- Open Access
Certain relationships among polygamma functions, Riemann zeta function and generalized zeta function
© Choi and Chen; licensee Springer 2013
- Received: 4 September 2012
- Accepted: 9 January 2013
- Published: 28 February 2013
Many useful and interesting properties, identities, and relations for the Riemann zeta function and the Hurwitz zeta function 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 by a more rapidly convergent series.
MSC:11M06, 33B15, 40A05, 26D07.
- Riemann zeta function
- Hurwitz zeta function
- gamma function
- polygamma functions
- Bernoulli numbers
- multiple Hurwitz zeta function
- Stirling numbers of the first kind
- asymptotic formulas
It is noted that, in many different ways, the Riemann zeta function and the Hurwitz zeta function can be continued meromorphically to the whole complex s-plane having simple poles only at with their respective residues 1 at this point.
where an empty sum is (elsewhere throughout this paper) understood to be nil.
we find that (; ). The assertion that all the non-trivial zeros of have real part is popularly known as the Riemann hypothesis which was conjectured (but not proven) in the memoir of Riemann . This hypothesis is still one of the most challenging mathematical problems today (see Edwards ), which was unanimously chosen to be one of the seven greatest unsolved mathematical puzzles of our time, the so-called millennium problems (see Devlin ).
We get no information about () from Riemann’s functional equation, since both members of (1.10) vanish upon setting (). In fact, until now no simple formula analogous to (1.14) is known for or even for any special case such as . It is not even known whether is rational or irrational, except that the irrationality of was proved recently by Apéry . But it is known that there are infinitely many which are irrational (see  and ).
- (i)If ,(1.15)
- (ii)If ,(1.16)
where is the j th Bernoulli number, and satisfy , and ∑′ means that when k is an odd number , the last term of the right-hand side in (1.16) is taken as .
In 1928, Hardy  proved (1.15). In 1970, Grosswald  proved (1.16). In 1970, Grosswald  gave another expression of . In 1983, Zhang  not only proved Ramanujan formulae (1.15) and (1.16), but also gave an explicit expression of . For various series representations for , see  and also see [, Chapter 4] and [, Chapter 4].
(see Theorem 3).
We first prove a relationship between polygamma functions and Riemann zeta functions asserted by Theorem 1.
which is equal to the right-hand side of (2.2). Hence, by the principle of mathematical induction, (2.1) is true for all . □
Next we give an interesting series representation for in terms of the generalized zeta functions ().
Now it is easy to find from (1.2) and (1.3) that (2.5) equals (2.4). □
We present a relationship between and asserted by the following theorem.
which is equal to the right-hand side of (2.6) replaced n by . Therefore, by the principle of mathematical induction, (2.6) holds true for all . □
We prove certain inequalities for affirmed by Theorem 3.
and are Bernoulli numbers.
Upon substituting from (2.9) into (2.10), we obtain the desired result. □
and taking the limit on each side of (3.1) as , we also get (2.4).
We find from (1.3) that the special cases ( and ) 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  and ). 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., [, Chapter 3], [, Chapter 3] and references therein).
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.
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).
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