## Abstract

The Riemann hypothesis is equivalent to the Li criterion governing a sequence of real constants that are certain logarithmic derivatives of the Riemann xi function evaluated at unity. A new representation of *λ*_{k} is developed in terms of the Stieltjes constants *γ*_{j} and the subcomponent sums are discussed and analysed. Accompanying this decomposition, we find a new representation of the constants *η*_{j} entering the Laurent expansion of the logarithmic derivative of the Riemann zeta function about *s*=1. We also demonstrate that the *η*_{j} coefficients are expressible in terms of the Bernoulli numbers and certain other constants. We determine new properties of *η*_{j} and *σ*_{j}, where 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 *λ*_{k} (Li 1997). It is an explicit expression in terms of the Stieltjes constants *γ*_{k} and is based upon the work of Matsuoka (1985*a*, 1986). Correspondingly, we find a new representation of the constants *η*_{j} entering the Laurent expansion of the logarithmic derivative of the Riemann zeta function about *s*=1. The subcomponent sums of *λ*_{k} are discussed and analysed. Based upon estimates for the Stieltjes constants, we determine new estimates on |*η*_{j}| and |*σ*_{j}|, where are the sums of reciprocal powers of the non-trivial zeros of the Riemann zeta function. Further properties of the sums *σ*_{j} are derived, including summation results. As the Li constants are the binomial transform of the sequence *σ*_{j}, this transform plays a role in some of our presentation.

In addition, we advance other subjects. We show that the *η*_{j} coefficients are expressible in terms of the Bernoulli numbers *B*_{j} 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 *η*_{j}, *σ*_{k} and the Li/Keiper constants with the statistical properties of Brownian motion.

Our new representation of *λ*_{k} is direct. It complements the arithmetic formula (Bombieri & Lagarias 1999; Coffey 2004, 2005*a*) 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 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 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 2005*a*, 2007*a*). 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 (2005*a*) 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 (2004*a*,*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 *c*_{n} (Coffey 2005*b*). Among the conclusions, we find that exactly *c*_{n}=*S*_{2}(*n*)/*n*, where *S*_{2}(*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 , 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 is defined by(2.1)The *λ*_{j} values are connected to sums over the non-trivial zeros of *ζ*(*s*) by way of (Keiper 1992; Li 1997)(2.2)

In the representation (Bombieri & Lagarias 1999; Coffey 2004, 2005*a*)(2.3)where *γ* is the Euler constant and the sum(2.4)has been characterized (Coffey 2005*a*),(2.5)Further bounds on *S*_{1}(*n*) have been developed by applying Euler–Maclaurin summations to all orders (Coffey 2005*a*). The summand of the quantity *S*_{1} can be written as a sum over the trivial zeros of *ζ* as(2.6)

The focus is then on the sum(2.7)where the constants *η*_{j} can be written as(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 *η*_{j} enter the expansion around *s*=1 of the logarithmic derivative of the zeta function(2.9)and the corresponding Dirichlet series valid for Re *s*>1 is(2.10)

## 3. New representation of *S*_{2}(*n*) and *η*_{j}

In an earlier work (Coffey 2004), we showed low-order examples of how *S*_{2}, and thus *λ*_{k}, could be written in terms of the Stieltjes constants *γ*_{j}. Here, we show how this can be generally done, based upon the work of Matsuoka (1985*a*, 1986). We then make some observations on the contributions of *γ*_{j} to these quantities.

In the Laurent expansion of the zeta function about *s*=1,(3.1)and the Stieltjes constants *γ*_{k} (Stieltjes 1905; Hardy 1912; Kluyver 1927; Briggs 1955; Mitrović 1962; Israilov 1979, 1981; Ivić 1985; Coffey 2006*a*) can be written in the form(3.2)

If we put(3.3)where the sum is over all the complex zeros of the zeta function, then(3.4)and we see from equation (2.2) that the connection between the values *λ*_{n} and the sequence {*σ*_{k}} is(3.5)It can be shown that (Zhang & Williams 1994; Coffey 2005*a*)(3.6)and(3.7)We have, from equation (3.5),(3.8)

Now, Matsuoka (1985*a*, 1986) has shown that(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.

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

(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 {*n*_{j}} sequence would follow. We prove this key fact alternatively in §4. (ii) As the value of *η*_{0} is −*γ*, the full summation term on the r.h.s. of equation (3.11) is equal toas per equation (2.7).

As examples of equation (3.10), we have(3.13a)

(3.13b)and(3.13c)

From equation (3.9), we see that there is always a term present in *σ*_{n} (coming from the *h*=1 term). In addition, there is always a *γ*^{n} term (coming from the *h*=*n* term). Therefore, in the expressions (3.10) or (3.11), there is always a contribution(3.14)In particular, the term *nγ* 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 *λ*_{j} contains the term . We see this term again arising from the *j*=*n* term in equation (3.10) for *λ*_{n}.

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

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

## 4. Further properties of the sequence *σ*_{k} and of *η*_{j}

Here, we present various further properties of the sums of equation (3.3). First, the inverse relation to equation (3.5), expressing the sums *σ*_{k} in terms of the Li constants, is(4.1)This equation follows from use of the orthogonality relation(4.2)where *δ*_{p,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:(4.3)

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

Taking the *j*th derivative of gives(4.9)When evaluated at *s*=1 this equation yields(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) itself(4.11)

We next show how equation (4.10) directly gives a summatory relation for the coefficients *η*_{j}, demonstrating proposition 4.1.

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

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

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

*The sequence* , *with η*_{0}=−*γ*,* has strict sign alternation. That is*, , *for some positive constants c*_{j}.

In regard to equation (4.17), we have(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 (2005*a*).

## 5. Summation representations of the Stieltjes constants and {*η*_{j}}

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 *B*_{j+1}. We then suspect that the *η*_{j} constants are expressible in terms of these numbers and other constants. In fact, we express here both the Stieltjes constants and the *η*_{k} 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:

(5.1)

We have for integers *p*≥1:

(5.2)*where L*_{p} *are logarithmic polynomials and*(5.3)The logarithmic polynomials are expressible in terms of partial Bell polynomials *B _{n,k}* as (Comtet 1974) .

For the proofs of these two propositions, we make use of the alternating zeta function(5.4)valid for Re *s*>0. We observe that(5.5)and(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 *ℓ*=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: . (iii) The convergence of equation (5.1) is such that more than 10 million terms over *n* are required to obtain the first few *γ*_{j} 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 (2006*b*).

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 logarithm(5.7)We then take into account the *ℓ*=0 term of the second line of equation (5.7) and compare with the integrated form of the defining expansion (2.9) for the *η*_{j} coefficients, giving(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 *η*_{k−1}, *σ*_{k}, *λ*_{k} and *S*_{2}(*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 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 *λ*_{k} becomes exponentially large and negative.

In fulfilling this criterion, the crux of the matter is the magnitude of the *η*_{n−1} contribution to *σ*_{n} or *S*_{2}. 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*_{ℓ} means that we have a partition of *k*−*h* over the non-negative integers. All such partitions are considered, meaning that their order does not matter. The number of such partitions isin *η*_{n−1}. Then, using equation (3.15), we are left with considering sums of the form(6.1)This is an overestimation of *η*_{n−1}, since the 4^{h} factor may be reduced, , otherwise is conservative.

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

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 *η*_{j−1} are away from uniformity, the less cancellation will occur in the sum *S*_{2}(*n*). We recall (Coffey 2005*a*) that for large enough *j*, the magnitudes |*η*_{j}| cannot increase more rapidly than 1/3^{j}.

## 7. A probabilistic setting for the *σ* and *η* values

We have previously argued heuristically (Coffey 2005*a*) concerning the sum *S*_{2} of equation (2.7) that for all *ϵ*>0. Indeed, very recent numerical calculations (Smith 1998) for the first approximately 10^{5} *c*_{n}=*S*_{2}(*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 *S*_{2}, 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 2005*a*, 2007*c*).

The Stieltjes constants enter as cumulants (Abramowitz & Stegun 1964; Ehm 2001) of a probability distribution built upon . Indeed, we have by equations (2.9) and (3.12) (Matsuoka 1985

*a*, 1986) that(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.In the elementary relation (Biane

*et al*. 2001; Coffey 2004, 2005*a*)(7.2a)with(7.2b)*k*_{n}is the*n*th cumulant of the random variable , where (Biane*et al*. 2001). Again, the cumulants are related to moments, . We also identify(7.3)Substitution of this relation into equation (7.2*a*) returns equation (3.5). In turn, equation (3.6) directly links*η*_{n−1}with*k*_{n}. Equation (7.3) is evident by either comparing with equation (3.4) or by operating on the Hadamard product for the xi function, . Equations (7.2*a*), (7.2*b*) 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*S*_{2}to, for instance, the asymptotic behaviour of random walks.A necessary and sufficient condition for the Riemann hypothesis to hold is that the integral , 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 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) . 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*S*_{2}(*n*) and therefore for the Li constants.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 of a certain diffusion equation is positive for some

*τ*_{0}>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 *λ*_{n} and *S*_{2}(*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 (2007*b*), wherein it is shown that the subsum *S*_{2}(*n*) may be further decomposed.

The relationship between the {*σ*_{k}} and {*λ*_{n}} sequences is that of the binomial transform, just as *S*_{2}(*n*) is essentially the binomial transform of the sequence . We have now shown the *η*_{j} sequence to be of strict sign alternation, verifying a conjecture of Coffey (2005*a*).

Therefore, we can now write the sum *S*_{2} of equation (2.7) as(8.1)We may note that, with the convention , this sum is an *n*th-order difference of the sequence . The alternating *η*_{k} 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 *λ*_{1} as one of their terms. We have also shown the connection between the *η*_{j} coefficients and the Lehmer sequence {*b*_{m}}.

We have given some reasons why the behaviour of the sum *S*_{2} 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 *λ*_{n} or *η*_{j} 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 (1985*a*, 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 {*b*_{n}}.

We begin with his conclusion, wherein he illustrated the numerical values(A1a)(A1b)and(A1c)where is defined as below equation (4.16). In equations (A 1*a*)–(A 1*c*), the sums are over all the non-trivial zeros of the zeta function. In equation (A 1*c*), 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 1*a*), we easily have(A2)Together, equations (A 1*a*) and (A 2) are equivalent to equation (4.6) of the text. Similarly, we have(A3a)and

(A3b)

Equations (A 1*c*) and (A 3*b*) can be restated as(A4)This equation follows directly from the differentiation of the relation (Lehmer 1988)(A5)Indeed, differentiating equation (A 4) *j* times with respect to *s* gives the identity 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 3*a*) and (A 3*b*). For this, we introduce the Glaisher constant *A*, given by(A6)We then have proposition A.1:

*For* ,(A7a)(A7b)*and*(A7c)*where γ is the Euler constant, and for p∈C*(A7d)

Differentiating equation (3.4) gives(A8)We then put *s*=1−*a*, yielding equation (A7*a*). At *a*=−1, we have the relations *ξ*(2)=*π*/6 and(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 , giving(A10)Using the explicit form of *σ*_{1} in equation (3.7) and noting the relation (A 3*a*) gives equations (A7*b*) and (A7*c*). From the functional equation of *ξ*′/*ξ* and equation (3.4) and multiplying by *s*^{p}, we have(A11)Differentiating this equation with respect to *s* and putting *s*=1 gives equation (A7*d*). ▪

In turn, equation (A 7*d*) at *p*=2 gives, as per equation (3.13*a*), . The result (A 7*d*) 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 1*c*), we have(A12)and from equation (A 7*a*),(A13)In particular, we find(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 have*(A15)*where Λ is the von Mangoldt function*.

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

By summation by parts, we have the alternative expression(A17)where *H*_{n} are the harmonic numbers.

*We have*(A18)*and*(A19)

From proposition A.2 and equation (2.9) at *s*=2, we have(A20)As *η*_{0}=−*γ*, equation (A 18) follows. Equation (A 19) follows by using equation (2.9) at *s*=0. ▪

Corollary A.3 gives the component sums(A21)and

(A22)

Many further summatory relations for the *η*_{j} coefficients may be obtained by proceeding similarly to proposition A.1 or by using equation (3.6).

Lehmer (1988) introduced the function(A23)In view of equation (2.9) and proposition 4.2 of the text, we easily find(A24)and(A25)Equations (A 24) and (A 25) are the examples of another binomial transformation pair. In a sense, *b*_{n} is a complement to the sum *S*_{2}(*n*) of equation (2.7). We have the consistent special cases(A26)In regard to this equation, we have the well-known values *ζ*(0)=−1/2, , and perhaps the not so well-known value (from equation (2.9)) . In the light of Lehmer's relation(A27)for *m*≥1 (Lehmer 1988) and equation (A 25), we have(A28)

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

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*, . In addition, from equation (4.16), we have proposition A.4.

*The sequence {b*_{n}} *has strict sign alternation*.

Illustration of the sign alternation and large-m behaviour of *b*_{m} is given in table 4 of Lehmer (1988). For instance, , as expected. Higher order approximations for large *m* are and , where . In turn, with these approximations, we have from equation (A 24) , which is the expected behaviour in this context. Such straightforward geometric dependence upon p arises when we neglect the complicating perturbations arising from the *σ*_{j} constants. In fact, if we ignore the *σ*_{m+1} term of equation (A 26) altogether, we can easily sum the resulting series(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 *δ*_{n} values. However, we shall not pursue this here, only noting the two special exact values *δ*_{0}=1/2 and *δ*_{1}=(1/2)ln 2*π*−1.

## Footnotes

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

- © 2008 The Royal Society