Research Article | Open Access
Michael S. Milgram, "Integral and Series Representations of Riemann's Zeta Function and Dirichlet's Eta Function and a Medley of Related Results", Journal of Mathematics, vol. 2013, Article ID 181724, 17 pages, 2013. https://doi.org/10.1155/2013/181724
Integral and Series Representations of Riemann's Zeta Function and Dirichlet's Eta Function and a Medley of Related Results
Contour integral representations of Riemann's Zeta function and Dirichlet's Eta (alternating Zeta) function are presented and investigated. These representations flow naturally from methods developed in the 1800s, but somehow they do not appear in the standard reference summaries, textbooks, or literature. Using these representations as a basis, alternate derivations of known series and integral representations for the Zeta and Eta function are obtained on a unified basis that differs from the textbook approach, and results are developed that appear to be new.
Dedicated to the memory of Leslie M. Saunders,
died September 8, 1972, age 29
(Physics Today obituary, issue of December 1972, page 63)
Riemann's Zeta function and its sibling Dirichlet's (alternating zeta) function play an important role in physics, complex analysis, and number theory and have been studied extensively for several centuries. In the same vein, the general importance of a contour integral representation of any function has been known for almost two centuries, so it is surprising that contour integral representations for both and exist that cannot be found in any of the modern handbooks (NIST, [1, Section ]; Abramowitz and Stegun, [2, Chapter 23]), textbooks (Apostol, [3, Chapter 12]; Olver, [4, Chapter 8.2]; Titchmarsh, [5, Chapter 4]; Whittaker and Watson, [6, Section ]), summaries (Edwards, , Ivić, [8, Chapter 4]; Patterson, ; Srivastava and Choi, [10, 11]), compendia (Erdélyi et al., [12, Section 1.12] and Chapter 17), tables (Gradshteyn and Ryzhik, , 9.512; Prudnikov et al, [14, Appendix ]), and websites [1, 15–19] that summarize what is known about these functions. One can only conclude that such representations, generally discussed in the literature of the late 1800s, have been long-buried and their significance has been overlooked by modern scholars.
It is the purpose of this work to disinter these representations and revisit and explore some of the consequences. In particular, it is possible to obtain series and integral representations of and that unify and generalize well-known results in various limits and combinations and reproduce results that are only now being discovered and investigated by alternate means. Additionally, it is possible to explore the properties of in the complex plane and on the critical line and obtain results that appear to be new. Many of the results obtained are disparate and difficult to categorize in a unified manner but share the common theme that they are all somehow obtained from a study of the revived integral representations. That is the unifying theme of this work. To maintain a semblance of brevity, most of the derivations are only sketched, with citations sufficient enough to allow the reader to reproduce any result for her/himself.
2. Contour Integral Representations of and
Throughout, I will use , where (complex), (reals), and and are positive integers.
The Riemann Zeta function and the Dirichlet alternating zeta function are well known and defined by (convergent) series representations: with It is easily found by an elementary application of the residue theorem that the following reproduces (1): and the following reproduces (2): Convert the result (4) into a contour integral enclosing the positive integers on the -axis in a clockwise direction, and provided that so that contributions from infinity vanish, the contour may then be opened such that it stretches vertically in the complex -plane, giving a result that cannot be found in any of the modern reference works cited (in particular ). It should be noted that this procedure is hardly novel—it is often employed in an educational context (e.g., in the special case ) to demonstrate the utility of complex integration to evaluate special sums ([10, Section 4.1]), ([21, Lemma 2.1]) and forms the basis for an entire branch of physics .
A similar application to (5), with some trivial trigonometric simplification, gives Equation (9) reduces, with , to a Jensen result , presented (without proof) in 1895, and, to the best of my knowledge, reproduced, again, but only in the special case , only once in the modern reference literature . In particular, although Lindelof's 1905 work  arrives at a result equivalent to that obtained here, attributing the technique to Cauchy (1827), it is performed in a general context, so that the specific results (7) and (9) never appear, particularly in that section of the work devoted to . In modern notation, for a meromorphic function , Lindelof writes ([24, Equation ]) in general where the contour of integration encloses all the singularities of the integrand but never specifically identifies . Much later, after the integral has been split into two by the invocation of identities for , he does make this identification ([24, Equation ]), arriving (with (3)) at ((26)—see below) a now-well-known result, thereby bypassing the contour integral representation (7). In a similar vein, Olver ([4, page 290]) reproduces (10), employing it to evaluate remainder terms, the Abel-Plana formula and a Jensen result but again never writes (7) explicitly. Perhaps the omission of an explicit statement of (7) and (9) from Chapter IV of Lindelof  is why these forms seem to have vanished from the historical record, although citations to Jensen and Lindelof abound in modern summaries. (Recently, Srivastava and Choi [11, pages 169–172], have reproduced Lindelof's analysis, also failing to arrive explicitly at (7); although they do write a general form of (8) ([11, Section 2.3, Equation ]), it is studied only for the case .) (Note added in proof: after a draft copy of this paper was posted, I was made aware that Ruijsenaars has previously obtained a generalized form of (7) for the Hurwitz zeta function ([25, Equation ]). He also notes (private communication) that he could not find prior references to his result in the literature).
At this point, it is worthwhile to quote several related integral representations that define some of the relevant and important functions of complex analysis, starting with Riemann's original 1859 contour integral representation of the zeta function : where the contour of integration encloses the negative -axis, looping from to enclosing the point .
The equivalence of (11) and (1) is well described in texts (e.g., ) and is obtained by reducing three components of the contour in (11). A different analysis is possible however (e.g., [7, Section 1.6])—open and translate the contour in (11) such that it lies vertically to the right of the origin in the complex -plane (and thereby vanishes) with and evaluate the residues of each of the poles of the integrand lying at to find This can be further reduced, giving the well-known functional equation for : which is valid for all by analytic continuation from . This demonstrates that (11) and (6) are fundamentally different representations of , since further deformations of (11) do not lead to (7).
Ephemerally, at one time it was noted in  that the following complex integral representation can be obtained by the invocation of the Cauchy-Schlömilch transformation . In fact, the representation (14) follows by the simple expedient of integrating (7) by parts in the special case (see (35)), as also noted in . Another result obtained in  on the basis of the Cauchy-Schlömilch transformation does not follow from (14) as erroneously claimed at one time, but later corrected, in .
Based on (11), a number of other representations of are well known. From the tables cited, the following are worth noting here:
3. Simple Reductions
3.1. The Case
The simplest reductions of (7) and (9) are obtained by uniting the two halves of their range corresponding to and . With the use of one finds (with and , resp.) the latter valid for all . In the case that , (22) reduces to equivalent to the well-known result (attributed to Jensen by Lindelof, and a special case of Hermite's result for the Hurwitz zeta function ([6, Section 13.2)]) (which also appears as an exercise in many textbooks—for example, [11, Equation ]): Similarly, (23) reduces to the known ([24, ]) result: Comparison of (24) and (25) immediately yields the known result ([13, Equation ]): Notice that (25) and (26) are valid for all . Simple substitution of in (23) gives which, after taking (3) into account, leads to displaying the existence of the trivial zeros of at , since the integral in (29) is clearly finite at each of these values of . The result (29) is equivalent to the known result (16) after taking (13) into account. Other possibilities abound. For example, set in (22) to obtain Further, a novel representation can be found by noting that (9) is invariant under the transformation of variables together with the choice along with the requirement that . This effectively allows one to choose , yielding the following complex representation for , as well as the the corresponding representation for because of (3), which is valid in the critical strip : Because (20) and (21) are valid for , it follows that (23) with is also a valid representation for , as well as the the corresponding representation for with recourse to (3), again only valid in the critical strip . Another interesting variation is the choice in (23), valid for , again applicable to the corresponding representation for because of (3). Such a representation may find application in the analysis of in the asymptotic limit . Finally, the choice leads to an interesting variation that is discussed in Section 9.
3.2. Simple Recursion via Partial Differentiation
As a function of , and are constant, as are their integral representations over a range corresponding to ; however the integrals change discontinuously at each new value of that corresponds to a new value of . Thus, the operator acting on (7) and (9) vanishes, except at nonnegative integers , when it becomes indefinite. Therefore, it is of interest to investigate this operation in the neighbourhood of as well as the limit .
Applied to (9) (and equivalent to integration by parts), the requirement that and identification of one of the resulting terms yields a new result with improved convergence at infinity. Similarly operating with on (7) yields (It should be noted that Srivastava et al.  study an incomplete form of (35) in order to obtain representations for sums both involving , and by itself, without commenting on the significance of the contour integral with infinite bounds (35).) Using (20) and (21) to reduce the integration range of (34) gives a fairly lengthy result for general values of . If , (34) becomes which, by letting , can be rewritten as and, with regard to (3), Similarly, ((37) and (38) are the first two entries in a family of recursive relationships that will be discussed elsewhere (in preparation)). In the case that , (35) reduces to the well-known result ([24, Equation ] (attributed to Jensen )) If in (34) (for the case in (35) see the following section), one finds a finite result (also equivalent to integrating (28) by parts): and, when , one finds the unusual forms: which is valid for any and all (particularly or ), after deformation of the contour of integration in (34) either to the left or to the right of the singularity at , respectively, belonging to a choice of where the symbol means zero or one as the case may be.
4. Special Cases
4.1. The Case
With respect to (35), setting together with some simplification gives reproducing (18), with , after taking (13) into account. This same case () naturally suggests that the integral in (6) be broken into two parts: where such that the contour in (44) bypasses the origin to the right. It is easy to show that the sum of the two integrals in (45) reduces to the latter being valid for all . One, of several commonly used possibilities of reducing (44), is to apply ([28, Equation ]) giving being convergent for all . Such sums appear frequently in the literature (e.g., [10, 29] Section ) and have been studied extensively when [10, 30]. In the case , (49) reduces to and (47) vanishes.
Along with (42) the integral in (35) can be similarly split, using the convergent series representation: where the Bernoulli numbers appearing in the original ([28, Equation (50.6.10)]) are related to by Thus the representation becomes (formally) after straightforward integration of (50) with . In (53), the integral converges, and the formal sum conditionally converges or diverges according to several tests. Howver, further analysis is still possible. Following the method of [10, Section 3.3] applied to (49), together with [10, Equation ], we have the following general result for : so that (formally), after premultiplying (55) with , then operating with , Notice that the right-hand side of (55) provides an analytic continuation of the left-hand side for and the right-hand side of (57) gives a regularization of the left-hand side. Putting together (53) and (57), incorporating the first term of (57) into the second term of (53), setting , and constraining eventually yields a new integral representation (compare with (121), [31, Equation ] and [25, Equation ]): which is valid for , a nonoverlapping analytic continuation of (42) that spans the critical strip. Additionally, using (13) followed by replacing gives a second representation: valid for , a non-overlapping analytic continuation of (18). Over the critical strip representations (58) and (59) are simultaneously true and augment the rather short list of other representations that share this property (, as cited in  (unnumbered equations immediately following , also reproduced in [11, Equations ].)). See Section 9 for further application of this result.
4.2. The Case
Interesting results can be obtained by translating the contour in (6) and (9) and adding the residues so-omitted. From (1) and (2) define the partial sums let , with and adjust all the above results accordingly by redefining . (If , an extra half-residue is removed and the contour deformed to pass to the right of the point .) This gives, for example, the generalization of (6): and (9): so (22) and (23) with become integral representations of the remainder of the partial sums and , the former corresponding to a special case of the Hurwitz zeta function .
In the case that , one finds with the latter two results being valid for all . Neither (64) nor (66) appear in references; (65) corresponds to a limiting case of Hermite's representation of the Hurwitz zeta function ([12, Equation ]). In analogy to the developments leading to (27), equations (64), (66), and (3), together with the following modified form of [13, Equation ] can be used to obtain the possibly new result: For corresponding to general partial sums, two possibilities are apparent. Based on (20), (21), (22), and (23) (for comparison, see also [4, Section 8.3], and [7, Section ] and Chapter 7), we findwhich together extend (24) and (26), respectively. Application of (34) and (35) with (20) and (21), also with , gives the remainders which are valid for all . For the case applying (62) and (63) gives the remainderswhich extend (64) and (66), the latter being valid for all . It should be noted that (74), valid for , generalizes well-established results (e.g., [13, Equation ]) that are only valid in the limiting case . (Although [13, Equation ] gives (with a missing minus sign) an integral representation for finite sums such as those appearing in (74), specified to be valid only for integer exponents (), that particular result in  also appears to apply after the generalization .) Omitting the integral term from (70) gives an upper bound for the remainder; similarly, omitting the integral term in (74) gives a lower bound for the remainder, provided that in both cases. In addition, many of the above results, in particular (70) and (74), appear to be new and should be compared to classical results such as [8, Theorem 1.8, page 23 and pages 99-100], and asymptotic approximations such as [33, Equation ].
5. Series Representations
5.1. By Parts
By simple manipulation of the integral representations given, it is possible to obtain new series representations. Starting from the integral representation (29), integrate by parts, giving where the hypergeometric function arises because of [34, Equation ]. Write the hypergeometric function as a (uniformly convergent) series, thereby permitting the sum and integral to be interchanged, apply a second integration by parts, and find Identify the integral in (77) using (18) giving which is valid for all values of by the ratio test and the principle of analytic continuation. This result, which appears to be new, can be rewritten by extracting the term, and applying (13) to yield a representation, valid for all values of , that is reminiscent of the following similar sum recently obtained by Tyagi and Holm ([31, Equation ]): (convergent by Gauss’ test). Equation (79) can also be rewritten in the form by replacing and applying (13).
With reference to (7), express the numerator factor as a difference of exponentials each of which in turn is written as a convergent power series in , and recognize that the resulting series can be interchanged with the integral and each term identified using (9) to find A more interesting variation of (82) can be obtained by applying (3) and explicitly identifying the term of the sum giving a convergent representation if (by Gauss' test). An immediate consequence is the identification because only the term in the sum contributes when . Additionally, when , (83) terminates at terms, becoming a recursion for in terms of —see Section 8 for further discussion. If the functional equation (13) is used to extend its region of convergence, (83) becomes which augments similar series appearing in [10, Sections 4.2 and 4.3], and corresponds to a combination of entries in [28, Equations and ].
5.2. By Splitting
5.2.1. The Lower Range
After applying (13), the integral in (42) is frequently (e.g., ) and conveniently split in two, defining with for arbitrary . Similar to the derivation of (53), after application of (13) we have which can be rewritten using (51) as Substitute (1) in (90), interchange the order of summation, and identify the resulting series to obtain a convergent representation that is valid for all .
Other representations are easily obtained by applying any of the standard hypergeometric linear transforms as in (118)—see Section 6. One such leads to a rapidly converging representation for small values of , valid for all . The inner sum of (92) can be written in several ways. By expanding the denominator as a series in with and interchanging the two series, we find the general form where the right-hand side continues the left if and . Alternatively, [28, Equation ] identifies so for and in (93), each term of the inner series (92) can be obtained analytically by taking the th derivative of the right-hand side with respect to the variable at , leading to a rational polynomial in coth and sinh. The first few terms are given in Table 1 for two choices of .
Application of (93) to (92) leads to a less rapidly converging representation that can be expressed in terms of more familiar functions: A second transform of the intermediate hypergeometric function yields another useful result: Both (95) and (96) can be reordered along a diagonal of the double sum (), allowing one of the resulting (terminating) series to be evaluated in closed form. In either case, both eventually give reducing to (49) in the case taking (13) into consideration. Series of the form (97) have been studied extensively [10, 11] when for special values of . These representations will find application in Section 8.
5.2.2. The Upper Range
Rewrite (87) in exponential form and expand the factor in a convergent power series, interchange the sum and integral, and recognize that the resulting integral is a generalized exponential integral usually defined in terms of incomplete Gamma functions ([1, 35], Section 8.19) as follows: giving a rapidly converging series representation: A useful variant of (99) is obtained by manipulating the well-known recurrence relation ([1, Equation ]) for to obtain Apply (100) repeatedly times to (99) and find in terms of logarithms and polylogarithms (), a result that loses numerical accuracy as increases, but which also, usefully, terminates if , . This is discussed in more detail in Section 8.
Another useful representation emerges from (99) by writing the function in terms of confluent hypergeometric functions. From [35, Equation ] and [36, Equation ], we find where the series on the right-hand side converges absolutely if independent of . Comparison of (99), (88), and (102) identifies which can be interpreted as a transformation of sums, or an analytic continuation, for the various guises of given in Section 5.2.1, if both, or either, of the two sides converge for some choice of the variables and/or , respectively. The fact that a term containing naturally appears in the expression (102) for suggests that there will be a severe cancellation of digits between the two terms and if (88) is used to attempt a numerical evaluation of , as has been reported elsewhere . Essentially, (88) is numerically useless (with common choices of arithmetic precision) for any choices of , even if is carefully chosen in an attempt to balance the cancellation of digits. However, with 20 digits of precision, using 20 terms in the series, it is possible to find the first zero of correct to 10 digits, deteriorating rapidly thereafter as increases. For any choice of , the convergence properties of the two sums representing and anti-correlate, as can be seen from Table 1. This can also be seen by rewriting (102) to yield the combined representation: In (104), the first inner term () vanishes like whereas the second term vanishes like , so for large values of cancellation of digits is bound to occur. (In fact, the coefficients of both asymptotic series prefaced by the exponential terms cancel exactly term by term.)
It should be noted that (102) is representative of a family of related transformations. For example, although setting in (102) does not lead to a new result, operating first with followed by setting leads to a transformation of a related sum: where the first term corresponding to has been isolated, since it contains the singularity that occurs due to divergence of the left-hand sum if . (In (86) and (87), is permitted by a fundamental theorem of integral calculus.) Similarly, operating with on (102) at leads to the identification (which can alternatively be obtained by writing the infinite sum as a product). A second application of on (102), again at using (106), leads to the following transformation of sums: