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# Certain relationships among polygamma functions, Riemann zeta function and generalized zeta function

- Junesang Choi
^{1}Email author and - Chao-Ping Chen
^{2}

**2013**:75

https://doi.org/10.1186/1029-242X-2013-75

© Choi and Chen; licensee Springer 2013

**Received:**4 September 2012**Accepted:**9 January 2013**Published:**28 February 2013

## Abstract

Many useful and interesting properties, identities, and relations for the Riemann zeta function $\zeta (s)$ and the Hurwitz zeta function $\zeta (s,a)$ have been developed. Here, we aim at giving certain (presumably) new and (potentially) useful relationships among polygamma functions, Riemann zeta function, and generalized zeta function by modifying Chen’s method. We also present a double inequality approximating $\zeta (2r+1)$ by a more rapidly convergent series.

**MSC:**11M06, 33B15, 40A05, 26D07.

## Keywords

- Riemann zeta function
- Hurwitz zeta function
- gamma function
- psi-function
- polygamma functions
- Bernoulli numbers
- multiple Hurwitz zeta function
- Stirling numbers of the first kind
- asymptotic formulas

## 1 Introduction

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 *s*-plane having simple poles only at $s=1$ with their respective residues 1 at this point.

*ψ*is defined by

*cf.*[[1], Eq. 1.2(54)] and [[2], Eq. 1.3(54)]):

where an empty sum is (elsewhere throughout this paper) understood to be nil.

*Euler’s formula*, which gives a relationship between the set of primes and the set of positive integers:

*Riemann’s functional equation*for $\zeta (s)$

we find that $\zeta (s)<0$ ($s\in \mathbb{R}$; $0<s<1$). The assertion that all the *non-trivial* 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 so-called millennium problems (see Devlin [5]).

*Basel problem*:

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]).

- (i)If $k\in \mathbb{N}\setminus \{1\}$,${\alpha}^{k}[\frac{1}{2}\zeta (1-2k)+\sum _{n=1}^{\mathrm{\infty}}\frac{{n}^{2k-1}}{{e}^{2n\alpha}-1}]={(-\beta )}^{k}[\frac{1}{2}\zeta (1-2k)+\sum _{n=1}^{\mathrm{\infty}}\frac{{n}^{2k-1}}{{e}^{2n\beta}-1}];$(1.15)
- (ii)If $k\in \mathbb{N}$,$\begin{array}{rcl}0& =& \frac{1}{{(4\alpha )}^{k}}[\frac{1}{2}\zeta (2k+1)+\sum _{n=1}^{\mathrm{\infty}}\frac{1}{{n}^{2k+1}({e}^{2n\alpha}-1)}]\\ -\frac{1}{{(-4\beta )}^{k}}[\frac{1}{2}\zeta (2k+1)+\sum _{n=1}^{\mathrm{\infty}}\frac{1}{{n}^{2k+1}({e}^{2n\beta}-1)}]\\ +\underset{j=0}{\overset{[\frac{k+1}{2}]}{{\sum}^{\mathrm{\prime}}}}\frac{{(-1)}^{j}{\pi}^{2j}{B}_{2j}{B}_{2k-2j+2}}{(2j)!(2k-2j+2)!}[{\alpha}^{k-2j+1}+{(-\beta )}^{k-2j+1}],\end{array}$(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 $2m-1$, the last term of the right-hand 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].

(see Theorem 3).

## 2 Main results

We first prove a relationship between polygamma functions and Riemann zeta functions asserted by Theorem 1.

**Theorem 1**

*For each*$r\in \mathbb{N}$,

*the following formula holds true*:

*Proof*We prove Eq. (2.1) by using the principle of mathematical induction on $n\in \mathbb{N}$. For $n=1$, in view of Eq. (1.6), it is found that both sides of (2.1) equal $1-\zeta (2r+1)$. Assume that (2.1) holds true for some $n\in \mathbb{N}$. Then we will show that (2.1) is true for $n+1$,

*i.e.*,

which is equal to the right-hand side of (2.2). Hence, by the principle of mathematical induction, (2.1) is true for all $n\in \mathbb{N}$. □

Next we give an interesting series representation for $\zeta (2r)$ in terms of the generalized zeta functions $\zeta (2r+1,j+1)$ ($j\in {\mathbb{N}}_{0}$).

**Theorem 2**

*The following formula holds true*:

*Proof*Applying the series representation (1.6) for ${\psi}^{(n)}(s)$ to the left-hand side of (2.1), we have

Now it is easy to find from (1.2) and (1.3) that (2.5) equals (2.4). □

We present a relationship between $\zeta (2r+1)$ and ${\psi}^{(2r)}(n+\frac{1}{2})$ asserted by the following theorem.

**Lemma**

*For each*$r\in \mathbb{N}$,

*the following formula holds true*:

*Proof*We prove Eq. (2.6) by using the principle of mathematical induction on $n\in \mathbb{N}$. When $n=1$, the left-hand side of (2.6) is clearly 1, and the right-hand side of (2.6) is seen to be 1 by using (1.6) and (1.3). Assume that (2.6) holds true for

*some*$n\in \mathbb{N}$. Then, by the induction hypothesis and using (1.7), we find

which is equal to the right-hand side of (2.6) replaced *n* by $n+1$. Therefore, by the principle of mathematical induction, (2.6) holds true for *all* $n\in \mathbb{N}$. □

We prove certain inequalities for $\zeta (2r+1)$ affirmed by Theorem 3.

**Theorem 3**

*Let*$r,n\in \mathbb{N}$

*and*$N\in {\mathbb{N}}_{0}$.

*Then the following inequalities hold true*:

and ${B}_{k}$ are Bernoulli numbers.

Upon substituting from (2.9) into (2.10), we obtain the desired result. □

## 3 Further observations and remarks

and taking the limit on each side of (3.1) as $n\to \mathrm{\infty}$, we also get (2.4).

*e.g.*, [19, 20], [[1], pp.85-86] and [[2], pp.151-153]): The

*n*-

*ple*(or, simply, the

*multiple*)

*Hurwitz zeta function*${\zeta}_{n}(s,a)$ is defined by

*a*:

*Stirling numbers of the first kind*which satisfy the following recurrence relations:

*n*:

We find from (1.3) that the special cases ($s=2r+1$ and $a=1$) of the first and second equations in (3.11) yield immediately (3.2) and (3.4), respectively.

An interesting historical introduction to the remarkably widely and extensively investigated subject of *closed-form* evaluation of series involving the zeta functions was presented (see [1] and [2]). A considerably large number of formulas have been derived, by using various methods and techniques, in the vast literature on this subject (see, *e.g.*, [[1], Chapter 3], [[2], Chapter 3] and references therein).

*cf.*, (1.8)):

and the relation in (1.6). But we record it in the form of a lemma in order to use it in the proof of Theorem 3.

## Declarations

### Acknowledgements

The first author is supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (2012-0002957).

## Authors’ Affiliations

## References

- Srivastava HM, Choi J:
*Series Associated with the Zeta and Related Functions*. Kluwer Academic, Dordrecht; 2001.MATHView ArticleGoogle Scholar - Srivastava HM, Choi J:
*Zeta and q-Zeta Functions and Associated Series and Integrals*. Elsevier, Amsterdam; 2012.MATHGoogle Scholar - Riemann B: Über die Anzahl der Primzahlen unter einer gegebenen Grösse.
*Monatsber. Akad. Berlin*1859, 1859: 671–680.Google Scholar - Edwards HM:
*Riemann’s Zeta Function*. Academic Press, New York; 1974. Dover Publications, Mineola, New York (2001)MATHGoogle Scholar - Devlin K:
*The Millennium Problems*. Basic Books, New York; 2002.MATHGoogle Scholar - Pengelley DJ: Dances between continuous and discrete: Euler’s summation formula. In
*Proceedings, Euler $2k+2$2k+2 Conference*Edited by: Bradley R, Sandifer E. 2003.Google Scholar - Tsumura H:An elementary proof of Euler’s formula for $\zeta (2m)$.
*Am. Math. Mon.*2004, 111: 430–431. 10.2307/4145270MATHMathSciNetView ArticleGoogle Scholar - Apéry R:Irrationalité de $\zeta (2)$ et $\zeta (3)$. 61. In
*Journées Arithmétiques de Luminy*. Soc. Math. France, Paris; 1979:11–13. (Colloq. Internat. CNRS, Centre Univ. Luminy, Luminy, 1978)Google Scholar - Rivoal T: La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs.
*C. R. Acad. Sci. Paris*2000, 331: 267–270. Série I 10.1016/S0764-4442(00)01624-4MATHMathSciNetView ArticleGoogle Scholar - Zudilin W:One of the numbers $\zeta (5)$, $\zeta (7)$, $\zeta (9)$, $\zeta (11)$ is irrational.
*Russ. Math. Surv.*2001, 56: 774–776. 10.1070/RM2001v056n04ABEH000427MATHMathSciNetView ArticleGoogle Scholar - Zhang N-Y: Ramanujan’s formulas and the values of the Riemann zeta function at odd positive integers.
*Adv. Math.*1983, 12: 61–71. (Chinese)MATHGoogle Scholar - Hardy GH: A formula of Ramanujan.
*J. Lond. Math. Soc.*1928, 3: 238–240. 10.1112/jlms/s1-3.3.238MATHView ArticleGoogle Scholar - Grosswald E: Comments on some formulae of Ramnujan.
*Acta Arith.*1972, 21: 25–34.MATHMathSciNetGoogle Scholar - Grosswald E: Die Werte der Riemannschen Zeta Function an ungeraden Argumenstellen.
*Gött. Nachr.*1970, 1970: 9–13.MATHMathSciNetGoogle Scholar - Dancs MJ, He T-X:An Euler-type formula for $\zeta (2k+1)$.
*J. Number Theory*2006, 118: 192–199. 10.1016/j.jnt.2005.09.005MATHMathSciNetView ArticleGoogle Scholar - Chen, C-P: Inequalities and series formula for ζ ( 3 ) . PreprintGoogle Scholar
- Abramowitz M, Stegun IA (Eds): National Bureau of Standards, Applied Mathematics Series 55 In Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. 10th edition. National Bureau of Standards, Washington; 1972. Reprinted by Dover Publications, New York (1965)Google Scholar
- Allasia G, Giordano C, Pećarić J: Inequalities for the gamma function relating to asymptotic expansions.
*Math. Inequal. Appl.*2002, 5(3):543–555.MATHMathSciNetGoogle Scholar - Choi J: Explicit formulas for the Bernoulli polynomials of order
*n*.*Indian J. Pure Appl. Math.*1996, 27: 667–674.MATHMathSciNetGoogle Scholar - Choi J, Srivastava HM: The multiple Hurwitz Zeta function and the multiple Hurwitz-Euler eta function.
*Taiwan. J. Math.*2011, 15: 501–522.MATHMathSciNetGoogle Scholar

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