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Mathematics 2019, 7(4), 369; https://doi.org/10.3390/math7040369
A Probabilistic Proof for Representations of the Riemann Zeta Function
School of Statistics, Qufu Normal University, Shandong 273165, China
Author to whom correspondence should be addressed.
Received: 4 April 2019 / Accepted: 19 April 2019 / Published: 24 April 2019
In this paper, we present a different proof of the well known recurrence formula for the Riemann zeta function at positive even integers, the integral representations of the Riemann zeta function at positive integers and at fractional points by means of a probabilistic approach.
Keywords:Bernoulli numbers; (half-) logistic distribution; integral representation; probabilistic approach; Riemann zeta function
The well known Riemann zeta function is defined bywhich can be continued meromorphically to the whole complex -plane, except for a simple pole at , see [1,2,3] for details. Finding recurrence formulas and integral representations of the zeta function zeta function has become an important issue in complex analysis and number theory. One of the famous formulas is the following recursion formula for positive even integerswhere and is the th Bernoulli number. Here is the set of positive integers. Several new proofs to (1) can be found in [4,5,6,7]. A new parameterized series representation of zeta function is derived in . However, no similar closed-form representation of at odd integers or fractional points can be found in literature. The Riemann zeta function for positive odd integer arguments can be expressed by series and integrals. One possible integral expression is established by  as followswhere are Bernoulli polynomials defined by the generating function 
The Bernoulli numbers are well-tabulated (see, for example, ):
More lists of Bernoulli numbers and their estimation can be found in the recent work by Qi .
The zeta function has many integral representations, one of which is the following  (P.172) (note that there is an extra 2 in (51) of  (P.172):
The aim of this note is to present a new proof of (1) for and deduce the integral representations for and . The proofs are based on the characteristic function and the moment generating function of logistic, half-logistic and elliptical symmetric logistic distributions in probability theory and mathematical statistics.
2. The Main Results and Their Proofs
In this section we present a new proof to the following results by using a probabilistic method. To the best of our knowledge, the result (5) is new.
For Riemann’s zeta function, we haveandwhereis theth Bernoulli number.
To prove the proposition, we need the following three lemmas.
We assume that random variablehas the standard logistic distribution with the probability density function (pdf)Then the moment generating function (mgf) ofis given by
See Johnson et al.  (Equation (23.10)). ☐
As far as we are aware, the formulas for characteristic and moment generating functions given in the following two lemmas are new.
We assume that random variable X has the standard-dimensional elliptically symmetric logistic distribution with pdfwhereThen the characteristic function ofis given bywhereis the Riemann zeta function.
Using the Taylor expansionin (9) can be written as
Noting that , we only need to determine the even-order moments. For we getwhere we have used the fact that
For any , we get the characteristic function of by performing the following calculations
This ends the proof of Lemma 2. ☐
We assume that random variablehas the standard half-logistic distribution with the pdfThen the mgf ofis given bywhereis the Riemann zeta function.
The mean of is given byUsing the expansionwe get, for any positive integer ,Then we havewhere we have used the factThis completes the proof of Lemma 3. ☐
Proof of Proposition 1.
The mgf of the standard logistic distribution can be written assee, for example, . Comparing (7) and (10) yields , whereand
Using the series expansion (see e.g., )where is the th Bernoulli numbers, we havefrom which we deduce that
This completes the proof of (4).
Now we prove (5). Denoted byandwhere is defined by (7). Taking th and th derivatives of the two functions with respect to , we getand
Note that for any real , and thus for any real and any positive integers . In particular, . However, , and from we havewhich concludes the proof of (5).
Finally, we prove (6). Using (12) one has
Taking the th derivative of both sides of (14) with respect to and then setting yields the desired result. ☐
These authors contributed equally to this work.
The research was supported by the National Natural Science Foundation of China (No. 11571198)
The authors wish to acknowledge the comments and suggestions made by the anonymous referees that helped in improving this version of the paper. The research was supported by the National Natural Science Foundation of China (No. 11571198)
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
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