda _{i}\right) , \end{aligned}$$
(1.2)
where
  • \(X\subseteq \mathbb {\mathbb {Z}}_{>0}\) and \(f:\mathbb {\mathbb {Z}}_{>0}\longrightarrow \mathbb {C}\) such that if \(n\not \in X\) then \(f\left( n\right) =0\);

  • \(\mathcal {P}_{X}\) is the set of partitions into elements of X;

  • \(\lambda \vdash n\) means \(\lambda \) is a partition of n, the size \(|\lambda |\) is the sum of the parts of \(\lambda \), i.e., the number n being partitioned, and \(\lambda _{i}\in \lambda \) means \(\lambda _{i}\in \mathbb {Z}_{>0}\) is a part of partition \(\lambda \).

Further define \(l\left( \lambda \right) :=k\), \(n_{\lambda }:=\lambda _{1}\cdots \lambda _{k}\) and denote \(\mathcal {P}:=\mathcal {P}_{\mathbb {Z}_{>0}}\). Noting \(\lambda _{1}\ge \cdots \ge \lambda _{k}\ge 1\), a partition-theoretic generalization of Riemann zeta function [6, eq. 11, pp. 4] is defined and identified as
$$\begin{aligned} \zeta _{\mathcal {P}}\left( \left\{ a\right\} ^{k}\right) :=\sum _{l\left( \lambda \right) =k}\frac{1}{n_{\lambda }^{a}}=\sum _{\lambda _{1}\ge \cdots \ge \lambda _{k}\ge 1}\frac{1}{\lambda _{1}^{a}\cdots \lambda _{k}^{a}}=S_{\varvec{a}_{k}}\left( \infty \right) , \end{aligned}$$
(1.3)
which leads to the generating function and the integral representation of \(S_{\varvec{m}_{k}}\left( \infty \right) \), presented in the next section.

2 Main results

We first apply (1.2) to the case \(X=\left\{ 1,2,\ldots ,N\right\} \) and \(f\left( n\right) :=\frac{t^{a}}{n^{a}}\), obtaining
$$\begin{aligned} \prod _{n=1}^{N}\frac{1}{1-\frac{t^{a}}{n^{a}}q^{n}}=\sum _{\lambda \in \mathcal {P}_{X}}q^{\left| \lambda \right| }\prod _{\lambda _{i}\in \lambda }\frac{t^{a}}{\lambda _{i}^{a}}=\sum _{\lambda \in \mathcal {P}_{X}}q^{\left| \lambda \right| }\frac{t^{l\left( \lambda \right) a}}{n_{\lambda }^{a}}, \end{aligned}$$
which, by further letting \(q\rightarrow 1\), yields the following generating function.

Theorem 1

The generating function of \(S_{\varvec{a}_{k}}\left( N\right) \) is given by
$$\begin{aligned} \sum _{k=0}^{\infty }S_{\varvec{a}_{k}}\left( N\right) t^{ak}=\prod _{n=1}^{N}\frac{n^{a}}{n^{a}-t^{a}}. \end{aligned}$$
(2.1)

Remark 2

The special case for \(a=1\) is [2, eq. 9, pp. 1272]
$$\begin{aligned} \sum _{k=0}^{\infty }t^{k}S_{\varvec{1}_{k}}\left( N\right) =\frac{N!}{\left( 1-t\right) \cdots \left( N-t\right) }=N\cdot B\left( N,1-t\right) , \end{aligned}$$
(2.2)
involving the beta function B, defined by
$$\begin{aligned} B\left( x,y\right) :=\int _{0}^{1}z^{x-1}\left( 1-z\right) ^{y-1}dz=\frac{\Gamma \left( x\right) \Gamma \left( y\right) }{\Gamma \left( x+y\right) }, \end{aligned}$$
(2.3)
where the integral representation holds for \(\text {Re}\left( x\right) \), \(\text {Re}\left( y\right) >0\).

Corollary 3

For \(m\in \mathbb {Z}_{>1}\), denote \(\xi _{m}:=\exp \left( \frac{2\pi \text {i}}{m}\right) \) with \(\text {i}^{2}=-1\). Then,
$$\begin{aligned} \sum _{k=0}^{\infty }S_{\varvec{m}_{k}}\left( \infty \right) t^{mk}=\prod _{j=0}^{m-1}\Gamma \left( 1-\xi _{m}^{j}t\right) . \end{aligned}$$
(2.4)

Proof

From (2.1) and (2.2), we have
$$\begin{aligned} \sum _{k=0}^{\infty }S_{\varvec{m}_{k}}\left( N\right) t^{mk}=\prod _{n=1}^{N}\frac{n^{m}}{\left( n-\xi _{m}^{0}t\right) \cdots \left( n-\xi _{m}^{m-1}t\right) }=\prod _{j=0}^{m-1}N\cdot B\left( N,1-\xi _{m}^{j}t\right) . \end{aligned}$$
Then, apply the limit (see [7, pp. 254, ex. 5]) \(\Gamma \left( z\right) =\underset{N\rightarrow \infty }{\lim }N^{z}B\left( N,z\right) \) to \(z_{j}=1-\xi _{m}^{j}t\), \(j=0,\ldots ,m-1\), by noting \(\xi _{m}^{0}+\cdots +\xi _{m}^{m-1}=0\), to complete the proof. \(\square \)

Remark 4

An alternative proof can be given by letting \(N=\infty \) in (2.1) and applying [3, Thm. 1.1, pp. 547].

Remark 5

For general \(a>0\), we failed to obtain a closed form of \(\overset{\infty }{\underset{n=1}{\prod }}\frac{n^{a}}{n^{a}-t^{a}}.\)

Example 6

When \(m=2\), we apply (2.4) to get
$$\begin{aligned} B\left( 1+t,1-t\right) =\Gamma \left( 1+t\right) \Gamma \left( 1-t\right) =\sum _{k=0}^{\infty }S_{\varvec{2}_{k}}\left( \infty \right) t^{2k}. \end{aligned}$$
From the integral representation (2.3), we obtain (also see Remark 7)
$$\begin{aligned} B\left( 1+t,1-t\right) =\int _{0}^{1}z^{t}\left( 1-z\right) ^{-t}dz=\sum _{k=0}^{\infty }\frac{t^{k}}{k!}\int _{0}^{1}\log ^{k}\left( \frac{z}{1-z}\right) dz. \end{aligned}$$
(2.5)
Then it follows, by comparing coefficients of t,
$$\begin{aligned} S_{\varvec{2}_{k}}\left( \infty \right) =\frac{1}{\left( 2k\right) !}\int _{0}^{1}\log ^{2k}\left( \frac{z}{1-z}\right) dz. \end{aligned}$$
In particular, \(k=1\) yields
$$\begin{aligned} \frac{\pi ^{2}}{6}=\zeta \left( 2\right) =S_{2}\left( \infty \right) =\frac{1}{2}\int _{0}^{1}\log ^{2}\left( \frac{z}{1-z}\right) dz. \end{aligned}$$

Remark 7

We may interchange the integral and the sum of the series in (2.5), by restricting t to a closed compact set, e.g., \(\left[ -\frac{1}{2},\frac{1}{2}\right] \), satisfying \(\text {Re}\left( 1-t\right) \), \(\text {Re}\left( 1+t\right) >0\) as that in (2.3), in order to guarantee uniform convergence of the integral representation. (Similar discussion is omitted for the multiple beta function, defined next.)

Definition 8

The multiple beta function [5, Ch. 49] is defined as
$$\begin{aligned} B\left( \alpha _{1},\ldots ,\alpha _{m}\right) :=\frac{\Gamma \left( \alpha _{1}\right) \cdots \Gamma \left( \alpha _{m}\right) }{\Gamma \left( \alpha _{1}+\cdots +\alpha _{m}\right) }=\int _{\Omega _{m}}\prod _{i=1}^{m}x_{i}^{\alpha _{i}-1}d\mathbf {x}, \end{aligned}$$
(2.6)
where \(\Omega _{m}=\left\{ \left( x_{1},\ldots ,x_{m}\right) \in \mathbb {R}_{>0}^{m}:x_{1}+\cdots +x_{m-1}<1,\ x_{1}+\cdots +x_{m}=1\right\} \) and the integral representation requires \(\text {Re}\left( \alpha _{1}\right) ,\ldots ,\text {Re}\left( \alpha _{m}\right) >0\).
Following the same idea as that in Example 6, we first have, from (2.4),
$$\begin{aligned} B\left( 1-\xi _{m}^{0}t,\ldots ,1-\xi _{m}^{m-1}t\right) =\frac{1}{\left( m-1\right) !}\sum _{k=0}^{\infty }S_{\varvec{m}_{k}}\left( \infty \right) t^{mk}. \end{aligned}$$
Then, apply the integral representation (2.6), expand the integrand as a power series in t, and compare coefficients of t, to obtain the following integral representation.

Theorem 9

For all \(m,\,k\in \mathbb {Z}_{>0}\) with \(m\ge 2\),
$$\begin{aligned} S_{\varvec{m}_{k}}\left( \infty \right) =\frac{\left( -1\right) ^{mk}\left( m-1\right) !}{\left( mk\right) !}\int _{\Omega _{m}}\log ^{mk}\left( \prod _{j=0}^{m-1}x_{j+1}^{\xi _{m}^{j}}\right) d\mathbf {x}. \end{aligned}$$

Corollary 10

In particular, the case \(k=1\) implies for integer \(m\in \mathbb {Z}_{>1}\) that
$$\begin{aligned} \zeta \left( m\right) =\frac{\left( -1\right) ^{m}}{m}\int _{\Omega _{m}}\log ^{m}\left( \prod _{j=0}^{m-1}x_{j+1}^{\xi _{m}^{j}}\right) d\mathbf {x}, \end{aligned}$$
or alternatively
$$\begin{aligned} \zeta \left( m\right)= & {} \frac{\left( -1\right) ^{m}}{m}\int _{0}^{1}\int _{0}^{1-x_{1}}\\&\cdots \int _{0}^{1-x_{1}-\cdots -x_{m-2}}\log ^{m}\left( x_{1}^{\xi _{m}^{0}}\cdots x_{m-1}^{\xi _{m}^{m-2}}\left( 1-x_{1}-\cdots -x_{m-1}\right) ^{\xi _{m}^{m-1}}\right) \\&\times dx_{m-1}\cdots dx_{1}. \end{aligned}$$

Acknowledgements

The author would like to thank Dr. Jakob Ablinger for his help on harmonic sums; Prof. Johannes Blümlein for his handwritten notes on the proof of (2.2); and especially his mentors, Prof. Peter Paule and Prof. Carsten Schneider, for their valuable suggestions. This work is supported by the Austrian Science Fund (FWF) Grant SFB F50 (F5006-N15 and F5009-N15).

Declarations

Open Access

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Authors’ Affiliations

(1)
Research Institute for Symbolic Computation, Johannes Kepler University

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