The Zeta and Related Functions : Recent Developments

The main object of this surveycum-expository article is to present an overview of some recent developments involving the Riemann Zeta function ζ(s), the Hurwitz (or generalized) Zeta function ζ(s, a), and the HurwitzLerch Zeta function Φ(z, s, a), which have their roots in the works of the great eighteenth-century Swiss mathematician, Leonhard Euler (1707 1783) and the Russian mathematician, Christian Goldbach (169

ζ(s, a), which are dened (for (s) > 1) by n−1 n s ( (s) > 0; s = 1) (1) and and for (s) 1; s = 1 by their meromorphic continuations (see, for details, the excellent works by Titchmarsh [72] and Apostol [4] as well as the monumental treatise by Whittaker and Watson [75]; see also [1,Chapter 23] and [57,Chapter 2]), so that (obviously) More generally, we have the following relationships: Generally speaking, Mathematics appeals to the intellect.In addition, however, great mathematics possesses a kind of perceptual quality which endows it with a beauty comparable to that of great art or great music.Much of the work of the 18th century Swiss mathematician, Leonhard Euler (17071783), belongs in this cat- A classical about three-century-old theorem of Christian Goldbach (16901764) was stated in a letter dated 1729 from Goldbach to Daniel Bernoulli (17001782).Goldbach's Theorem (or, equivalently, the Goldbach-Euler Theorem) was revived and revisited recently as the following problem in many publications such as (for example) [a] J. D. Shallit and K. Zikan, A theorem of Goldbach, Amer.Math.Monthly 93 (1986), where S denotes the set of all nontrivial integer kth powers, that is, What does Goldbach's Theorem (5) have to do with the Riemann Zeta function ζ(s) dened by (1)?
In order to answer this question, let T denote the set of all positive integers that are not in S other than 1, that is, T := {τ : τ / ∈ S and τ ∈ N \ {1}} .
We then nd that Thus, in terms of the Riemann Zeta function ζ(s) dened by (1), Goldbach's Theorem (5) is easily seen to assume the following elegant form: Since ζ(s) is a decreasing function of its argument s for s 2, we have the above alternative form (6) of Goldbach's Theorem (5) can also be rewritten as follows: where f(x) := x − [x] = The fractional part of x ∈ R.
As a matter of fact, it is fairly straightforward to show also that and so on.
Several extensions and generalizations of Goldbach's Theorem (5) have been investigated.For example, we recall the following generalizations given in [c] J. Choi (p ∈ N), respectively.This last pair of the Goldbach-Euler type sums poses a natural question as the following open problem.
Open Problem.For each of the following double sums:   Another result that has attracted fascinatingly and tantalizingly large number of seemingly independent solutions is the so-called Basler Problem or Basel Problem: which was used above in (7).The city of Basel in Switzerland was one of many free cities in Europe.By the 17th century, it had become an important center of trade and commerce.The University of Basel became a noted institution in Europe, largely through the fame of an extraordinary family, namely, the Bernoullis.This family had come from Antwerp to Basel.The founder of the Bernoulli mathematical dynasty was Nicolaus (Nicholas) Bernoulli who was a painter and Alderman of Basel.He had 3 sons; two of his sons, Jakob [often referred to as James] (16541705) and Johann [often referred to as John] (16671748), became noted mathematicians.Both were pupils of Gottfried Wilhelm von Leibniz (16461716) with whom Johann Bernoulli carried on an extensive correspondence and with whose work both Jacob Bernoulli and Johann Bernoulli became familiar.Jacob Bernoulli was a professor at the University of Basel until his death in 1705.Johann Bernoulli, who had been a professor at the University of Groningen in the (present-day) Netherlands, replaced his brother at the University of Basel.Johann Bernoulli had 3 sons.Two of them, Nicholas II (16951726) and Daniel (17001782), were mathematicians who befriended Euler.They both went to the Academy in St. Petersburg in 1725 and they both had a high regard for their younger colleague, Euler.After some eort, Daniel wrote to Euler that he had secured for him a stipend in the Academy.The appointment for Euler was actually in the physiology section, but Euler quickly drifted into the mathematics section.He thus left Basel for St. Petersburg in 1727 and remained there until 1741 when he was summoned by Frederick the Great of Prussia to the Berlin Academy.Euler was in Berlin until 1766 when he was summoned back to the Academy in St. Petersburg where he remained until his death in 1783.Euler carried on an extensive correspondence with various mathematicians, especially with Christian Goldbach (16901764).He also wrote a series of letters on various subjects in natural philosophy and addressed these letters to a German princess.The quality of all his letters reects Euler's pleasant personality.
It is easily observed that Euler's series in (9) converges faster than the dening series for ζ (3), but obviously not as fast as the series in (10).In fact, the order estimates for their general terms are given as follows: It is especially remarkable that Euler was already blind when he performed the breathtaking calculations leading to his result (9) rather mentally.
Evaluations of such Zeta values as ζ (3), ζ (5), et cetera are known to arise naturally in a wide variety of applications such as those in Elastostatics, Quantum Field Theory, et cetera (see, for example, Tricomi [73], Witten [77], and Nash and O'Connor [39], [40]).On the other hand, in the case of even integer arguments, we already have the following computationally useful relationship: (n ∈ N 0 := N ∪ {0}) with the well-tabulated Bernoulli numbers dened by the following generating function: as well as by the familiar recursion formula: (n ∈ N \ {1}).as well as its various interesting extensions and generalizations (see Section 6).

Series Representations for
The following simple consequence of the binomial theorem and the denition (1): yields, for a = 1 and t = ±1/m, a useful the series identity in the form: where (λ) ν denotes the general Pochhammer symbol or the shifted factorial, since which is dened, in terms of the familiar Gamma function, by it being understood conventionally that (0) 0 := 1 and assumed tacitly that the Γ-quotient exists (See, for details, [48] and [57]).

Making use of the familiar harmonic numbers
H n given by the following set of series representations for ζ (2n + 1) (n ∈ N) were proven by Srivastava [51] by appealing appropriately to the series identity (15) in its special cases when m = 2, 3, 4, and 6, and also to many other properties and characteristics of the Riemann Zeta function such as the familiar functional equation for ζ(s) (which was discovered by Euler around 1749, that is, almost 110 years before Riemann): the familiar derivative formula: so that and each of the following limit relationships: First Series Representation: Second Series Representation: Third Series Representation: c 2019 Journal of Advanced Engineering and Computation (JAEC) Fourth Series Representation: Here, as well as elsewhere in this presentation, an empty sum is understood (as usual) to be zero.
The rst series representation ( 22) is markedly dierent from each of the series representations for ζ (2n + 1), which were given earlier by Zhang and Williams [79, p. 1590, Equation (54)] and (subsequently) by Cvijovi¢ and Klinowski [14, p. 1265, Theorem A] (see also [80] and [81]).Since ζ (2k) → 1 as k → ∞, the general term in the series representation (22) has the following order estimate: whereas the general term in each of the aforecited earlier series representations has the order estimate given below: In case we suitably combine ( 22) and ( 24), we readily obtain the following series representation: where n ∈ N.Moreover, in terms of the Bernoulli numbers B n and the Euler polynomials E n (x) dened by the generating functions (12) and respectively, it is known that (cf., e.g., [37, p. 29]) (28) Thus, by combining (28) with the identity (11), we nd that where n ∈ N. If we apply the relationship (29), the series representation (26) can immediately be put in the following alternative form: where n ∈ N, which is a slightly modied and corrected version of a result proven, using a signicantly dierent technique, by Tsumura [74, p.

383, Theorem B].
One other interesting combination of the series representations (22) and ( 24) leads us to the following variant of Tsumura's result (26) or (30): where n ∈ N, which is essentially the same as the A number of other similar combinations of the series representations (22) to (25) would yield some interesting companions of Ewell's result (31).
Next, by setting t = 1/m and dierentiating both sides with respect to s, we nd from the following obvious consequence of the series identity (14): where m ∈ N \ {1}.In the particular case when m = 2, (33) immediately yields Upon letting s → −2n − 1 (n ∈ N) in the further special of this last identity (34) when a = 1, which, in light of the elementary identity: would combine with the result (22) to yield the following series representation: This last series representation (37)  ] where an obviously more complicated (asymptotic) version of (37) was proven similarly).
In light of another elementary identity: where n, k ∈ N, we can obtain the following yet another series representation for ζ (2n + 1) by applying ( 22) and ( 35): where n ∈ N and the coecients Ω n,k where n ∈ N; k ∈ N 0 .

Other Families of Series
Representations for In this section, we start once again from the identity ( 14) with (of course) a = 1, t = ±1/m, and s replaced by s + 1.Thus, by applying (15), we nd yet another class of series identities including, for example, and where m ∈ N \ {1, 2} .In fact, it is the series identity (42) which was rst applied by Zhang and Williams [79] (and, subsequently, by Cvi-jovi¢ and Klinowski [14]) with a view to proving two (only seemingly dierent) versions of the series representation (37).Indeed, if we appeal to (43) with m = 4, we can derive the following much more rapidly convergent series representation for ζ (2n + 1) (see [50, p. 9, Equation ( 41)]): where n ∈ N and (and in what follows) a prime denotes the derivative of ζ (s) or ζ (s, a) with respect to s.
By virtue of the identities (36) and (38), the results ( 24) and ( 44) would lead us eventually to the following additional series representations for ζ (2n + 1) (n ∈ N) (see [50,p. 10,Equations (42) and ( 43)]): where n ∈ N, and where n ∈ N. Out of the four seemingly analogous results (24), (44), (45), and (46), the innite series in (45) would obviously converge most rapidly, with its general term having the order estimate: From the work by Srivastava and Tsumura [69], we recall the following three new members of the class of the series representations ( 24) and (45): and where n ∈ N. The general terms of the innite series occurring in these three members ( 47), (48), and ( 49) have the order estimates: which exhibit the fact that each of these last three series representations (47), (48), and (49) converges more rapidly than Wilton's result (35) and two of them [cf.Equations ( 48) and ( 49)] at least as rapidly as Srivastava's result (45).
In passing, we nd it worthwhile to remark that an integral representation for ζ (2n + 1), which is easily seen to be equivalent to the series representation (58), was given by D abrowski [16, p. 203, Equation ( 16)], who [16, p. 206] mentioned the existence of (but did not fully state) the series representation (59) as well.The series representation (58) was derived also in a paper by Borwein et al. (cf.[6, p. 269, Equation ( 57)]).

Numerical Verications and Symbolic Computations
Based Upon Mathematica (Version 4.0) In this section, we rst summarize the results of numerical verications and symbolic computations with the series in ( 73) by using Mathematica (Version 4.0) for Linux: where Out [3] = 0.0372903 Out [4] = 0.0372903 Out [2] evidently validates the series representation (73) symbolically.Furthermore, our numerical computations in Out [3], Out [4], and Out [5], together, exhibit the fact that only 50 terms (k = 1 to k = 50) of the series in (73) can produce an accuracy of as many as seven decimal places.
Our symbolic computations and numerical verications with the series in (74)  In fact, since the general term of the series in (74) has the following order estimate: for getting p exact digits, we must have xe x = a.
Some relevant details about the symbolic computations and numerical verications with the series in (74) using Mathematica (Version 4.0) for Linux are being summarized below.
In particular, when ν = σ = 1, the fractional derivative formula (89) would reduce at once to the following form: which exhibits the hitherto unnoticed fact that the function Φ λ,µ;ν (z, s, a) studied by Garg et al.  28)], were proven by Lin and Srivastava [35].
A multiple (or, simply, n-dimentional) Hurwitz-Lerch Zeta function Φ n (z, s, a) was studied recently by Choi et al. [9, p. 66, Eq. ( 6)].R ducanu and Srivastava (see [43] and the references cited therein), on the other hand, made use of the Hurwitz-Lerch Zeta function Φ(z, s, a) in dening a certain linear convolution operator in their systematic investigation of various analytic function classes in Geometric Function Theory in Complex Analysis.Furthermore, Gupta et al. [26] revisited the study of the familiar Hurwitz-Lerch Zeta distribution by investigating its structural properties, reliability properties and statistical inference.These investigations by Gupta et al. [26] and others (see, for example, [53], [57], [60] and [61]), fruitfully using the Hurwitz-Lerch Zeta function Φ(z, s, a) and some of its above-mentioned generalizations, motivated Srivastava et al. [67] to present a further generalization and analogous investigation of a new family of Hurwitz-Lerch Zeta functions dened in the following form [67,p. 491,Equation (1.20)]: For the above-dened function in (92), Srivastava et al. [67] established various integral representations, relationships with the H-function which is dened by means of a Mellin-Barnes type contour integral (see, for example, [65] and [67]), fractional derivative and analytic continuation formulas, as well as an extension of the generalized Hurwitz-Lerch Zeta function Φ (ρ,σ,κ) λ,µ;ν (z, s, a) in (92).This natural further extension and generalization of the function Φ (ρ,σ,κ) λ,µ;ν (z, s, a) was indeed accomplished by introducing essentially arbirary numbers of numerator and denominator parameters in the definition (92).For this purpose, in addition to the symbol ∇ * dened by dened by (see, for details, [17, p. 183] and [64, p. 21]; see also [33, p. 56], [38, p. 30] in terms of the generalized hypergeometric function p F q (p, q ∈ N 0 ).
where E := (1 − λ 1 , ρ 1 ; 1), • • • , (1 − λ p , ρ p ; 1), (1 − a, 1; s) (0, 1), (1 − µ 1 , σ 1 ; 1), • • • , (1 − µ q , σ q ; 1), (−a, 1; s), provided that both sides of the assertions (99), (100) and (101) exist, the path of integration L in (101) being a Mellin-Barnes type contour in the complex ξ-plane, which starts at the point −i∞ and terminates at the point i∞ with indentations, if necessary, in such a manner as Indeed, for the sake of the interested reader, we recall from Srivastava's work [56]  (s, 1), 0, where λ > 0 and H m,n p,q [•] denotes Fox's Hfunction (see, for details, [63]. The theory and applications of the various special as well as limit cases of the λ-generalized in addition to those mentioned above, can be found in (for example) the recent works [56] and [66], and indeed also in many of the earlier references which are cited in each of these recent works.Remarkably, just as its such aforementioned special cases as the Hurwitz-Lerch Zeta dened by ( 105) is potentially useful and is currently being applied in many areas of the mathematical, statistical, physical and engineering sciences.The relevant details of such developments are easily accessible in the current literature on the subject.
egory.Euler's work on ζ(s) began around 1730 with approximations to the value of ζ(2), continued with the evaluation of ζ(2n) (n ∈ N), and resulted around 1749 in the discovery of the celebrated functional equation for ζ(s) almost 110 years before the remarkably inuential German mathematician, Georg Friedrich Bernhard Riemann (18261866).

342 c 2019
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Just as the celebrated Riemann Hypothesis dated 1859 that all nontrivial zeros of ζ(s) lie on the critical line: Goldbach's conjecture has not been proven as yet.Interestingly, not too long ago in the year 2001, on the occasion of the publication of the following (very funny, tender, charming, and irresistible) novel:Uncle Pedros and Goldbach's Conjecture: A Novel of Mathematical Obsession (by Apostolos Doxiadis), Faber and Faber, London, 2001. the British publisher (Faber and Faber) had oered a reward of one million U.K. Pounds to anyone who can prove Goldbach's Conjecture.
provides a signicantly simpler (and much more rapidly convergent) version of the following other main result of Cvijovi¢ and Klinowski [14, p. 1265, Theorem B]: c 2019 Journal of Advanced Engineering and Computation (JAEC) which