# New Zero-Free Regions for Hypergeometric Zeta and Fractional Hypergeometric Zeta Functions

Authors: Demessie Ergabus Birmechu; Hunduma Legesse Geleta
Journal: Journal of Mathematics
 (2022)

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## Abstract

We describe and demonstrate zero-free regions for the families called “hypergeometric zeta functions” and “fractional hypergeometric zeta functions”. These zero-free regions are vertical strips in the right half of the complex plane. In the proofs, we utilize positivity results on the oscillatory integrals and monotonicity of real-valued functions to demonstrate these zero-free regions for both families.


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## Body

## 1. Introduction

Many papers have been devoted to the study of the locations of zeros and zero-free regions of zeta function [1–3] and its generalizations such as the Hurwitz zeta function [4–6], multiple zeta functions [7], hypergeometric zeta functions [8], and fractional hypergeometric zeta functions [9, 10]. The families of hypergeometric zeta functions and fractional hypergeometric zeta functions are known only in their integral representations as generalizations of the classical Riemann zeta function via integral representation. A study of these families of zeta functions was initiated in [10, 11], respectively. In the same papers [10, 11], the authors discovered that both the hypergeometric zeta functions of order N, ζNs and the fractional hypergeometric zeta functions of order a, ζas can be continued analytically to the whole complex plane, except for finite simple poles on the real axis at 1,0,…,2−N for the hypergeometric zeta functions and except for infinite simple poles on the real axis at 1,0,−1,−2,… for fractional hypergeometric zeta functions. It was also discovered in [10] that ζas has also simple zeros on the real axis at 1−a,−a,−1+a,−2+a,… where a is a fixed positive real number. These zeros are called the trivial zeros for fractional hypergeometric zeta function. Regarding these families, the following open problems were raised in [8–10] as follows:(1)
Do the families admit the Dirichlet series?(2)
Do the families admit functional equation?(3)
Do they admit product formula?(4)
Do they have nontrivial zeros?(5)
Do they have zero-free regions?Concerning zero-free regions for hypergeometric zeta function of order 2, some results were known, and some problems were left open, see [8, 9].In this paper, we investigate zero-free regions for the families of hypergeometric zeta functions of orderN denoted by ζNs, where N is a positive integer, and the families of fractional hypergeometric zeta functions of order a denoted by ζas, where a is a positive real number, which is defined by the integral representations, respectively, as follows:(1)ζNs=1Γs+N−1∫0∞xs+N−2ex−TN−1xdx.Forℜs>1, where TNx=∑n=0Nxn/n! is the Taylor (Maclaurin) polynomial of the exponential function ex and(2)ζas=Γa+1Γs+a−1∫0∞xs+a−2e−xaγa,xdx.Forℜs>1 and for all positive real number a, where Γs is the Gamma function defined by(3)Γs=∫0∞xs−1e−xdx.Forℜs>0 and γa,x is the lower incomplete Gamma function of the form(4)γa,x=∫0xe−tta−1dt.We observe that ifa is a natural number, both families are the same, and in particular, if N=a=1, the families represent the classical Riemann zeta function.Finding zeros and zero-free regions for these families becomes difficult as compared to the classical Riemann zeta functions which are known in different representations, such as Dirichlet series, Euler product formula, and functional equations. In the absence of such different representations, finding zeros and zero-free regions lie entirely on their integral representations. Therefore, in the present paper, we describe and demonstrate zero-free regions forζNs for fixed but arbitrary natural number N and ζas for fixed but arbitrary a such that 0<a<1 in the right half of the complex plane by utilizing positivity results on oscillatory integrals and monotonicity of real-valued functions.Our main results are the following.Theorem 1.
For fixed natural numberN, let VN=s=σ+it∈ℂ:1≤σ<2. Then, ζNs≠0 on VN.Theorem 2.
For fixed positive real numbera with 0<a<1, let Va=s=σ+it∈ℂ:1≤σ<2−a. Then, ζas≠0 on Va.The structure of the present work is as follows. In Section2, we review some of the main results obtained so far regarding zeta function, hypergeometric zeta functions, fractional hypergeometric zeta functions, and positivity results on oscillatory integrals and monotonicity of real-valued functions. In Section 3, we reveal and prove our main results and demonstrate that both ζNs and ζas are zero free on the right half-plane in vertical strips VN and Va, respectively. In Section 4, we give some concluding remarks and discussion.


## 2. Preliminaries

As our work is a continuation of the hypergeometric zeta functions [11] and fractional hypergeometric zeta functions [10], first, we review some basic results concerning these families of zeta functions, and second, we review positivity results on oscillatory integrals and monotonicity of real-valued functions, in order to state and prove our results in perspective.


### 2.1. An Overview of Some Families of Zeta Functions

Definition 1.
The Riemann zeta functionζs is a function of a complex variable s=σ+it defined as an infinite series (Dirichlet series) given, convergent for all σ>1,(5)ζs=∑n=1∞1ns.Forσ>1, the classical zeta function ζs is also defined by the integral given(6)ζs=1Γs∫0∞xs−1ex−1dx,where Γs=∫0∞xs−1e−xdx is the Gamma function. For σ>1, ζs is also defined by(7)ζs=∏p1−p−s−1,where the product runs over all prime numbers p. This product formula is called Euler’s product formula for the Riemann’s zeta function. Since a convergent infinite product of nonzero factors is not zero, the zeta function does not vanish in the right half of the complex plane. Therefore, this product formula is one of the important tools to show that the Riemann zeta function is zero free in the right half of the complex plane (actually for σ>1).The celebrated functional equation of the Riemann zeta function is also another important representation to locate the zeros of the zeta function on the left half of the complex plane. It is given by(8)ζs=22πs−1ζ1−sΓ1−ssinπs2.Definition 2.
The pointss=−2,−4,−6,… are called the “trivial” zeros of the zeta function ζs, and the strip s∈ℂ:0≤σ≤1 is called the critical strip.Regarding the zeros inside the critical strip, it is conjectured that the nontrivial zeros all must be located on the “critical line”ℜs=1/2. This conjecture is known as Riemann’s hypothesis. As a generalization of the Riemann zeta function ζs via integral representation, we have the following definition.Definition 3.
(see [11], Definition 2.1). Let N be a natural number and TNx=∑n=0Nxn/n! be the Taylor (Maclaurin) polynomial of the exponential function ex. For σ>1, the Nth order hypergeometric zeta functions ζNs is defined as(9)ζNs=1Γs+N−1∫0∞xs+N−2ex−TN−1xdx.Observe that whenN=1, we get the classical zeta function ζs. It has been shown in [11] that the hypergeometric zeta function ζNs can be extended analytically to the entire complex plane, except for N simple poles at s=1,0,−1,…,2−N. The hypergeometric zeta functions have zeros on the negative real axis, and these zeros are called the trivial zeros for the hypergeometric zeta functions.Concerning zero-free regions for hypergeometric zeta function of order 2, the following results are known.Theorem 3.
(see [8], Theorem 1.1). ζ2s has no zeros in the left-half complex plane s=σ+it∈ℂ:σ<σ2, except for infinitely many “trivial” zeros on the negative real axis, one in each of the intervals(10)Sm=σm+1,σm,where m≥2 is a positive integer, zk=rkeiθk are the nonzero roots of ez−TN−1x and σm=1−π/π−θ1m.Theorem 4.
(see [9], Theorem 3.4.2 and Corollary 3.4.3). Let s=σ+it with σ>1 and C>1. Then, ζ2s≠0 for s<Cσ.Definition 4.
(see [10], Definition 2.1). The fractional hypergeometric zeta function ζas is defined for all positive real number a and σ>1 as(11)ζas=Γa+1Γs+a−1∫0∞xs+a−2e−xaγa,xdx,where Γs is the Gamma function defined by(12)Γs=∫0∞xs−1e−xdx,and γa,x is the lower incomplete Gamma function of the form(13)γa,x=∫0xe−tta−1dt.Observe that whena=N, natural number, we get the Hypergeometric zeta functions ζNs.


### 2.2. Positivity Properties of Integrals

Proposition 1.
(see [1], Proposition 3.1). Let the function hr≥0 on 0,∞,h∈Lloc10,∞,h decreasing and strictly decreasing on some open subinterval of the form kπ/t,k+1π/t for some nonnegative integer k, and t positive real number. Then,(14)∫0∞hrsintrdr>0.
Moreover,(15)∫0Thrsintrdr>0.
For anyT>0 provided, h satisfies the above assumptions with 0,∞ replaced by 0,T.Corollary 1.
(see [1], Corollary 3.2). Let t be a positive real number and x˜t,k≥1 be such that tlnx˜t,k=2πk, for some positive integer k. Let h≥0 on x˜t,k,∞,h∈Lloc10,∞,h decreasing, strictly decreasing on some interval of the form jπ/t,j+1π/t for some positive integer j such that jπ/t≥x˜t,k. Then,(16)∫x˜t,k∞hrsintrdr>0.
Moreover,(17)∫x˜t,kThrsintrdr>0.
For someT>x˜t,k if T>j+1π/t≥x˜t,k.Remark 1.
Givent>0, we can take a y=e2π/t≥1 such that(18)∫lny∞hrsintrdr>0,where h≥0 on lny,∞,h∈Lloc10,∞,h decreasing, strictly decreasing on some interval of the form jπ/t,j+1π/t for some j∈ℕ such that jπ/t≥x˜t,k.Corollary 2.
(see [1], Corollary 3.3). Let t>0 and x˜t,k≥1 be such that tlnx˜t,k=2πk, for some positive integer k and let g≥0 on x˜t,k,∞,g∈Lloc1x˜t,k,∞ such that x↦xgx is decreasing on x˜t,k,∞, strictly decreasing on lnjπ/t,lnj+1π/t for some positive integer j. Then,(19)∫x˜t,k∞gxsintlnxdx>0.
Moreover,(20)∫x˜t,kTgxsintlnxdx>0.
For anyT>x˜t,k, whenever xgx is decreasing in x˜t,k,T, strictly decreasing on lnjπ/t,lnj+1π/t for some positive integer j such that lnjπ/t≥x˜t,k and lnj+1π/t≤T.Lemma 1.
Letfx and gx be nonnegative differentiable real-valued functions on some domain D such that fx is decreasing on D and gx is increasing on D. Then, the function(21)hx=fxgx,is decreasing on D.Proof.
Sincef is decreasing and g is increasing, we have fx≤0 on D and g′x≥0 on D, respectively. Thus,(22)f′x=fxgx′=f′xgx−fxg′xgx2≤0.
OnD, because f′xgx≤0,fxg′x≥0 and their difference f′xgx−fxg′x≤0. Hence, hx is decreasing on the domain D.Remark 2.
Iffx and gx are strictly decreasing and strictly increasing, respectively, which are nonnegative on some domain D, then hx=fx/gx is also strictly decreasing on the same domain D.


## 2.1. An Overview of Some Families of Zeta Functions

Definition 1.
The Riemann zeta functionζs is a function of a complex variable s=σ+it defined as an infinite series (Dirichlet series) given, convergent for all σ>1,(5)ζs=∑n=1∞1ns.Forσ>1, the classical zeta function ζs is also defined by the integral given(6)ζs=1Γs∫0∞xs−1ex−1dx,where Γs=∫0∞xs−1e−xdx is the Gamma function. For σ>1, ζs is also defined by(7)ζs=∏p1−p−s−1,where the product runs over all prime numbers p. This product formula is called Euler’s product formula for the Riemann’s zeta function. Since a convergent infinite product of nonzero factors is not zero, the zeta function does not vanish in the right half of the complex plane. Therefore, this product formula is one of the important tools to show that the Riemann zeta function is zero free in the right half of the complex plane (actually for σ>1).The celebrated functional equation of the Riemann zeta function is also another important representation to locate the zeros of the zeta function on the left half of the complex plane. It is given by(8)ζs=22πs−1ζ1−sΓ1−ssinπs2.Definition 2.
The pointss=−2,−4,−6,… are called the “trivial” zeros of the zeta function ζs, and the strip s∈ℂ:0≤σ≤1 is called the critical strip.Regarding the zeros inside the critical strip, it is conjectured that the nontrivial zeros all must be located on the “critical line”ℜs=1/2. This conjecture is known as Riemann’s hypothesis. As a generalization of the Riemann zeta function ζs via integral representation, we have the following definition.Definition 3.
(see [11], Definition 2.1). Let N be a natural number and TNx=∑n=0Nxn/n! be the Taylor (Maclaurin) polynomial of the exponential function ex. For σ>1, the Nth order hypergeometric zeta functions ζNs is defined as(9)ζNs=1Γs+N−1∫0∞xs+N−2ex−TN−1xdx.Observe that whenN=1, we get the classical zeta function ζs. It has been shown in [11] that the hypergeometric zeta function ζNs can be extended analytically to the entire complex plane, except for N simple poles at s=1,0,−1,…,2−N. The hypergeometric zeta functions have zeros on the negative real axis, and these zeros are called the trivial zeros for the hypergeometric zeta functions.Concerning zero-free regions for hypergeometric zeta function of order 2, the following results are known.Theorem 3.
(see [8], Theorem 1.1). ζ2s has no zeros in the left-half complex plane s=σ+it∈ℂ:σ<σ2, except for infinitely many “trivial” zeros on the negative real axis, one in each of the intervals(10)Sm=σm+1,σm,where m≥2 is a positive integer, zk=rkeiθk are the nonzero roots of ez−TN−1x and σm=1−π/π−θ1m.Theorem 4.
(see [9], Theorem 3.4.2 and Corollary 3.4.3). Let s=σ+it with σ>1 and C>1. Then, ζ2s≠0 for s<Cσ.Definition 4.
(see [10], Definition 2.1). The fractional hypergeometric zeta function ζas is defined for all positive real number a and σ>1 as(11)ζas=Γa+1Γs+a−1∫0∞xs+a−2e−xaγa,xdx,where Γs is the Gamma function defined by(12)Γs=∫0∞xs−1e−xdx,and γa,x is the lower incomplete Gamma function of the form(13)γa,x=∫0xe−tta−1dt.Observe that whena=N, natural number, we get the Hypergeometric zeta functions ζNs.


## 2.2. Positivity Properties of Integrals

Proposition 1.
(see [1], Proposition 3.1). Let the function hr≥0 on 0,∞,h∈Lloc10,∞,h decreasing and strictly decreasing on some open subinterval of the form kπ/t,k+1π/t for some nonnegative integer k, and t positive real number. Then,(14)∫0∞hrsintrdr>0.
Moreover,(15)∫0Thrsintrdr>0.
For anyT>0 provided, h satisfies the above assumptions with 0,∞ replaced by 0,T.Corollary 1.
(see [1], Corollary 3.2). Let t be a positive real number and x˜t,k≥1 be such that tlnx˜t,k=2πk, for some positive integer k. Let h≥0 on x˜t,k,∞,h∈Lloc10,∞,h decreasing, strictly decreasing on some interval of the form jπ/t,j+1π/t for some positive integer j such that jπ/t≥x˜t,k. Then,(16)∫x˜t,k∞hrsintrdr>0.
Moreover,(17)∫x˜t,kThrsintrdr>0.
For someT>x˜t,k if T>j+1π/t≥x˜t,k.Remark 1.
Givent>0, we can take a y=e2π/t≥1 such that(18)∫lny∞hrsintrdr>0,where h≥0 on lny,∞,h∈Lloc10,∞,h decreasing, strictly decreasing on some interval of the form jπ/t,j+1π/t for some j∈ℕ such that jπ/t≥x˜t,k.Corollary 2.
(see [1], Corollary 3.3). Let t>0 and x˜t,k≥1 be such that tlnx˜t,k=2πk, for some positive integer k and let g≥0 on x˜t,k,∞,g∈Lloc1x˜t,k,∞ such that x↦xgx is decreasing on x˜t,k,∞, strictly decreasing on lnjπ/t,lnj+1π/t for some positive integer j. Then,(19)∫x˜t,k∞gxsintlnxdx>0.
Moreover,(20)∫x˜t,kTgxsintlnxdx>0.
For anyT>x˜t,k, whenever xgx is decreasing in x˜t,k,T, strictly decreasing on lnjπ/t,lnj+1π/t for some positive integer j such that lnjπ/t≥x˜t,k and lnj+1π/t≤T.Lemma 1.
Letfx and gx be nonnegative differentiable real-valued functions on some domain D such that fx is decreasing on D and gx is increasing on D. Then, the function(21)hx=fxgx,is decreasing on D.Proof.
Sincef is decreasing and g is increasing, we have fx≤0 on D and g′x≥0 on D, respectively. Thus,(22)f′x=fxgx′=f′xgx−fxg′xgx2≤0.
OnD, because f′xgx≤0,fxg′x≥0 and their difference f′xgx−fxg′x≤0. Hence, hx is decreasing on the domain D.Remark 2.
Iffx and gx are strictly decreasing and strictly increasing, respectively, which are nonnegative on some domain D, then hx=fx/gx is also strictly decreasing on the same domain D.


## 3. Zero-Free Regions forζNs and ζas

In this section, we demonstrate zero-free regions for hypergeometric zeta functionζNs and fractional hypergeometric zeta function ζas.Theorem 5.
LetN be a fixed natural number. Then, ζNs≠0 on the vertical strip(23)VN=s=σ+it∈ℂ:1≤σ<2.Proof.
Forσ>1,(24)ζNs=1Γs+N−1∫0∞xs+N−2ex−TN−1xdx.
Rearranging the above equation, we get(25)Γs+N−1ζNs=∫0∞xs−2ex−TN−1x/xNdx.
To apply positivity properties of oscillatory integrals, puthx=fx/gx, where(26)fx=xσ−2andgx=ex−TN−1xxN.
Since the functionfx=xσ−2 is decreasing on 0,∞ and the function gx=ex−TN−1x/xN is increasing on 0,∞ for σ−2<0, by Lemma 1, hx is decreasing on 0,∞. Moreover, the function hx is nonnegative, hx∈Lloc10,∞ and strictly decreasing on any subinterval of 0,∞ as it is actually strictly decreasing on 0,∞. Thus, for any positive real number t,(27)∫0∞xσ−2ex−TN−1x/xNsintlnxdx>0.
This implies that(28)ℑΓs+N−1ζNs>0.
Hence,ζNs≠0. But, then ζNs=ζNs¯¯, so that it holds also for t<0.
Fort=0,(29)Γσ+N−1ζNσ=∫0∞xσ−2ex−TN−1x/xNdx.
We know that(30)xσ−2ex−TN−1x/xN>0.
For allx∈0,∞ and continuous on 0,∞. Hence,(31)Γσ+N−1ζNσ=∫0∞xσ−2ex−TN−1x/xNdx>0.
Therefore,ζNs≠0 for any real number t where 1<σ<2. Therefore, ζNs≠0 on VN=s=σ+it∈ℂ:1≤σ<2. This vertical strip is shown roughly in Figure 1.Figure 1
Zero-free region for hypergeometric zeta function of orderN.Theorem 6.
Let0<a<1 be fixed. Then, ζas≠0 in the vertical strip(32)Va=s=σ+it∈ℂ:1≤σ<2−a.Proof.
Forσ>1,(33)ζas=Γa+1Γs+a−1∫0∞xs+a−2e−xaγa,xdx,is defined and analytic to the right half of σ=1. Rearranging the above equation, we get(34)Γs+a+1Γaζas=∫0∞xs+a−2e−xγa,xdx.
To apply positivity properties of the oscillatory integral, forx∈0,∞, put(35)hx=xσ+a−2exγa,x>0,and consider the case where 1<σ<2−a.
In this case, the functionfx=xσ+a−2 is decreasing on 0,∞, and the function gx=exγa,x is increasing on 0,∞. Hence, by Lemma 1, hx is decreasing (actually strictly decreasing) on 0,∞. Moreover, the function hx is nonnegative, hx∈Lloc10,∞, and strictly decreasing on any subinterval of 0,∞. Thus, for any t>0,(36)∫0∞xσ+a−2e−xγa,xsintlnxdx>0.
This implies that(37)ℑΓs+a+1Γaζas>0,which implies that ζas≠0. But, then ζas=ζas¯¯, so that it holds also for t<0.
Fort=0,(38)Γσ+a+1Γaζaσ=∫0∞xσ+a−2e−xγa,xdx.
We know thatxσ+a−2/γa,xex>0 for all x∈0,∞ and continue on 0,∞. Thus,(39)∫0∞xσ+a−2e−xγa,xdx>0.
Hence,ζas≠0, for any t∈ℝ and 1<σ<2−a. Therefore, ζas≠0 on Va=s=σ+it∈ℂ:1≤σ<2−a. This vertical strip is shown roughly in Figure 2.Figure 2
Zero-free region for fractional hypergeometric zeta function of ordera.Remark 3.
The widths of the vertical stripsVN and Va have the following properties.(1)
The width of the vertical stripVN is independent of N for N>1. For N=a=1, we have the classical Riemann zeta function, and in this case, actually, we have a zero-free region to the right of σ=1.(2)
The width of the vertical stripVa depends on a. Observe that as “a” increases from 0 to 1 excluding both zero and one, the vertical strip becomes narrower and narrower and eventually becomes a vertical line.(3)
Asa decreases from 1 to 0 excluding both zero and one, the vertical strip Va becomes wider and wider, and eventually, the vertical strip Va becomes s=σ+it∈ℂ:1≤σ<2.


## 4. Conclusion

In this paper, zero-free regions were described and demonstrated for both families of hypergeometric zeta functionsζNs and fractional hypergeometric zeta functions ζas. The zero-free regions are vertical strips VN for ζNs and Va for ζas. The width of VN is independent of the parameter N, and the width of Va depends on the parameter a. We have some evidence but no proof at the present time that it is possible to extend the zero-free regions for both ζNs and ζas to both the left half and right half of the complex plane. We do not know whether these families of functions have nontrivial zeros, but we have some evidence but no proof at present especially the fractional hypergeometric zeta functions may have nontrivial zeros which may coincide with the nontrivial zeros of the classical Riemann zeta functions.



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