## Abstract

We study coefficients *b*_{n}, expressible as sums over the Li/Keiper constants *λ*_{j}, that contain information on the Riemann xi function. We present a number of relations for and representations of *b*_{n}. These include the expression of *b*_{n} 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 .

## 1. Introduction

Let *ζ* denote the Riemann zeta function, and *ξ*(*s*)=(*s*/2)(*s*−1)*π*^{−s/2}*Γ*(*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 *b*_{n} 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 *λ*_{j}’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, *λ*_{n} is absolutely convergent, while for *λ*_{1} 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, 2007*a*, 2008, in press).

In particular, consider the expansions (He *et al.* 2009)
1.3
The middle expansion in terms of the Li/Keiper constants *λ*_{n} holds for |*z*|<*δ*_{1}≤1, where *δ*_{1}=1 corresponds to the RH. Similarly, the right-most expansion holds for |*z*+1|<*δ*_{2}≤2. Thus, expansion (1.3) has overlapping domains of expansion, allowing analytic continuation.

In He *et al.* (2009), the coefficients *b*_{n} of expansion (1.3) are simply an inessential device. However, we treat their properties, including a ‘curious identity’ *b*_{2}=*b*_{3}, 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 *b*_{n}’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 *μ*_{1}. Integrating by parts, we have
1.6
where . For , we have *θ*=1/*μ*+*O*(1/*μ*^{3}), , 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/(4*μ*^{2}+1). Furthermore, , where *U*_{k} is the *k*th 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 *λ*_{n} 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 *δ*_{jk} 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 *x*^{α}*e*^{−x} with *α*>−1, and possess the generating function
1.13

## 2. Relationships and representations of *b*_{n}

We have:

### Proposition 2.1.

*For n≥1 we have*
2.1

### Corollary 2.2.

*We have b*_{1}=0.

### Corollary 2.3.

*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.

We have:

### Corollary 2.4.

2.2
*Thus*
2.3

We have:

### Proposition 2.5.

*The coefficient b*_{2n+1} *is always expressible as a rational linear combination of b*_{2n}*, b*_{2n−2}*,…, b*_{2}*.*

### Examples

We have *b*_{3}=*b*_{2}, *b*_{5}=2*b*_{4}−*b*_{3}, and *b*_{7}=3*b*_{3}−5*b*_{4}+3*b*_{6}.

Let .

We have:

### Corollary 2.6.

*We have the relation for n even*
2.4

We have:

### Proposition 2.7.

*We have the summation relation for n≥2,*
2.5

### Corollary 2.8.

*In particular, we have b*_{2}=*b*_{3}.

We have the following representation.

### Proposition 2.9.

*For n≥1, we have*
2.6

### Corollary 2.10.

On the RH, with , and *t*_{j} is real, we have
2.7

Write (Li 1997)
2.8
with on the RH. The rapid asymptotic growth of *a*_{j} with *j* has been described in Coffey (2007*b*). We have:

### Proposition 2.11.

*We have the recurrence relation for m≥1*
2.9

### Proposition 2.12.

*On the RH, we have, for n≥1,*
2.10

### Corollary 2.13.

On the RH, we have, for *n*≫1,
2.11

### Proposition 2.14.

*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.

We substitute the sum (1.2) into equation (1.11), 3.1 The interchange of sums is justified by the absolute convergence of the sum (1.2). ■

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 2.4 (2.2) follows by binomial expansion in the sum (2.1).

### Remarks

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*, *Σ*_{2n} may be eliminated between *b*_{2n} and *b*_{2n+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). ■

### Examples

We have *Σ*_{4}=8(−4*b*_{4}+3*b*_{3}), *b*_{5}=2*b*_{4}−*b*_{3}, *Σ*_{6}=−12(25*b*_{3}−40*b*_{4}+16*b*_{6}), 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 *b*_{1}=0. Relation (2.5) follows. ■

### Proof of proposition 2.9.

We use equation (1.11) and proposition 1 of Coffey (2007*b*)
3.13
so that
3.14
We use the representation (Andrews *et al.* 1999, p. 286) with Bessel function *J*_{α} 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 _{1}*F*_{1}
3.17

Here, we have used Kummer’s first transformation for the function _{1}*F*_{1} (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. ■

### Remarks

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).

The theory of Laguerre polynomials is pervasive in formulating the Li criterion (Coffey 2005, 2007*b*, in press).

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.

We have from equations (1.11) and (1.8) 3.27 We then apply the sum at . Then simple manipulations yield equation (2.10). ■

### Proof of corollary 2.13.

In equation (2.10) we change variable to *x* = (*n* − 1)*θ*/2. At the leading order in *n* we obtain
3.28
Performing the integral gives equation (2.11). ■

### Remarks

It is evident that equation (2.10) includes the cases *b*_{1}=0, *b*_{2}=*b*_{3} and *b*_{5}=2*b*_{4}−*b*_{3}.

The asymptotic result (2.11) is consistent with the right-most expansion in equation (1.3) having a radius of convergence at most 2.

Simply from the sum representation (2.1) one may suspect an asymptotic form *b*_{n}∼(1/2^{n})1/*μ*_{1}. One could also estimate *b*_{n} 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 10^{5} non-trivial zeta zeroes (Odlyzko 2010) we find *b*_{1000}≃9.21×10^{−302}, while from equation (2.11) we have *b*_{1000}≃9.22×10^{−302}. See figure 1 for a semilog plot of the first 1000 values of *b*_{n} versus *n* (with *b*_{1} 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.

*Method 1.* Similarly to corollary 2.13, using (1.8), we have
3.31
Therefore, at the leading order we have
3.32
with the error incurred being *o*(*n*). Using equation (3.30) gives the proposition.

*Method 2.* We alternatively use the expression (1.9) and have the expansions
3.33
and . We have
3.34
We then have
3.35
Using equation (3.30) we find equation (2.12). ■

### 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
■

### Remarks

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*(*n*^{1/2+ϵ}) for any *ϵ*>0.

Generally, alternating binomial sums may be difficult to estimate, but we have done so in recovering corollary 2.13.

Regarding equations (3.36) and (3.37), it is possible to use an even more accurate approximation to , with , but at the cost of a more complicated integral to perform.

Suppose that *λ*_{j} has a subdominant term close to . Then we expect there to be a correction term in *b*_{n} close to
3.38
Here, _{2}*F*_{1} is the Gauss hypergeometric function (Gradshteyn & Ryzhik 1980; Andrews *et al*. 1999), and by transformation rules (Gradshteyn & Ryzhik 1980, p. 1043), the _{2}*F*_{1} function in equation (3.38) is the same as
3.39

The above argument extends so that, if *λ*_{j} has a subdominant term *j*^{1/2+ϵ}, we expect in *b*_{n} a term close to
3.40

## Appendix A. Values of *b*_{n}

Exact expressions for *b*_{n} can be written in terms of , polygammic constants , and the derivatives . Table 1 gives approximate numerical values for the initial *b*_{n}’s.

The values *b*_{0},…,*b*_{15} 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 10^{5} complex zeta zeroes (Odlyzko 2010; equation (2.1)).

- Received February 8, 2010.
- Accepted May 5, 2010.

- © 2010 The Royal Society