Pure and Applied Mathematics Journal
Volume 4, Issue 5-1, October 2015, Pages: 6-14

The Higher Derivation of the Hurwitz Zeta-function

Qianli Yang

Department of Mathematics and Information Science, Weinan Normal University, Weinan, P. R. China

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To cite this article:

Qianli Yang. The Higher Derivation of the Hurwitz Zeta-function. Pure and Applied Mathematics Journal. Special Issue: Mathematical Aspects of Engineering Disciplines. Vol. 4, No. 5-1, 2015, pp. 6-14. doi: 10.11648/j.pamj.s.2015040501.12


Abstract: In this paper, the Euler-Maclaurin Summation formula was researched, the purpose of research is to promote the application of the Hurwitz Zeta-function; Combination method of number theory special a function and Euler-Maclaurin Summation Formula was been used; By three derivatives of the the Euler-Maclaurin Summation formula, three formulas of Hurwitz Zeta-function were been given.

Keywords: Hurwitz Zeta-function, Euler Maclauring Summation, Logarithmic Derivative


1. Introduction and Main Results

In paper [1], many authors give many integral formulas of some Series by Zeta function, In paper [2], prefessor S. Kanemitsu exhibit the importance and usefulness of the Euler Maclaurn Summation formula by applying it to the sum.

In a Similar setting which appeared in the pursuit of the divisor problem [2] . In paper [3], Author give the formula  and many beautiful formulas. In paper [5] we gave the formula . In this paper, we can give  by an integral formula, then we give , and . The papers [8-10] also give some results about Zeta function.

We use the following notation.

Notation  -- the complex variable -- the gamma function (). -- the digamma funtion.

Both of which are meromorphically continued to the whole complex plane with simple poles at non-positive integers;

 -- Hurwitz Zeta function, , , the power taking the princial value -- the Riemann Zeta function, both of which are continued meromorphically over the complex plane with a simple pole at ,

 the generalized Bernoulli polynomial of degree  in , defined through the generating function satisfying the addition formula

(1)

([1],Formula(24),p.61)with the properties

 , , , where  and  are the th Bernoulli polynomial and the th Bernoulli number defined by (1) with .

 the th periodic Bernoulli polynomial.

Theorem1. for any  with , . we have

(2)

, ,

Corollary 1. For ,  and

. We have

(3)

Corollary 2. For ,  and

. We have

(4)

Corollary 3. For ,  and

. We have

(5)

2. Lemmas and Proof

Lemma 1[6]. Suppose that  is of the Class  in the closed interval .then

(6)

Lemma 2. Let . Then, for any  with , we have

(7)

Also the asymptotic formula

(8)

hods true as . On the other hand, formula (7) with  yields the integral representation

(9)

Which is true for all ,and  can be any natural number satisfying ; the integral being absolutely convergent in the region , where it is analytic except at .

Proof of Lemma 2.

Since the th derivative of  is

(10)

We derive from (6) that

(11)

For any natural number .

Now we note that the last integral is in (11) clearly  by the mean value theorem for partial integration. Thus, taking the limit as  the case when , we conclude that the constant term on the right-hand side of (11) must coincide with the left-hand side.

That is, (9) follows.

Then, by analytic continuation; (9) can be show  to hold true for all .

At the same time, this gives a generic definition of ;

(12)

Where  satisfies .

On the other hand, for . Formula (11) implies that the constant term is

,

Which must be equal to

,

Which we denote by .

Upon replacing the constant term by  in (11) gives (7), which then entails (8) on replacing the integral by the above estimate . this completes the proof of Lemma 2.

Lemma 3.[3] For any complex  and

(13)

Lemma 4.5 For any complex  and

(14)

Lemma 5.7 For any  and , we have

(15)

Which is the most convenient for most cases. but if , the right-hand side is table interpreted as .

3. The Proof of Theorem and Corollaries

First, we now go on to the proof Theorem.

From lemma 4 (14), when . Let

So . We give derivation

(16)

When .

Let ,

(17)

(18)

(19)

Put (17), (18) and (19) in (16), we can get the poof of theorem with

When  let

In Lemma 1 (6)

(20)

(21)

,

,

(22)

(23)

We can us the same why to get

(24)

Put (21),(22),(23) and (24) in (6), we get the proof with

We now complete the proof theorem.

Second, we go on to the Proof of Corollary 1.

Since those terms of the sum with  for

 have singularities at non-positive integers, we have to take the limit as the limit as  of

 or

as the case may be, on noting that other terms vanish because of simple zeros of  at non-positive integers.

By the fundamental difference equation satisfied by the gamma function we deduce for  that

(25)

We contend that

(26)

To prove this it suffices to note from (20) that in taking the limit

Only one term counts, , it is equal to

Which is  on recalling the fact

, which in fact follows from (25). Hence

(27)

and

Hence we conclude that

(28)

In order to deduce (3) from (28) we have recourse to Lemma 5 and its counterpart

for ([7,Formula(2.6)]). Substituting the formulas

and

in (28), we now complete the proof of Corollary 1.

Because proof method of Corollary 2 and Corollary 3 is similar. So, in here, we only give proof of Corollary 3.

From [4]

(29)

We any easily deduce that

(30)

So

,

,

, ,

, .

Let , ,

When

Then

We now complete the proof of Corollary 3.

Acknowledgement

Project of Science and Technology Agency of Shaanxi Province (No.2013JM1016)

Project of civil military integration research of Shaanxi Province ( No.15JMR19)


References

  1. H.M.Srivastava and J.Choi, Seies Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, Boston, and London,2001.
  2. Shigeru Kanemitsu and Haruo Tsukada, Vistas of Special Functions, World Scientific Publishing Co.Pte.Lid.2007
  3. H-L.Li and M.Toda,"Elaboration of Some Results of Srivastava and Choi". J.Anal.Appl 25(2006),517-533.
  4. Hailong Li,Number Theory and Special Functions, Science Press Beijing 2011".
  5. Yang Qian-Li, Wang Yong-Xing and Lihai-Long,On Derivate of a Number Theory and It’s Application Mathem Atics in Practice and Theory. Vol.35,No.8 (2005)194-199.
  6. Kanemitsu S, Kumagai H. and Srivastava HM. Some Integral and Asymptotic Formulas Associate with the Hurwitz Zate Function [J] Appl. Mathcomput,2004(1)31-37.
  7. Ishibashi,M. and Kanemitsu, S., Fractional Part Sums and Divisor Function. In: Number Theory and Combinatorics (Japan 1984; eds: J.Akiyama et al.). Singapore: World Sci.1985,pp.119-183.
  8. Adamchik, V. S., Polygamma functions of negative order. J. Comput. Appl.Math. 100 (1998), 191-199.
  9. Berndt, B. C., On the Hurwitz zeta-function. Rocky Mount. J. Math. 2 (1972),151-157.
  10. Elizalde, E., An asymptotic expansion for the derivative of the generalized Riemann zeta-function. Math. Comput. 47 (1986), 347 - 350.

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