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A new generalization of the Riemann zeta function and its difference equation
Advances in Difference Equations volumeÂ 2011, ArticleÂ number:Â 20 (2011)
Abstract
We have introduced a new generalization of the Riemann zeta function. A special case of our generalization converges locally uniformly to the Riemann zeta function in the critical strip. It approximates the trivial and nontrivial zeros of the Riemann zeta function. Some properties of the generalized Riemann zeta function are investigated. The relation between the function and the general Hurwitz zeta function is exploited to deduce new identities.
1 Introduction
The family of zeta functions including Riemann, Hurwitz, Lerch and their generalizations constantly find new applications in different areas of mathematics (number theory, analysis, numerical methods, etc.) and physics (quantum field theory, string theory, cosmology, etc.). A useful generalization of the family is expected to have wide applications in these areas as well. Some extensions of the FermiDirac (FD) and BoseEinstein (BE) functions have been introduced in [1]. The extended FermiDirac (eFD)
and the extended BoseEinstein (eBE) functions
provide a unified approach to the study of the zeta family. These functions proved useful in providing simple and elegant proofs of some known results and yielding new results.
The HurwitzLerch zeta function
has the integral representation ([[2]2, p. 27, (1.6)(3)])
If a cut is made from 1 to âˆž along the positive real zaxis, Î¦ is an analytic function of z in the cut zplane provided that Ïƒ > 0 and â„œ(a) > 0. A class of functions can be expressed in terms of the function Î¦. For example the polylogarithm function
Hurwitz's zeta function
and the Riemann zeta function
are special cases of this function. The HurwitzLerch zeta function is related to the above
eFD and eBE functions via
and shows the extension of the variable x to the complex domain as described in (1.4). The Weyl transform representation of the functions (1.1) and (1.2) leads to new identities for the family of the zeta functions [1].
Here we provide a new generalization of the Riemann zeta function that is also related to the eFD and eBE functions and to the HurwitzLerch zeta function. We study its properties and relations with other special functions. Before defining the new function, it is worth putting the family of zeta functions in perspective for our purpose. Riemann proved that the zetafunction
has a meromorphic continuation to the complex plane, which satisfies the functional equation [[3], p. 13 (2.1.1)]
From the equation (1.11) it is obvious that s = 2, 4, 6,..., are simple zeros of the Riemann zeta function. They are called the trivial zeros. It is noted that the simple zero of the sine function on the RHS of (1.11) at s = 0 is canceled by the simple pole of the zeta function Î¶(1  s) and the simple zeros of the sine function at s = 1, 2, 3,... are canceled by the simple poles of the gamma function Î“(1  s) at these points. All other zeros of the Riemann zeta function, which are infinitely many as proven by Hardy [4â€“6], are called the nontrivial zeros, are symmetric about the critical line Ïƒ = 1/2 in the critical strip 0 â‰¤ Ïƒ â‰¤ 1. For the detailed properties of the family of zeta functions we refer to [3â€“5, 7â€“12]. There have been several generalizations of the Riemann zeta function.
Truesdell [13] studied the properties of the de JonquiÃ¨re's function or the polylogarithm
that generalizes the Riemann zeta function and has the integral representation
Note that if x lies anywhere except on the segment of real axis from 1 to âˆž, where a cut is imposed (1.12) defines an analytic function of x for Ïƒ > 0. However (1.12) coincides with the zeta function in Ïƒ > 1 for x = 1 as we have
The FermiDirac (FD) function â„‘_{ s1}(x) defined by [[14], p. 20 (25)]
and the BoseEinstein (BE) function defined by [[14], p. 449 (9)]
are also related to the zeta family by
From (1.1), we find that the weighted function
converges uniformly to Î³(s)(1  2^{1s})Î¶(s) as Î½ â†’ 0^{+} in every substrip 0 < Ïƒ _{1} â‰¤ Ïƒ â‰¤ Ïƒ _{2} < 1 of the critical strip 0 < Ïƒ < 1. However, for x = 0 in (1.2) we get
which converges to the Riemann zeta function in the region Ïƒ â‰¥ Ïƒ _{1} > 1 as Î½ â†’ 0^{+}. However, the function (1.20) is not even defined in the critical strip 0 < Ïƒ < 1 as the integral is divergent there. So it is desirable to have a generalization of the Riemann and Hurwitz zeta functions in the critical strip, which converges locally uniformly at least. A special case of our new generalization converges to the Riemann zeta function locally uniformly in the critical strip and gives a unified approach, not only to the study of Riemann, Hurwitz, HurwitzLerch zeta functions, but also of the FD and BE functions along with their extensions. An important feature of our approach is the desired simplicity of the proofs using Weyl's fractional transform.
The article is organized as follows. For completeness, in Sect. 2 we state some preliminaries and a general representation formula proved earlier in [1]. In Sect. 3, we define the extended Riemann zeta function and prove its series representation. A connection of the function with the eFD and eBE functions is shown in the next section. In Sect. 5, we prove functional relations of the generalized Riemann zeta function. Some concluding remarks and discussion are given in the last section.
2 Some preliminaries, Mellin and Weyl's transforms
The function spaces H(Îº; Î») and H(âˆž; Î») are defined as follows (see [1]).
A function f âˆˆ C ^{âˆž} (0, âˆž) is said to be a member of H(Îº; Î») if:

1.
f(t) is integrable on every finite subinterval [0, T] (0 < T < âˆž) of ;

2.
f(t) = O(t ^{Î»} ) (t â†’ 0^{+});

3.
f(t) = O(t ^{Îº} ) (t â†’ âˆž).
Furthermore, if the above relation f(t) = O(t ^{Îº} ) (t â†’ âˆž) is satisfied for every exponent , then the function f (t) is said to be in the class H(âˆž; Î»). It is noted that . Clearly, we have
The Mellin transform of f âˆˆ H(Îº; Î») is defined by (see [[15], p. 83])
The Weyl transform (or Weyl's fractional integral) of order s of Ï‰ âˆˆ H(Îº; 0) is defined by (see [[9], Vol. II, p. 181] and [[16], p. 237]),
For Ïƒ â‰¤ 0, we define the Weyl transform (or Weyl's fractional derivative) of order s of Ï‰ âˆˆ H(Îº; 0) as follows (see [[16], p. 241]),
where n is the smallest positive integer greater than or equal to Ïƒ provided that Ï‰(0) is well defined and that
We can rewrite Weyl's fractional derivative (2.4) alternately as
where n is the smallest positive integer greater than or equal to Ïƒ. In particular for s = n (n = 0, 1, 2, 3,...) in (2.6), we find that
Notice that {W ^{s} } is a multiplicative group [[16], p. 245] and satisfies
The notations â„œ_{ s }{f(t); x} and are also used to represent the Weyl transform (see [[9], Vol. II, p. 181] and [1]). Following the above terminology it was proved in [1] that
Note that for the case Ïƒ = 0, (2.9) yields
which is HardyRamanujan's master theorem (see [[10], p. 186 (B)]. Some special cases of (2.10) include
which shows that z _{ a } (x) âˆˆ H(1; 0) (0 â‰¤ a < 1). Similarly, we have (see [[15], p. 91 (3.3.6)])
which shows that cos(2Ï€x) âˆˆ H(1/2, 0).
3 The generalized Riemann zeta function Îž_{ Î½ }(s; x)
The eFD and the eBE functions defined by (1.1) and (1.2) provide a unified approach to the study of the zeta family. The weighted function Î“(s)(1  2^{1s}) Î˜_{Î½}(s; 0) converges uniformly to Î“(s)(1  2^{1s})Î¶(s) as Î½ â†’ 0^{+} in every substrip 0 < Ïƒ _{1} â‰¤ Ïƒ â‰¤ Ïƒ _{2} < 1 of the critical strip 0 < Ïƒ < 1. However, the function Î“(s)Î¨_{ Î½ }(s; 0) is not even defined in the critical strip as the integral representation (1.1) is divergent in 0 < Ïƒ < 1. It is desirable to have a function that converges uniformly to the Riemann zeta function in some sense and connects the eFD and eBE functions. We assume that Î½ is real and 0 â‰¤ Î½ < 1 in the rest of the article and use analytic continuation [[6], pp. 2223] to introduce the extended Hurwitz zeta function as follows:
For x = 0 and Î½ = 0 in (3.1) [[6], p. 22]
Theorem 3.1 The generalized Riemann zeta function (3.1) is well defined and the weighted function Î“(s)Îž_{ Î½ }(s; 0) converges uniformly to the weighted Riemann zeta function Î“(s)Î¶(s) as Î½ â†’ 0^{+} in every substrip 0 < Ïƒ _{1} â‰¤ Ïƒ â‰¤ Ïƒ _{2} < 1 of the critical strip 0 < Ïƒ < 1.
Proof. First we note that
which shows that the generalized Riemann zeta function (3.1) is well defined. Second, that the difference integral representation (as 1  e ^{Î½t} â‰¤ 1, 0 â‰¤ Î½ < 1, 0 â‰¤ t < âˆž),
is absolutely convergent shows that the limit as Î½ â†’ 0^{+} and the integral in (3.4) are reversible. Letting Î½ â†’ 0^{+} in (3.4) we find that the convergence
is uniform. â–
Theorem 3.2 (Connection with the Hurwitzzeta function)
Proof. We assume that 0 < Î½ < 1 and Ïƒ > 1. In this case, from (3.1)
Note that the RHS in (3.7) remains well defined for 0 < Ïƒ < 1 and 0 < Î½ < 1. Moreover, for Î½ = 0, we have the well known integral representation (3.2) (see [[6], p. 22]) for 0 < Ïƒ < 1. Hence the proof. â–
Remark 3.3 The representation (3.6) of the generalized Riemann zeta function shows that the function is meromorphic. For Î½ âˆˆ (0, 1) the function has a removable singularity at s = 1 as the residue of the function is zero. However, for Î½ = 0 the function has a simple pole at s = 1 with residue 1. We can rewrite (3.6) as
Putting s =  n and using [[7], p. 264]
we find that the function is related to the Bernoulli's polynomials via (see [[7], p. 264, (17)])
Using the relations (see [[11], pp. 2628])
and
we obtain the closed form
which shows that
Thus the generalized Riemann zeta function approximates the trivial zeros (s = 2, 4, 6,...) of the Riemann zeta function as Î½ â†’ 0^{+}. The relation (3.17) gives the rate at which these zeros are approached. One needs to see if all the zeros can be approximated uniformly. Since 2nÎ¶(1  2n) â†’ âˆž as n â†’ âˆž, by setting
we have
which shows that all the nontrivial zeros can, indeed, be approximated uniformly.
Remark 3.4 It is worth visualizing the behavior of the function near Î½ = 0 for large n more generally. Though Îž_{ Î½ }(n, 0) is a function of one continuous and one discrete variable, conceive it as if it were a sheet over the strip Î½ âˆˆ (0, 1), n âˆˆ (0, âˆž) in the (Î½, n)plane. At every n the sheet approaches the naxis arbitrarily closely, but it does not do so for all n, since the sheet rises increasingly more sharply for larger values of n. The asymptotic formula for Î¶(1  2n) (see [[11], pp. 2628]) can be used in conjunction with Stirlings formula to give the coefficient of Î½ (for small Î½)
The function of the discrete variable can be thought of as the parts of the sheet lying over the grid lines of the integer values of n. The sequence where the curve intersects the grid lines gives a path. The nontrivial zeros are then clearly uniformly approximated by paths approaching Î½ = 0 lying between Î½ = 1/ 2nÎ¶(1  2_{ n }) and the naxis.
4 Connection with the eFD and eBE integral functions
Theorem 4.1 The generalized Riemann zeta function is related to the eBE integral functions and the incomplete gamma function via
Proof. We have the identity
By taking the Weyl's transform of both sides in (4.2) we obtain
However, we have (see [[9], pp. 255, 266])
From (4.3) and (4.4) we arrive at (4.1). â–
Corollary 4.2
Proof. This follows from (4.1) when we take Î½ = 0 and use (1.20). â–
Theorem 4.3
Proof. First we note that
Therefore, following the general expansion result (2.9), we arrive at (4.6). â–
Remark 4.4 A very interesting special case of (4.6) arises when Î½ = s = 0. In this case we have the wellknown result proved by Hardy and Littlewood [5]
Equations (4.5) and (4.6) lead to the useful representation
Theorem 4.5 The generalized Riemann zeta and the eFD integral functions are related by
Proof. We have the identity
Taking the Weyl transform of both sides in (4.11), we find that
However, we have
The substitution t = Ï„/2 in (4.13) leads to
From (4.12), (4.13), and (4.14) we arrive at (4.10). â–
Remark 4.6 It is useful to write (4.10) in the form
Putting v = x = 0 in (4.15) we find the classical integral representation
for the weighted Riemann zeta function. Note that the simple pole of the zeta function at s = 1 is cancelled by the (simple) zero of the factor 1  2^{1s}such that the product Î˜_{0}(s, 0) = (1  2^{1s})Î¶(s) remains well defined in the sense of the Riemann removable singularity theorem. Moreover using the relations (1.8) and (1.9) we can rewrite (4.10) in terms of the HurwitzLerch zeta function as
This can be extended to a function of the complex variable z as given in (1.4).
Corollary 4.7 (Connection with the FD functions)
Proof. This follows from (1.17) and (1.18) and from (4.10) when we put Î½ = 0. â–
5 Difference equation for the generalized Riemann zeta function
Functional relations arising from difference equations are useful for the study of special functions. For example, the Bernoulli polynomials satisfy the difference equation ([[7], p. 265 (18)])
Do the generalized Riemannzeta functions also satisfy such relations? The next theorem gives the answer to this question.
Theorem 5.1 The generalized Riemann zeta function Îž_{ Î½ }(s; x) satisfies the difference equation
Proof. We have the identity
Applying the Weyl transform on both sides in (5.3) we obtain
However (see [[9], Vol. II, p. 202]),
and (see [[2], p. 262 (6.9.2)(21)])
From (5.4), (5.5), and (5.6) we arrive at (5.2). â–
6 Concluding remarks and discussion
According to Bombieri (see [[17], (2)]), the formula
was proved by Tchebychev, from which he deduced that (s  1)Î¶(s) has limit one as s â†’ 1. He used the above formula in his first memoir to prove the asymptotic formula for the number of primes less than a given number. Putting Î½ = 1 in (3.6) we have
which is exactly Tchebychev's formula. This shows that a very special case of our new generalized Riemann zeta function appeared earlier in the work of Tchebychev in the study of the Riemann zeta function and the location of the nontrivial zeros. However, the general case of the function and its relation with the zeta family does not seem to have been realized so far. We studied the properties and functional relations of the new function. It achieves the desired simplification of the cumbersome proofs of elegant properties of the HurwitzLerch zeta function. This simplification can be expected to lead to other results that may have remained unproven due to the complexity of their proofs.
It was seen that the generalized Riemann zeta function has simple connections with the recently defined eBE and eFD functions, which have also found another use in Physics. For certain problems arising in condensed matter theory quasiparticles that are neither fermions nor bosons had been proposed. They had been called "anyons" (see references in [18] for the literature on these particles). The eBE and eFD functions had been put forward as possible candidates for the anyon function as they interpolate very naturally between the BE and FD functions. The above connections enable us to obtain an asymptotic expansion of the function in the critical strip. It turns out that the new function is also related to the Bernoulli polynomials via (3.12) and approximates the nontrivial zeros of the Riemann zeta function as well.
Abbreviations
 BE:

BoseEinstein
 eFD:

extended FermiDirac
 FD:

FermiDirac.
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Acknowledgements
Two of the authors (MAC and AQ) are grateful to the King Fahd University of Petroleum and Minerals for providing excellent research facilities. AT acknowledges her indebtedness to the Higher Education Commission of Pakistan for the Indigenous PhD Fellowship.
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Chaudhry, M.A., Qadir, A. & Tassaddiq, A. A new generalization of the Riemann zeta function and its difference equation. Adv Differ Equ 2011, 20 (2011). https://doi.org/10.1186/16871847201120
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DOI: https://doi.org/10.1186/16871847201120