We study coefficients bn, expressible as sums over the Li/Keiper constants λj, that contain information on the Riemann xi function. We present a number of relations for and representations of bn. These include the expression of bn as a sum over non-trivial zeroes of the Riemann zeta function, as well as integral representations. Conditional on the Riemann hypothesis, we provide the asymptotic form of .
Let ζ denote the Riemann zeta function, and ξ(s)=(s/2)(s−1)π−s/2Γ(s/2)ζ(s) the classical completed zeta function, where Γ is the gamma function (Riemann 1859–1860; Edwards 1974; Ivić 1985; Titchmarsh 1986). Within the critical strip , 0<Re s<1, the complex zeroes of ζ and ξ coincide, and we denote them by ρ. The ξ-function is entire, of order 1, and of maximal type.
Herein, we mainly investigate certain sums bn over complex zeta function zeroes. We provide various representations and properties of these sums, with the main specific results stated in §2. We also supply some remarks on the Li criterion Li (1997) for the Riemann hypothesis (RH).
We recall that the Li equivalence for the RH results as a necessary and sufficient condition that the logarithmic derivative of the function ξ[1/(1−z)] be analytic in the unit disc. This obtains from a conformal map of the critical strip to this disc. The equivalence Li (1997) states that a necessary and sufficient condition for the non-trivial zeroes of the ζ-function to lie on the critical line Re is that constants are non-negative for every integer k. The sequence can be defined by 1.1 The λj’s are connected to sums over the non-trivial zeroes of ζ(s) by way of (Keiper 1992; Li 1997) 1.2 For n>1, λn is absolutely convergent, while for λ1 the sum should be taken over complex conjugate pairs of zeroes of increasing imaginary part. For further discussion of the Li criterion, its application, and results on series expansion of the ξ function, see, for example, Bombieri & Lagarias (1999) and Coffey (2004, 2005, 2007a, 2008, in press).
In particular, consider the expansions (He et al. 2009) 1.3 The middle expansion in terms of the Li/Keiper constants λn holds for |z|<δ1≤1, where δ1=1 corresponds to the RH. Similarly, the right-most expansion holds for |z+1|<δ2≤2. Thus, expansion (1.3) has overlapping domains of expansion, allowing analytic continuation.
In He et al. (2009), the coefficients bn of expansion (1.3) are simply an inessential device. However, we treat their properties, including a ‘curious identity’ b2=b3, in the next section. The latter relationship is simply the beginning of an infinite set of relations that we make explicit. In the appendix, we record approximate numerical values for the early coefficients. He et al. (2009) contains a physical discussion, including attempting to regard ξ as a quantum mechanical wave function.
To emphasize that the bn’s are not required for the purposes of He et al. (2009), we have the following argument. Let N(T) be the count of non-trivial zeta zeroes for 0<Im ρ<T. We have , and it is well known that, as , . Suppose that the RH holds. Then, we have (e.g. He et al. 2009, 2.14) 1.4 where and . We can rewrite equation (1.4) as 1.5 where the lower limit just as well may be taken as μ1. Integrating by parts, we have 1.6 where . For , we have θ=1/μ+O(1/μ3), , and the required limit on the right-hand side of equation (1.6) is zero. We therefore obtain the equivalent exact forms 1.7 and 1.8 We can note that, in equation (1.8), π/2 may be replaced by , and therefore the range of integration is a relatively narrow one.
We may quickly rewrite equation (1.7) as dθ/dμ=−4/(4μ2+1). Furthermore, , where Uk is the kth Chebyshev polynomial of the second kind, and . We obtain 1.9 thereby recovering equation (3.13) of He et al. (2009). As observed there, on the RH, the values λn are indeed non-negative.
Implicit in He et al. (2009, p. 8) are the relationships 1.10 and, for n≥1, 1.11 That equation (1.11) holds may be easily verified by using He et al. (2009, 3.4) and the orthogonality relationship 1.12 with δjk the Kronecker symbol. We shall have recourse to these relations in the following developments.
The Laguerre polynomials play a role in the following, and we let be the Laguerre polynomial of degree n and parameter α (Andrews et al. 1999). We recall that these polynomials are orthogonal on with weight function xαe−x with α>−1, and possess the generating function 1.13
2. Relationships and representations of bn
For n≥1 we have 2.1
We have b1=0.
We have .
In equation (2.1) the sum includes zeroes ρ along with (1−ρ). (Owing to the functional equation of the ξ or ζ-functions.) We write when the companion zero (1−ρ) is explicitly taken into account.
2.2 Thus 2.3
The coefficient b2n+1 is always expressible as a rational linear combination of b2n, b2n−2,…, b2.
We have b3=b2, b5=2b4−b3, and b7=3b3−5b4+3b6.
We have the relation for n even 2.4
We have the summation relation for n≥2, 2.5
In particular, we have b2=b3.
We have the following representation.
For n≥1, we have 2.6
On the RH, with , and tj is real, we have 2.7
We have the recurrence relation for m≥1 2.9
On the RH, we have, for n≥1, 2.10
On the RH, we have, for n≫1, 2.11
On the RH, we have 2.12
A corollary of proposition 2.14 is corollary 2.13.
In the next section, proofs are supplied, as well as some discussion.
3. Proofs of propositions
Proof of proposition 2.1.
Corollary 2.2 immediately follows as we have 3.2
From the Hadamard product for the ξ-function, we have 3.3 Therefore, , implying corollary 2.3.
Corollary 2.3 recovers what otherwise may be found by applying the functional equation of the ζ function.
Indeed, all odd-order derivatives of ξ are zero at 1/2.
Proof of proposition 2.5.
We use equation (2.3) and put . Then, for each n, Σ2n may be eliminated between b2n and b2n+1, and the result follows. ■
Proof of corollary 2.6.
We have for n even from corollary 2.4 3.4a and 3.4b Therefore, from equation (3.4a) we have 3.5 We insert this equation into equation (3.4b) written in the form 3.6 We find 3.7 Finally, we use a recursion relation for the binomial coefficient, , and obtain equation (2.4). ■
We have Σ4=8(−4b4+3b3), b5=2b4−b3, Σ6=−12(25b3−40b4+16b6), and . Of course, .
Proof of proposition 2.7.
The result is a consequence of the functional equation of the ξ function, so that 3.8 We have 3.9 using 3.10 Then 3.11 Comparing with the expansion (1.3), we obtain 3.12 where we have used b1=0. Relation (2.5) follows. ■
Proof of proposition 2.9.
We use equation (1.11) and proposition 1 of Coffey (2007b) 3.13 so that 3.14 We use the representation (Andrews et al. 1999, p. 286) with Bessel function Jα for α>−1, 3.15 giving 3.16 The integral is first evaluated (Gradshteyn & Ryzhik 1980, p. 716) in terms of the confluent hypergeometric function 1F1 3.17
Here, we have used Kummer’s first transformation for the function 1F1 (Andrews et al. 1999, p. 191) as well as the relation 3.18 The insertion of equation (3.17) into equation (3.14) gives the proposition. ■
By integrating by parts we may verify that equation (2.6) returns the sum representation of proposition 2.1. We have 3.19 The integral converges since necessarily Re ρ>0. We use the Laplace transform of a Laguerre polynomial (Gradshteyn & Ryzhik 1980, p. 844), and we recover equation (2.1).
By multiply differentiating equation (3.17), we have a family of summations, 3.20 Otherwise, we may follow the steps as above and find for α>−1 and z≠1 3.21
We have the contour integral representation 3.22 where the contour encircles the origin in the positive direction and closes at Re . This gives 3.23 It may be verified that the residue at t=u/2 gives .
The defining sums of proposition 2.1 may be recovered from corollary 2.10 in the following way. Write on the RH 3.24
Then use Gradshteyn & Ryzhik (1980, p. 846) for n even and odd to evaluate the integrals.
Proof of proposition 2.11.
This follows from the identity (φ′/φ)φ=φ′. We make use of 3.25 and similarly 3.26 With some further series manipulations we obtain equation (2.9). ■
Proof of proposition 2.12.
Proof of corollary 2.13.
It is evident that equation (2.10) includes the cases b1=0, b2=b3 and b5=2b4−b3.
Simply from the sum representation (2.1) one may suspect an asymptotic form bn∼(1/2n)1/μ1. One could also estimate bn from 3.29 where the contour encircles z=−1.
For one of the integrals in equation (3.28), we first have for −2<Re α<0. Then performing logarithmic differentiation and taking α→−1 we obtain 3.30
Numerically from equation (2.1) as a sum over the first 105 non-trivial zeta zeroes (Odlyzko 2010) we find b1000≃9.21×10−302, while from equation (2.11) we have b1000≃9.22×10−302. See figure 1 for a semilog plot of the first 1000 values of bn versus n (with b1 omitted). These numerical values, suggesting that indeed the right-most expansion in equation (1.3) has radius of convergence 2, could be taken as evidence that the RH holds.
Proof of proposition 2.14.
Proof of corollary 2.13.
We now re-prove this corollary as a result of proposition 2.13. We have by equation (1.11), 3.36 In order to accurately approximate the summand, we use the digamma function ψ=Γ′/Γ, and have for j≫1. Then we have, using an integral representation for ψ (Gradshteyn & Ryzhik 1980, p. 943), 3.37 ■
The result (2.12) is not new (Coffey 2005; Voros 2006), but we include it and the method of proof as a companion to proposition 2.12 and corollary 2.13. We suspect that the o(n) terms in equation (2.12) are of size O(n1/2+ϵ) for any ϵ>0.
Generally, alternating binomial sums may be difficult to estimate, but we have done so in recovering corollary 2.13.
Suppose that λj has a subdominant term close to . Then we expect there to be a correction term in bn close to 3.38 Here, 2F1 is the Gauss hypergeometric function (Gradshteyn & Ryzhik 1980; Andrews et al. 1999), and by transformation rules (Gradshteyn & Ryzhik 1980, p. 1043), the 2F1 function in equation (3.38) is the same as 3.39
The above argument extends so that, if λj has a subdominant term j1/2+ϵ, we expect in bn a term close to 3.40
Appendix A. Values of bn
Exact expressions for bn can be written in terms of , polygammic constants , and the derivatives . Table 1 gives approximate numerical values for the initial bn’s.
The values b0,…,b15 have been obtained in Mathematica by series expansion of the φ function (1.3). The remaining values have been found in Matlab by summing over the first 105 complex zeta zeroes (Odlyzko 2010; equation (2.1)).
- Received February 8, 2010.
- Accepted May 5, 2010.
- © 2010 The Royal Society