Royal Society Publishing

New results concerning power series expansions of the Riemann xi function and the Li/Keiper constants

Mark W Coffey


The Riemann hypothesis is equivalent to the Li criterion governing a sequence of real constants Embedded Image that are certain logarithmic derivatives of the Riemann xi function evaluated at unity. A new representation of λk is developed in terms of the Stieltjes constants γj and the subcomponent sums are discussed and analysed. Accompanying this decomposition, we find a new representation of the constants ηj entering the Laurent expansion of the logarithmic derivative of the Riemann zeta function about s=1. We also demonstrate that the ηj coefficients are expressible in terms of the Bernoulli numbers and certain other constants. We determine new properties of ηj and σj, where Embedded Image are the sums of reciprocal powers of the non-trivial zeros of the Riemann zeta function.

1. Introduction

The Riemann hypothesis (Riemann 1859) and its extensions are well recognized to be among the most important problems of mathematical physics and analytic number theory. The apparent pair and higher correlation properties of the zeta zeros have served to intensify the study of random matrices and other areas of mathematical physics (Bogomolny & Keating 1995, 1996). This paper is concerned with the Li equivalence (Keiper 1992; Li 1997, 2004) of the Riemann hypothesis that is described below.

In this paper, we develop a new representation of the Li (Keiper 1992) constants λk (Li 1997). It is an explicit expression in terms of the Stieltjes constants γk and is based upon the work of Matsuoka (1985a, 1986). Correspondingly, we find a new representation of the constants ηj entering the Laurent expansion of the logarithmic derivative of the Riemann zeta function about s=1. The subcomponent sums of λk are discussed and analysed. Based upon estimates for the Stieltjes constants, we determine new estimates on |ηj| and |σj|, where Embedded Image are the sums of reciprocal powers of the non-trivial zeros of the Riemann zeta function. Further properties of the sums σj are derived, including summation results. As the Li constants are the binomial transform of the sequence σj, this transform plays a role in some of our presentation.

In addition, we advance other subjects. We show that the ηj coefficients are expressible in terms of the Bernoulli numbers Bj and certain other constants. We also (see appendix A) obtain analytic results and other properties for Lehmer (1988) sums over the non-trivial zeros of the zeta function. Moreover, we describe possible methods for linking ηj, σk and the Li/Keiper constants with the statistical properties of Brownian motion.

Our new representation of λk is direct. It complements the arithmetic formula (Bombieri & Lagarias 1999; Coffey 2004, 2005a) obtainable from the Guinand–Weil explicit formula (Guinand 1948; Weil 1980) or by other means.

The Riemann hypothesis is equivalent to the Li criterion governing the sequence of real constants Embedded Image that are certain logarithmic derivatives of the Riemann xi function evaluated at unity. This equivalence results from a necessary and sufficient condition that the logarithmic derivative of the function ξ[1/(1−z)] be analytic in the unit disc, where ξ is the Riemann xi function. The Li equivalence (Li 1997) states that a necessary and sufficient condition for the non-trivial zeros of the Riemann zeta function to lie on the critical line Re s=1/2 is that Embedded Image is non-negative for every integer k.

It is possible to use our approach also in pursuit of confirmation of the extended and generalized Riemann hypotheses. The corresponding λ constants have been defined for Dirichlet & Hecke L-functions and other zeta functions (Li 2004), and the same leading behaviour O(j ln j) has been found (Coffey 2005a, 2007a). Our attention here is only with the classical zeta function.

There has been significant interest in the last few years in the Li/Keiper constants (Keiper 1992; Li 1997, 2004) and the accompanying criterion for the Riemann hypothesis. We mention such literature that relates to the concrete application of this criterion. The lengthy reference by Coffey (2005a) proposed such an application, extracted the leading order of the Li/Keiper constants, conjectured their subdominant behaviour and presented limited numerical evidence and many other analytic results, including families of sums over the classical zeta function. An early version of these results was communicated to Lagarias, who has incorporated similar analysis into a setting of automorphic L-functions (Lagarias 2004). In the area of numerical calculation, contributions have been made by Maślanka (2004a,b), which supplement the original analytic and numerical efforts of Keiper (1992). In addition, Smith (1998) proposed a power series development closely related to the Keiper & Li ideas, and we have just recently analysed his coefficients cn (Coffey 2005b). Among the conclusions, we find that exactly cn=S2(n)/n, where S2(n), a subdominant summatory contribution to the Li/Keiper constants, is defined and discussed in the following.

2. Preliminary relations

The function ξ is determined from ζ by the relation Embedded Image, where Γ is the gamma function (Riemann 1859; Edwards 1974; Ivić 1985; Titchmarsh 1986; Karatsuba & Voronin 1992; Davenport 2000) and satisfies the functional equation ξ(s)=ξ(1−s). The sequence Embedded Image is defined byEmbedded Image(2.1)The λj values are connected to sums over the non-trivial zeros of ζ(s) by way of (Keiper 1992; Li 1997)Embedded Image(2.2)

In the representation (Bombieri & Lagarias 1999; Coffey 2004, 2005a)Embedded Image(2.3)where γ is the Euler constant and the sumEmbedded Image(2.4)has been characterized (Coffey 2005a),Embedded Image(2.5)Further bounds on S1(n) have been developed by applying Euler–Maclaurin summations to all orders (Coffey 2005a). The summand of the quantity S1 can be written as a sum over the trivial zeros of ζ asEmbedded Image(2.6)

The focus is then on the sumEmbedded Image(2.7)where the constants ηj can be written asEmbedded Image(2.8)and Λ is the von Mangoldt function (Riemann 1859; Edwards 1974; Ivić 1985; Titchmarsh 1986; Karatsuba & Voronin 1992; Ivić 1993), such that Λ(k)=ln p when k is a power of a prime p and Λ(k)=0 otherwise.

The constants ηj enter the expansion around s=1 of the logarithmic derivative of the zeta functionEmbedded Image(2.9)and the corresponding Dirichlet series valid for Re s>1 isEmbedded Image(2.10)

3. New representation of S2(n) and ηj

In an earlier work (Coffey 2004), we showed low-order examples of how S2, and thus λk, could be written in terms of the Stieltjes constants γj. Here, we show how this can be generally done, based upon the work of Matsuoka (1985a, 1986). We then make some observations on the contributions of γj to these quantities.

In the Laurent expansion of the zeta function about s=1,Embedded Image(3.1)and the Stieltjes constants γk (Stieltjes 1905; Hardy 1912; Kluyver 1927; Briggs 1955; Mitrović 1962; Israilov 1979, 1981; Ivić 1985; Coffey 2006a) can be written in the formEmbedded Image(3.2)

If we putEmbedded Image(3.3)where the sum is over all the complex zeros of the zeta function, thenEmbedded Image(3.4)and we see from equation (2.2) that the connection between the values λn and the sequence {σk} isEmbedded Image(3.5)It can be shown that (Zhang & Williams 1994; Coffey 2005a)Embedded Image(3.6)andEmbedded Image(3.7)We have, from equation (3.5),Embedded Image(3.8)

Now, Matsuoka (1985a, 1986) has shown thatEmbedded Image(3.9)Therefore, by combining this equation with equations (3.7) and (3.8) and using the definition (2.4), we obtain proposition 3.1.

Embedded Image(3.10)Then, comparing with equations (2.3) and (2.7), we identifyEmbedded Image(3.11)Similarly, by comparing equation (3.6) with (3.9), we obtain proposition 3.2.

Embedded Image(3.12)

(i) We note that if one were able to show that the right-most sum in equation (3.12) is always of one sign, then strict sign alternation of the {nj} sequence would follow. We prove this key fact alternatively in §4. (ii) As the value of η0 is −γ, the full summation term on the r.h.s. of equation (3.11) is equal toEmbedded Imageas per equation (2.7).

As examples of equation (3.10), we haveEmbedded Image(3.13a)

Embedded Image(3.13b)andEmbedded Image(3.13c)

From equation (3.9), we see that there is always a term Embedded Image present in σn (coming from the h=1 term). In addition, there is always a γn term (coming from the h=n term). Therefore, in the expressions (3.10) or (3.11), there is always a contributionEmbedded Image(3.14)In particular, the term present in equation (3.11) is always cancelled by the corresponding term in this equation. We may note that already in Coffey (2004) we had deduced that λj contains the term Embedded Image. We see this term again arising from the j=n term in equation (3.10) for λn.

It has been proved that (Berndt 1972)Embedded Image(3.15)which has been improved to (Zhang & Williams 1994)Embedded Image(3.16)In addition, Matsuoka (1985a, 1986) gave an upper bound of Embedded Image on |σn|. This bound is not of direct use to us because it gives an exponentially large upper bound on λn. With the use of equation (3.15) and arguing as in Matsuoka (1985a, 1986), we are able to slightly improve the upper bound on |σn| and |ηn−1| to Embedded Image.

It may be of interest to interchange the two finite sums over j and h in equation (3.11),Embedded Imageand see if useful information can be obtained in this way. We are leaving this investigation to future effort.

4. Further properties of the sequence σk and of ηj

Here, we present various further properties of the sums of equation (3.3). First, the inverse relation to equation (3.5), expressing the sums σk in terms of the Li constants, isEmbedded Image(4.1)This equation follows from use of the orthogonality relationEmbedded Image(4.2)where δp,q is the Kronecker delta. When inserting equation (2.3) into equation (4.1) in order to rederive equation (3.5), the following orthogonality property is useful:Embedded Image(4.3)

The functional equation for the Riemann xi function generates summation relations for the sequence {σj} (Zhang & Williams 1994). When evaluated at s=1 (or the symmetric point s=0), the equation Embedded Image and equation (3.4) giveEmbedded Image(4.4)A further example is the corresponding evaluation at s=2Embedded Image(4.5)Similarly, the logarithmic derivative of equation (3.4) givesEmbedded Image(4.6)andEmbedded Image(4.7)Thus, the combination of equations (4.4) and (4.6) yieldsEmbedded Image(4.8)In turn, equation (4.8) implies that Embedded Image. As numerical examples, we haveEmbedded ImageEmbedded ImageEmbedded ImageEmbedded ImageandEmbedded ImageFrom high precision values of the Stieltjes constants, one may also find the ηj coefficients to high precision from a recursion relation (e.g. Coffey 2004; appendix). Then, equation (3.6) delivers the sums σk. The values of σk decrease quickly with k as described below, and equation (4.8) illustrates the relation Embedded Image.

Taking the jth derivative of Embedded Image givesEmbedded Image(4.9)When evaluated at s=1 this equation yieldsEmbedded Image(4.10)When inserted into equation (3.5), we have yet another representation for the Li constants, which must be equivalent to equation (3.5) itselfEmbedded Image(4.11)

We next show how equation (4.10) directly gives a summatory relation for the coefficients ηj, demonstrating proposition 4.1.

For j≥1Embedded Image(4.12)We insert equation (3.6) into both sides of equation (4.10) and use the sumEmbedded Image(4.13)For equation (4.13), we applied the summation (Coffey 2005a)Embedded Image(4.14)Rearrangement of the terms givesEmbedded Image(4.15)and then (4.12).

In appendix A, we give an alternative method for deriving the summation results of this section for the σj values. We also establish connections with sums considered by Lehmer (1988).

Matsuoka (1985b) has also shown the following:Embedded Image(4.16)where Embedded Image is the first complex zero of ζ(s) and Embedded Image is the second such complex zero. For numerical purposes, we recall that Embedded Image. Equation (4.16), together with equation (3.6) and the fact that for k≥2Embedded Image(4.17)gives proposition 4.2.

The sequence Embedded Image, with η0=−γ, has strict sign alternation. That is, Embedded Image, Embedded Image for some positive constants cj.

In regard to equation (4.17), we haveEmbedded Image(4.18)where ζ(s, a) is the Hurwitz zeta function (cf. equation (2.6)). That is, equation (4.17) holds for all real k>1.

The new result, proposition 4.2, was obtained by the author several years ago, as announced in Coffey (2005a).

5. Summation representations of the Stieltjes constants and {ηj}

If we examine the relation (4.12) for j an odd integer, then the l.h.s. vanishes, while on the r.h.s., ζ(j+1) is expressible in terms of the Bernoulli number Bj+1. We then suspect that the ηj constants are expressible in terms of these numbers and other constants. In fact, we express here both the Stieltjes constants and the ηk values in terms of Bernoulli numbers. Such relations are all the more important because the Bernoulli numbers have many known arithmetic and number theoretic properties.

We have for integers k≥0:

Embedded Image(5.1)

We have for integers p≥1:

Embedded Image(5.2)where Lp are logarithmic polynomials andEmbedded Image(5.3)The logarithmic polynomials are expressible in terms of partial Bell polynomials Bn,k as (Comtet 1974) Embedded Image.

For the proofs of these two propositions, we make use of the alternating zeta functionEmbedded Image(5.4)valid for Re s>0. We observe thatEmbedded Image(5.5)andEmbedded Image(5.6)with the latter expansion based upon the generating function for the Bernoulli numbers. We insert equations (5.5) and (5.6) into the r.h.s. of equation (5.4), and multiply and manipulate the infinite series. The first term on the r.h.s. of equation (5.6) provides the polar term of the zeta function, while the =0 term of a product of sums gives simply a factor of ln 2. Comparing to the defining expansion (3.1) for the Stieltjes constants yields equation (5.1).

(i) It is probable that proposition 5.1 may be obtained by several other methods. (ii) For k=0 in equation (5.1), a rather classical expression for the Euler constant is obtained: Embedded Image. (iii) The convergence of equation (5.1) is such that more than 10 million terms over n are required to obtain the first few γj values to a handful of significant digits. A much faster converging, but more complicated, extension of equation (5.1) is given in proposition 6.1 of Coffey (2006b).

For the proof of proposition 5.2, we insert equations (5.5) and (5.6) into the r.h.s. of equation (5.4) and take the logarithmEmbedded Image(5.7)We then take into account the =0 term of the second line of equation (5.7) and compare with the integrated form of the defining expansion (2.9) for the ηj coefficients, givingEmbedded Image(5.8)By making use of the defining expansion for the logarithmic polynomials (Comtet 1974, p. 140), equations (5.2) and (5.3) follow.

Either directly through proposition 5.2 or by way of the Stieltjes constants, we see that the Li/Keiper constants of equation (2.3) are ultimately expressible in terms of the Bernoulli numbers and certain other constants such as powers of ln n where n≥2.

6. A possible implication

The importance of having explicit representations for ηk−1, σk, λk and S2(k), as in equations (3.9)–(3.12), rather than simply knowing that they exist, should not be underestimated. In this section we describe a possible path for the verification of the Riemann hypothesis based upon criterion (c) of Bombieri & Lagarias (1999). This criterion states that we need to show that there exists a constant c(ϵ) such that Embedded Image for every fixed positive ϵ and each positive integer n in order for the Riemann hypothesis to hold. The essence of this criterion seems to be that we need to show that every Li/Keiper constant is bounded away from −∞. Indeed, the Riemann hypothesis can fail under the Li criterion only if a λk becomes exponentially large and negative.

In fulfilling this criterion, the crux of the matter is the magnitude of the ηn−1 contribution to σn or S2. We describe a possible route for such estimation that depends heavily on known and needed estimates for the Stieltjes constants.

On the r.h.s. of equation (3.12), the constrained sum over the indices j means that we have a partition of kh over the non-negative integers. All such partitions are considered, meaning that their order does not matter. The number of such partitions isEmbedded Imagein ηn−1. Then, using equation (3.15), we are left with considering sums of the formEmbedded Image(6.1)This is an overestimation of ηn−1, since the 4h factor may be reduced, Embedded Image, otherwise Embedded Image is conservative.

In addition, the use of inequality (3.16) gives an improved estimation. For instance, equation (6.1) can be replaced with the expressionEmbedded Image(6.2)where, by the duplication formula satisfied by the gamma function, Embedded Image. Thus, for large k, we may expect an improvement over equation (6.1) by a factor of Embedded Image.

Given proposition 4.2, verification of the Riemann hypothesis may be boiled down once again to estimation of the alternating binomial sum in equation (2.7). The more the values of ηj−1 are away from uniformity, the less cancellation will occur in the sum S2(n). We recall (Coffey 2005a) that for large enough j, the magnitudes |ηj| cannot increase more rapidly than 1/3j.

7. A probabilistic setting for the σ and η values

We have previously argued heuristically (Coffey 2005a) concerning the sum S2 of equation (2.7) that Embedded Image for all ϵ>0. Indeed, very recent numerical calculations (Smith 1998) for the first approximately 105 cn=S2(n)/n values appear to confirm this conjecture. Needless to mention, rigorous confirmation of this statement would provide verification of the Riemann hypothesis. In fact, this conjecture implying the Riemann hypothesis is stronger than it.

In this section we provide yet another perspective. We describe probabilistic connections of the power series expansions employed in this paper. In particular, there are at least four, albeit related, points of view that could connect the η coefficients of equations (2.9) and (3.12), other series coefficients and the sum S2, with the theories of diffusion, Brownian motion and random walk. Underlying much of this connection is the fact that a Jacobi theta function (Coffey 2002), a solution of the heat equation, provides a basis for a (Mellin transform) representation of the Riemann xi function (Riemann 1859; Biane et al. 2001; Coffey 2005a, 2007c).

  1. The Stieltjes constants enter as cumulants (Abramowitz & Stegun 1964; Ehm 2001) of a probability distribution built upon Embedded Image. Indeed, we have by equations (2.9) and (3.12) (Matsuoka 1985a, 1986) thatEmbedded Image(7.1)We recall that the cumulants have direct statistical meaning. As examples, the first cumulant gives the mean of a distribution, the second cumulant the variance, the third cumulant the third moment or skewness and the fourth cumulant is related to the kurtosis.

  2. In the elementary relation (Biane et al. 2001; Coffey 2004, 2005a)Embedded Image(7.2a)withEmbedded Image(7.2b)kn is the nth cumulant of the random variable Embedded Image, where Embedded Image (Biane et al. 2001). Again, the cumulants are related to moments, Embedded Image. We also identifyEmbedded Image(7.3)Substitution of this relation into equation (7.2a) returns equation (3.5). In turn, equation (3.6) directly links ηn−1 with kn. Equation (7.3) is evident by either comparing Embedded Image with equation (3.4) or by operating on the Hadamard product for the xi function, Embedded Image. Equations (7.2a), (7.2b) and (7.3) provide a tight framework for linking the non-negative random variable Y to properties of Brownian excursions (Biane et al. 2001). In this approach, it appears preferable to use supplementary relations such as equation (3.6), as equation (7.3) by itself displays none of the substructure of the Li constants. This may provide an avenue for relating the η values and S2 to, for instance, the asymptotic behaviour of random walks.

  3. A necessary and sufficient condition for the Riemann hypothesis to hold is that the integral Embedded Image, and this formula may be interpreted in terms of Brownian motion also (Balazard et al. 1999). Specifically, for two-dimensional Brownian motion starting from 0 (or from 1), one may put Embedded Image as the first point of impact upon the critical line. Then, the vanishing of the integral just cited is equivalent to the zero expectation value (Balazard et al. 1999) Embedded Image. This interpretation may provide another clue that the application of the theory of Brownian motion, for instance concerning first passage time, could yield very useful information for S2(n) and therefore for the Li constants.

  4. The Riemann hypothesis, and accordingly the Li criterion, are equivalent to the positivity of a certain Weil inner product (e.g. theorem 3.1 of Bombieri (2003)). This can lead to the investigation of the properties of iterated kernels of the diffusion equation. If the Green function Embedded Image of a certain diffusion equation is positive for some τ0>0, the non-negativity of a linear functional follows, and then so does the Riemann hypothesis. An even better approach may be a direct study of the iterates of related kernels of the diffusion equation. In this case, what is needed is a demonstration of positivity of the asymptotic form of an integral that may be interpreted as over Brownian motion on the real line (Bombieri 2003, §8).

8. Summary and brief discussion

Our formulae for λn and S2(n) such as (2.3), (3.8) and (3.11) are effective. One can accordingly algorithmically generate the values for the Li/Keiper constants and all associated sequences and sums. The efficacy of our approach is further shown by the very recent work of Coffey (2007b), wherein it is shown that the subsum S2(n) may be further decomposed.

The relationship between the {σk} and {λn} sequences is that of the binomial transform, just as S2(n) is essentially the binomial transform of the sequence Embedded Image. We have now shown the ηj sequence to be of strict sign alternation, verifying a conjecture of Coffey (2005a).

Therefore, we can now write the sum S2 of equation (2.7) asEmbedded Image(8.1)We may note that, with the convention Embedded Image, this sum is an nth-order difference of the sequence Embedded Image. The alternating ηk values are very far from arbitrary. Indeed, the functional equation of the xi function enforces the summatory relation of equation (4.12). We may expect this equation to have further analytic applications and be a possible useful check for numerical calculations.

We have considered Lehmer sums (Lehmer 1988) over the non-trivial zeros of the zeta function. The Lehmer sums contain the first Li/Keiper constant λ1 as one of their terms. We have also shown the connection between the ηj coefficients and the Lehmer sequence {bm}.

We have given some reasons why the behaviour of the sum S2 may be tied up with the theory of one- or two-dimensional Brownian motion. Indeed, the properties of cumulants and other statistical arguments provide another course for generating positivity results.

The importance of an explicit formula for λn or ηj should not be overlooked. For instance, in principle, only improved estimation of the Stieltjes constants prevents the verification of the Riemann hypothesis by way of either the Li criterion (Li 1997) or criterion (c) of Bombieri & Lagarias (1999). Based upon the work of Matsuoka (1985a, 1986), we have been able to, yet again, derive an arithmetic formula for the Li/Keiper constants, without recourse to the Guinand–Weil explicit formula.

Lehmer sums over the non-trivial zeros of ζ

Lehmer (1988) considered the numerical calculation of the reciprocal-power sums of equation (3.3). We give a number of apparently novel summation relations in connection with his presentation. We also relate the η values of equation (2.9) specifically to his sequence of coefficients {bn}.

We begin with his conclusion, wherein he illustrated the numerical valuesEmbedded Image(A1a)Embedded Image(A1b)andEmbedded Image(A1c)where Embedded Image is defined as below equation (4.16). In equations (A 1a)–(A 1c), the sums are over all the non-trivial zeros of the zeta function. In equation (A 1c), we have introduced a condition, apparently not supplied by Lehmer, which ensures absolute convergence of the sums with parameter a.

In regard to equation (A 1a), we easily haveEmbedded Image(A2)Together, equations (A 1a) and (A 2) are equivalent to equation (4.6) of the text. Similarly, we haveEmbedded Image(A3a)and

Embedded Image(A3b)

Equations (A 1c) and (A 3b) can be restated asEmbedded Image(A4)This equation follows directly from the differentiation of the relation (Lehmer 1988)Embedded Image(A5)Indeed, differentiating equation (A 4) j times with respect to s gives the identity Embedded Image when evaluated at s=0 and relation (4.10) when evaluated at s=1.

The following proposition provides further closed form results for Lehmer's sums (A 3a) and (A 3b). For this, we introduce the Glaisher constant A, given byEmbedded Image(A6)We then have proposition A.1:

For Embedded Image,Embedded Image(A7a)Embedded Image(A7b)andEmbedded Image(A7c)where γ is the Euler constant, and for p∈CEmbedded Image(A7d)

Differentiating equation (3.4) givesEmbedded Image(A8)We then put s=1−a, yielding equation (A7a). At a=−1, we have the relations ξ(2)=π/6 andEmbedded Image(A9)We then use the functional equation of the zeta function to change from ζ′(2) to ζ′(−1), recalling that ζ(−1)=−1/12, and use the well-known relation Embedded Image, givingEmbedded Image(A10)Using the explicit form of σ1 in equation (3.7) and noting the relation (A 3a) gives equations (A7b) and (A7c). From the functional equation of ξ′/ξ and equation (3.4) and multiplying by sp, we haveEmbedded Image(A11)Differentiating this equation with respect to s and putting s=1 gives equation (A7d). ▪

In turn, equation (A 7d) at p=2 gives, as per equation (3.13a), Embedded Image. The result (A 7d) may be extended by the multiple differentiations of equation (A 11) and putting s=0 and 1. It may be additionally extended by multiplying successive differentiated relations by powers of s and evaluating at s=0 and 1.

Many other summatory relations follow from the above. For example, from equation (A 1c), we haveEmbedded Image(A12)and from equation (A 7a),Embedded Image(A13)In particular, we findEmbedded Image(A14)where again the functional equation of ζ could be applied on the r.h.s.

In the course of proposition A.1, we used the value ζ′(2). This leads to proposition A.2.

We haveEmbedded Image(A15)where Λ is the von Mangoldt function.

We have by the Dirichlet series (2.10)Embedded Image(A16)and the proposition follows. ▪

By summation by parts, we have the alternative expressionEmbedded Image(A17)where Hn are the harmonic numbers.

We haveEmbedded Image(A18)andEmbedded Image(A19)

From proposition A.2 and equation (2.9) at s=2, we haveEmbedded Image(A20)As η0=−γ, equation (A 18) follows. Equation (A 19) follows by using equation (2.9) at s=0. ▪

Corollary A.3 gives the component sumsEmbedded Image(A21)and

Embedded Image(A22)

Many further summatory relations for the ηj coefficients may be obtained by proceeding similarly to proposition A.1 or by using equation (3.6).

Lehmer (1988) introduced the functionEmbedded Image(A23)In view of equation (2.9) and proposition 4.2 of the text, we easily findEmbedded Image(A24)andEmbedded Image(A25)Equations (A 24) and (A 25) are the examples of another binomial transformation pair. In a sense, bn is a complement to the sum S2(n) of equation (2.7). We have the consistent special casesEmbedded Image(A26)In regard to this equation, we have the well-known values ζ(0)=−1/2, Embedded Image, and perhaps the not so well-known value (from equation (2.9)) Embedded Image. In the light of Lehmer's relationEmbedded Image(A27)for m≥1 (Lehmer 1988) and equation (A 25), we haveEmbedded Image(A28)

As another example, we have from equation (A 24), differentiating equation (2.9) and evaluating at s=0,Embedded Image(A29)Since Embedded Image from equation (A 27), we see once again that Embedded Image.

Although it does not appear to be stated in Lehmer (1988), the expansion (A 23) holds in the disc |s|<2, the radius of convergence is 2. Consistent with this, we see from equation (A 27) that for large values of m, Embedded Image. In addition, from equation (4.16), we have proposition A.4.

The sequence {bn} has strict sign alternation.

Illustration of the sign alternation and large-m behaviour of bm is given in table 4 of Lehmer (1988). For instance, Embedded Image, as expected. Higher order approximations for large m are Embedded Image and Embedded Image, where Embedded Image. In turn, with these approximations, we have from equation (A 24) Embedded Image, which is the expected behaviour in this context. Such straightforward geometric dependence upon p arises when we neglect the complicating perturbations arising from the σj constants. In fact, if we ignore the σm+1 term of equation (A 26) altogether, we can easily sum the resulting seriesEmbedded Image(A30)In obtaining equation (A 28), we substituted the Dirichlet series of the zeta function, interchanged the resulting double sums and used again the series expression for the zeta function. It is the phase information in the σ values that leads to fine structure in the η values.

We could further discuss the other sequences introduced in Lehmer (1988). For instance, there is another binomial transform pair between the Stieltjes constants and Lehmer's δn values. However, we shall not pursue this here, only noting the two special exact values δ0=1/2 and δ1=(1/2)ln 2π−1.


    • Received September 10, 2007.
    • Accepted November 28, 2007.


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