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