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

Research in Number Theory20173:10

  • Received: 14 November 2016
  • Accepted: 25 January 2017
  • Published:


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.


  • 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 \(a_{1}=\cdots =a_{k}=a>0\)
$$\begin{aligned} S_{\varvec{a}_{k}}\left( N\right) :=S_{\underset{k}{\underbrace{a,\ldots ,a}}}\left( N\right) =\sum _{N\ge n_{1}\ge \cdots \ge n_{k}\ge 1}\frac{1}{\left( n_{1}\cdots n_{k}\right) ^{a}}, \end{aligned}$$
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]
$$\begin{aligned} \prod _{n\in X}\frac{1}{1-f\left( n\right) q^{n}}=\sum _{\lambda \in \mathcal {P}_{X}}q^{\left| \lambda \right| }\prod _{\lambda _{i}\in \lambda }f\left( \lambda _{i}\right) , \end{aligned}$$
  • \(X\subseteq \mathbb {\mathbb {Z}}_{>0}\) 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
$$\begin{aligned} \zeta _{\mathcal {P}}\left( \left\{ a\right\} ^{k}\right) :=\sum _{l\left( \lambda \right) =k}\frac{1}{n_{\lambda }^{a}}=\sum _{\lambda _{1}\ge \cdots \ge \lambda _{k}\ge 1}\frac{1}{\lambda _{1}^{a}\cdots \lambda _{k}^{a}}=S_{\varvec{a}_{k}}\left( \infty \right) , \end{aligned}$$
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
$$\begin{aligned} \sum _{k=0}^{\infty }S_{\varvec{a}_{k}}\left( N\right) t^{ak}=\prod _{n=1}^{N}\frac{n^{a}}{n^{a}-t^{a}}. \end{aligned}$$

Remark 2

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


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)
$$\begin{aligned} B\left( 1+t,1-t\right) =\int _{0}^{1}z^{t}\left( 1-z\right) ^{-t}dz=\sum _{k=0}^{\infty }\frac{t^{k}}{k!}\int _{0}^{1}\log ^{k}\left( \frac{z}{1-z}\right) dz. \end{aligned}$$
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
$$\begin{aligned} B\left( \alpha _{1},\ldots ,\alpha _{m}\right) :=\frac{\Gamma \left( \alpha _{1}\right) \cdots \Gamma \left( \alpha _{m}\right) }{\Gamma \left( \alpha _{1}+\cdots +\alpha _{m}\right) }=\int _{\Omega _{m}}\prod _{i=1}^{m}x_{i}^{\alpha _{i}-1}d\mathbf {x}, \end{aligned}$$
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
$$\begin{aligned} \zeta \left( m\right)= & {} \frac{\left( -1\right) ^{m}}{m}\int _{0}^{1}\int _{0}^{1-x_{1}}\\&\cdots \int _{0}^{1-x_{1}-\cdots -x_{m-2}}\log ^{m}\left( x_{1}^{\xi _{m}^{0}}\cdots x_{m-1}^{\xi _{m}^{m-2}}\left( 1-x_{1}-\cdots -x_{m-1}\right) ^{\xi _{m}^{m-1}}\right) \\&\times dx_{m-1}\cdots dx_{1}. \end{aligned}$$


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).


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Authors’ Affiliations

Research Institute for Symbolic Computation, Johannes Kepler University, 4040 Linz, Austria


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