The Riemann zeta function \zeta (s) is defined by
\zeta (s):=\sum _{n=1}^{\mathrm{\infty}}\frac{1}{{n}^{s}}\phantom{\rule{1em}{0ex}}(\mathrm{\Re}(s)>1).
(1.1)
The Hurwitz (or generalized) zeta function \zeta (s,a) is defined by
\zeta (s,a):=\sum _{k=0}^{\mathrm{\infty}}{(k+a)}^{s}\phantom{\rule{1em}{0ex}}(\mathrm{\Re}(s)>1;a\in \mathbb{C}\setminus {\mathbb{Z}}_{0}^{}),
(1.2)
where ℂ and {\mathbb{Z}}_{0}^{} denote the sets of complex numbers and nonpositive integers, respectively. It is easy to see from the definitions (1.1) and (1.2) that
\zeta (s)=\zeta (s,1)={({2}^{s}1)}^{1}\zeta (s,\frac{1}{2})=1+\zeta (s,2).
(1.3)
It is noted that, in many different ways, the Riemann zeta function \zeta (s) and the Hurwitz zeta function \zeta (s,a) can be continued meromorphically to the whole complex splane having simple poles only at s=1 with their respective residues 1 at this point.
The polygamma functions {\psi}^{(n)}(s) (n\in \mathbb{N}) are defined by
{\psi}^{(n)}(s):=\frac{{\mathrm{d}}^{n+1}}{\mathrm{d}{z}^{n+1}}log\mathrm{\Gamma}(s)=\frac{{\mathrm{d}}^{n}}{\mathrm{d}{s}^{n}}\psi (s)\phantom{\rule{1em}{0ex}}(n\in {\mathbb{N}}_{0};s\in \mathbb{C}\setminus {\mathbb{Z}}_{0}^{}),
(1.4)
where ℕ denotes the set of positive integers and {\mathbb{N}}_{0}:=\mathbb{N}\cup \{0\}, and \mathrm{\Gamma}(s) is the familiar gamma function and psi (or digamma) function ψ is defined by
\psi (s):=\frac{\mathrm{d}}{\mathrm{d}s}log\mathrm{\Gamma}(s)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\psi}^{(0)}(s)=\psi (s).
(1.5)
Here is a wellknown (useful) relationship between the polygamma functions {\psi}^{(n)}(s) and the generalized zeta function \zeta (s,a):
It is also easy to have the following expression (cf. [[1], Eq. 1.2(54)] and [[2], Eq. 1.3(54)]):
{\psi}^{(n)}(s+m)={\psi}^{(n)}(s)+{(1)}^{n}n!\sum _{k=1}^{m}\frac{1}{{(s+k1)}^{n+1}}\phantom{\rule{1em}{0ex}}(m,n\in {\mathbb{N}}_{0}),
(1.7)
which, in view of (1.6), can be expressed in the following form:
\zeta (n,s+m)=\zeta (n,s)\sum _{k=1}^{m}\frac{1}{{(s+k1)}^{n}}\phantom{\rule{1em}{0ex}}(n\in \mathbb{N}\setminus \{1\};m\in {\mathbb{N}}_{0}),
(1.8)
where an empty sum is (elsewhere throughout this paper) understood to be nil.
The Riemann zeta function \zeta (s) in (1.1) plays a central role in the applications of complex analysis to number theory. The numbertheoretic properties of \zeta (s) are exhibited by the following result known as Euler’s formula, which gives a relationship between the set of primes and the set of positive integers:
\zeta (s)=\prod _{p}{(1{p}^{s})}^{1}\phantom{\rule{1em}{0ex}}(\mathrm{\Re}(s)>1),
(1.9)
where the product is taken over all primes. It is readily seen that \zeta (s)\ne 0 (\mathrm{\Re}(s)=\sigma \geqq 1), and the Riemann’s functional equation for \zeta (s)
\zeta (s)=2{(2\pi )}^{s1}\mathrm{\Gamma}(1s)sin\left(\frac{1}{2}\pi s\right)\zeta (1s)
(1.10)
shows that \zeta (s)\ne 0 (\sigma \leqq 0) except for the trivial zeros in
\zeta (2n)=0\phantom{\rule{1em}{0ex}}(n\in \mathbb{N}).
(1.11)
Furthermore, in view of the following known relation:
\zeta (s)=\frac{1}{1{2}^{1s}}\sum _{n=1}^{\mathrm{\infty}}\frac{{(1)}^{n1}}{{n}^{s}}\phantom{\rule{1em}{0ex}}(\mathrm{\Re}(s)>0;s\ne 1),
(1.12)
we find that \zeta (s)<0 (s\in \mathbb{R}; 0<s<1). The assertion that all the nontrivial zeros of \zeta (s) have real part \frac{1}{2} is popularly known as the Riemann hypothesis which was conjectured (but not proven) in the memoir of Riemann [3]. This hypothesis is still one of the most challenging mathematical problems today (see Edwards [4]), which was unanimously chosen to be one of the seven greatest unsolved mathematical puzzles of our time, the socalled millennium problems (see Devlin [5]).
Leonhard Euler (17071783), in 1735, computed the Basel problem:
1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+\cdots =\zeta (2)=\frac{{\pi}^{2}}{6}
(1.13)
to 20 decimal places with only a few terms of his powerful summation formula discovered in the early 1730s, now called the EulerMaclaurin summation formula. This probably convinced him that the sum in (1.13) equals {\pi}^{2}/6, which he proved in the same year 1735 (see [6]). Euler also proved
\zeta (2n)={(1)}^{n+1}\frac{{(2\pi )}^{2n}}{2(2n)!}{B}_{2n}\phantom{\rule{1em}{0ex}}(n\in {\mathbb{N}}_{0}),
(1.14)
where {B}_{n} are the Bernoulli numbers (see [[1], Section 1.6]; see also [[2], Section 1.7]). Subsequently, many authors have proved the Basel problem (1.13) and Eq. (1.14) in various ways (see, e.g., [7]).
We get no information about \zeta (2n+1) (n\in \mathbb{N}) from Riemann’s functional equation, since both members of (1.10) vanish upon setting s=2n+1 (n\in \mathbb{N}). In fact, until now no simple formula analogous to (1.14) is known for \zeta (2n+1) or even for any special case such as \zeta (3). It is not even known whether \zeta (2n+1) is rational or irrational, except that the irrationality of \zeta (3) was proved recently by Apéry [8]. But it is known that there are infinitely many \zeta (2n+1) which are irrational (see [9] and [10]).
The following formulae involving \zeta (2k+1) were given by Ramanujan (see [11]):

(i)
If k\in \mathbb{N}\setminus \{1\},
{\alpha}^{k}[\frac{1}{2}\zeta (12k)+\sum _{n=1}^{\mathrm{\infty}}\frac{{n}^{2k1}}{{e}^{2n\alpha}1}]={(\beta )}^{k}[\frac{1}{2}\zeta (12k)+\sum _{n=1}^{\mathrm{\infty}}\frac{{n}^{2k1}}{{e}^{2n\beta}1}];
(1.15)

(ii)
If k\in \mathbb{N},
\begin{array}{rcl}0& =& \frac{1}{{(4\alpha )}^{k}}[\frac{1}{2}\zeta (2k+1)+\sum _{n=1}^{\mathrm{\infty}}\frac{1}{{n}^{2k+1}({e}^{2n\alpha}1)}]\\ \frac{1}{{(4\beta )}^{k}}[\frac{1}{2}\zeta (2k+1)+\sum _{n=1}^{\mathrm{\infty}}\frac{1}{{n}^{2k+1}({e}^{2n\beta}1)}]\\ +\underset{j=0}{\overset{[\frac{k+1}{2}]}{{\sum}^{\mathrm{\prime}}}}\frac{{(1)}^{j}{\pi}^{2j}{B}_{2j}{B}_{2k2j+2}}{(2j)!(2k2j+2)!}[{\alpha}^{k2j+1}+{(\beta )}^{k2j+1}],\end{array}
(1.16)
where {B}_{j} is the j th Bernoulli number, \alpha >0 and \beta >0 satisfy \alpha \beta ={\pi}^{2}, and ∑^{′} means that when k is an odd number 2m1, the last term of the righthand side in (1.16) is taken as \frac{{(1)}^{m}{\pi}^{2m}{B}_{2m}^{2}}{{(m!)}^{2}}.
In 1928, Hardy [12] proved (1.15). In 1970, Grosswald [13] proved (1.16). In 1970, Grosswald [14] gave another expression of \zeta (2k+1). In 1983, Zhang [11] not only proved Ramanujan formulae (1.15) and (1.16), but also gave an explicit expression of \zeta (2k+1). For various series representations for \zeta (2n+1), see [15] and also see [[1], Chapter 4] and [[2], Chapter 4].
Many useful and interesting properties, identities, and relations for \zeta (s) and \zeta (s,a) have been developed, for example, the formulas recalled above. Here, we aim at presenting certain (presumably) new and (potentially) useful relationships among polygamma functions, Riemann zeta function, and generalized zeta function by mainly modifying Chen’s method [16]. We also give a double inequality approximating \zeta (2r+1) (r\in \mathbb{N}) by a more rapidly convergent series, that is, the Dirichlet lambda function \lambda (s) defined by
\lambda (s)=\sum _{k=1}^{\mathrm{\infty}}\frac{1}{{(2k1)}^{s}}\phantom{\rule{1em}{0ex}}(\mathrm{\Re}(s)>1)
(1.17)
(see Theorem 3).