Royal Society Publishing

Symmetrization of the Hurwitz zeta function and Dirichlet L functions

Ross C McPhedran, Lindsay C Botten, Nicolae-Alexandru P Nicorovici, I John Zucker


We consider the Hurwitz zeta function ζ(s,a), and form two parts ζ+ and ζ by symmetric and antisymmetric combinations of ζ(s,a) and ζ(s,1−a). We consider the properties of ζ+ and ζ, and then show that each may be decomposed into parts denoted by Embedded Image and Embedded Image, each of which obeys a functional equation of the Dirichlet L type, with a multiplicative factor of −1 for the functions Embedded Image. We show the results of this procedure for rational a=p/q, with q=1, 2, 3, 4, 5, 6, 7, 8, 10, and demonstrate that the functions Embedded Image and Embedded Image have some of the key properties of Dirichlet L functions.


1. Introduction

The Hurwitz zeta function (Whittaker & Watson 2004, p. 265),Embedded Image(1.1)is a generalization of the Riemann zeta function, which can be used in the analysis of many one-dimensional sums. It depends on a complex parameter, s, and a real parameter, a, and for specific values of a between 0 and 1, it is connected with Dirichlet L functions (Zucker & Robertson 1976)Embedded Image(1.2)

These in turn are of importance in number theory, and in the evaluation of singly and doubly periodic lattice sums (Zucker & Robertson 1976; Glasser & Zucker 1980).

The L functions are defined by a set of characters Χk(n) (modulo k, an integer), with Χk(n)=±1 for real characters (Zucker & Robertson 1976). Their connection with lattice sums arises from the reflection formula or functional equation they obey,Embedded Image(1.3)andEmbedded Image(1.4)

The construction of L functions from sets of real characters is limited by the multiplicative properties required for sets of characters and, in fact, is only possible for certain values of the degree k (Zucker & Robertson 1975, 1976). On the other hand, we will show that sets of solutions of the functional equations (1.3) and (1.4) can be constructed for arbitrary integers k, so that it is interesting to investigate for applications whether the determining useful feature of the L functions is their link with sets of real characters or their solution of specific functional equations.

Doubly periodic sums involve products of each of the two types of L functions in equations (1.3) and (1.4). For example, Lorenz (1871) and Hardy (1920) showed thatEmbedded Image(1.5)

However, the L functions based on real characters are not adequate to express all double sums in terms of simple combinations. We have found that it becomes very difficult to find explicit expressions for double sums of the formEmbedded Image(1.6)with p1,p2,r integers and p1r/2, p2r/2, as soon as r exceeds 4.

We propose here to investigate the construction of a more general class of functions than the L functions, including of course all L functions with real characters. We will base the construction around the choice of functions obeying functional equations very similar to equations (1.3) and (1.4), in order to preserve the connection with lattice sums. We will show that for rational parameters a=p/q, with q being a positive integer, we can always construct a set of q functions obeying the functional equations, and that these functions share important properties of the L functions. Rather than just two types of function, of positive and negative orders, we divide the functions of both positive and negative order into the functions of positive and negative type, with the result that all the four sets of functions can be made either purely real or purely imaginary on the critical line Re(s)=1/2. We will illustrate the constructional procedure in the main text for the cases q=1–5, and will relegate the results for q=6, 7, 8 and 10 in the electronic supplementary material appendices.

Note that references to the standard formulae include readily available texts, and also an online repository of mathematical formulae (, useful for those who may not have access to a specialized mathematical library.

2. First symmetrization of the Hurwitz zeta function

We wish to consider one-dimensional sums along the line related to the Lerch Phi function and the Hurwitz zeta function. These are customarily defined in different ways, with the former existing in both half- and full-range versions, and the latter only in the half-range form. We need to define two full-range forms for the Hurwitz zeta function.

The Lerch Phi function is, in the half-range form (,Embedded Image(2.1)which converges absolutely if |z|<1, and if |z|=1, it converges for Re(s)>1. The definition requires (−a) not to be zero or a positive integer. We introduce a definition for the full-range version of this function,Embedded Image(2.2)which in this form converges only if |z|=1. We adopt here, and hereafter, the convention in sums that if a term in the sum is singular, it is omitted; thenEmbedded Image(2.3)

The half-range Hurwitz zeta function is ( Image(2.4)

We define two full-range zeta functions, differing in the way the negative k terms are folded into the positive k terms,Embedded Image(2.5)andEmbedded Image(2.6)

Then ζ+(s, a) is a periodic function of a, with unit period. It is even about a=0 and 1/2. It satisfies the partial differential equationEmbedded Image(2.7)

The function ζ(s,a) is odd about a=1/2, differs from one unit interval to the next according toEmbedded Image(2.8)and satisfies the partial differential equationEmbedded Image(2.9)

Hence, we can form Taylor seriesEmbedded Image(2.10)andEmbedded Image(2.11)

The functions ζ+(s, a) and ζ(s, a) have alternating zeros for s equal to a negative integer, for arbitrary a. Indeed, from Abramowitz & Stegun (1972, eqn 23.1.5), the Bernoulli polynomials satisfy the relationEmbedded Image(2.12)and with the Hurwitz zeta expressed in the form (Whittaker & Watson 2004, p. 267)Embedded Image(2.13)for m, a non-positive integer, we haveEmbedded Image(2.14)so thatEmbedded Image(2.15)

Here, Bm(a) denotes the Bernoulli polynomial.

Note that ζ(s, a) has a pole at s=1,Embedded Image(2.16)(

Hence, ζ+(s, a) also has a pole at s=1 with residue 2,Embedded Image(2.17)while ζ(s, a) is regular, thereEmbedded Image(2.18)

We also haveEmbedded Image(2.19)

Note that the case a=1 is exceptional,Embedded Image(2.20)

3. Reflection formulae for ζ+ and ζ

The Lerch Phi function and the Hurwitz zeta function are closely related (,Embedded Image(3.1)

Hence, with p=q=1,Embedded Image(3.2)

This may be written asEmbedded Image(3.3)

The Lerch Phi function also satisfies the relation ( Image(3.4)where p≥0 and q>0 are integers.

We put t=p/q in equation (3.2) and compare it with (3.4), to obtain the identityEmbedded Image(3.5)

We now replace the left-hand side of equation (3.5) with one half of the original sum and the sum with Embedded Image replaced by Embedded Image, and then separate out the exponential into its cos and sin parts. On the right-hand side, we also separate the exponential into cos and sin parts. The result isEmbedded Image(3.6)

The identity (3.6) is true for arbitrary s and, in particular, for real s. Since the ζ functions are real for s real, we can separate equation (3.6) into two parts,Embedded Image(3.7)andEmbedded Image(3.8)

The identity (3.7) contains interdependent terms, since from equation (2.5),Embedded Image(3.9)

Similarly in equation (3.8),Embedded Image(3.10)

In equation (3.7), the result for p=q isEmbedded Image(3.11)so that, using the reflection equation for ζ(s) (Abramowitz & Stegun 1972, eqn 23.2.6),Embedded Image(3.12)we can writeEmbedded Image(3.13)

Further, from equation (3.13),Embedded Image(3.14)

We expand the equation (3.14) in the neighbourhood of s=1 using equation (2.17),Embedded Image(3.15)

Consider now the behaviour of ζ+(s, a) and ζ(s, a) in the neighbourhood of s=1 and 0. We use ( Image(3.16)so thatEmbedded Image(3.17)

We combine the particular expressions (Whittaker & Watson 2004, p. 271)Embedded Image(3.18)andEmbedded Image(3.19)to obtain the Maclaurin seriesEmbedded Image(3.20)which adds the next term to the result (2.19). The behaviour of ζ±(s, a) in the neighbourhood of s=0 follows directly from equation (3.20), i.e.Embedded Image(3.21)

Embedded Image(3.22)

In figures 1 and 2, we illustrate the asymptotic formulae (3.17), (3.21) and (3.22). Note the change in sign of ζ+(δ, a) for a=1/6.

Figure 1

(a) The two functions of a, ζ+(−δ, a)/δ (solid line) and log[2 sin(πa)] (dashed line), (b) the two functions ζ+(1+δ, a)−2/δ (solid line) and −[ψ(1−a)+ψ(a)] (dashed line). In both graphs δ=10−7.

Figure 2

(a) The two functions of a, ζ(−δ, a)+2a−1 (solid line) and δ log[Γ(a)/Γ(1−a)] (dashed line), (b) the two functions ζ(1+δ, a) (solid line) and π cot(πa) (dashed line). In both graphs δ=10−7.

4. Eigenfunctions of the reflection formulae

Consider the reduced form of equation (3.8), taking into account (3.10). The number of independent functions Embedded Image is given by the floor function ⌊(q−1)/2⌋, and Embedded Image. Hence, equation (3.8) becomesEmbedded Image(4.1)for p=1,2, …, ⌊(q−1)/2⌋. Here,Embedded Image(4.2)so thatEmbedded Image(4.3)

We next arrange our functions Embedded Image with Embedded Image in a vector [ζ(s)], and write equation (4.1) in matrix form asEmbedded Image(4.4)

Here, the matrix Embedded Image is independent of s,Embedded Image(4.5)

From equation (4.4),Embedded Image(4.6)so thatEmbedded Image(4.7)

The matrix Embedded Image thus has eigenvalues, which are either ±1. The number of positive and negative eigenvalues is controlled by the traceEmbedded Image(4.8)

This trace then corresponds to a Gauss sum (Berndt & Evans 1981), the value of which is either zero if q≡1,2(mod4) or unity if q≡3,4(mod4). In the former case, ⌊(q−1)/2⌋ is even, and the numbers of positive and negative eigenvalues match. In the latter case, ⌊(q−1)/2⌋ is odd, and there is one more positive than negative eigenvalue.

We write the matrix Embedded Image in the Jordan canonical form of a pre-factor similarity matrix S times a matrix J in Jordan canonical form, times the inverse of S,Embedded Image(4.9)orEmbedded Image(4.10)

If the matrix J is diagonal, with the eigenvalues ±1 arranged along the diagonal, then equation (4.9) takes the form of the reflection equations for [ζ(s)], with S−1[ζ(s)] giving the appropriate linear combinations of Embedded Image entering each of the reflection equations. This will become clearer from the following examples.

One of our goals here is to show how the systems of sums corresponding to the various values of q can be solved in terms of basis functions Embedded Image and Embedded Image. These are split into basis functions of positive (+) order, which correspond to Dirichlet L functions of positive subscript, and negative (−) order, which correspond to Dirichlet functions of negative subscript. They all obey reflection equations of the type to be expected from functions L±q, with a multiplying factor +1 for Embedded Image and −1 for Embedded Image, and all are normalized, thus their leading coefficient is unity for Re(s)→∞. The reflection equations they will obey areEmbedded Image(4.11)Embedded Image(4.12)Embedded Image(4.13)andEmbedded Image(4.14)

Note that L1(s) may be used to construct solutions of equations (4.11) and (4.12) for arbitrary q,Embedded Image(4.15)

Note also that if q has divisors, say q=q1q2, then solutions of the reflection equations for q1 or q2 may be used to construct solutions for q. For example, if Embedded Image and Embedded Image satisfy equations (4.11) and (4.12), respectively, then we can construct solutions for equation (4.11) of the formEmbedded Image(4.16)

We can also construct solutions for equation (4.12) of the formEmbedded Image(4.17)

Naturally, we can interchange q1 and q2, and obtain further solutions. These same remarks apply equally to solutions of equations (4.13) and (4.14).

Given that two solutions of equation (4.11) areEmbedded Image(4.18)then another solution isEmbedded Image(4.19)

A similar argument leads to another solution of equation (4.12),Embedded Image(4.20)

We comment that the functions in equations (4.15)–(4.20) have pre-factors multiplying functions of the Dirichlet type. The zeros of the pre-factors all lie on the critical line Re(s)=1/2, being of the form Embedded Image or Embedded Image, with variously Embedded Image.

Note that the reflection equations (4.11)–(4.14) admit a simplification. We defineEmbedded Image(4.21)and then the reflection equations for these functions are simplyEmbedded Image(4.22)

This shows that these two functions are, respectively, purely real and purely imaginary on the critical line Re(s)=1/2. The corresponding definitions for the other pair of functions areEmbedded Image(4.23)withEmbedded Image(4.24)

If q is such that there exist one or more Dirichlet L functions L±q, then these of course satisfy equation (4.11) for functions of positive subscript, or equation (4.13) for functions of negative subscript. The conditions on q for this to occur have been given by Zucker & Robertson (1976); let P denote an odd integer, which has no squares in its prime factorization. Then, if q=P and P≡1(mod4), there is an Lq, whereas if P≡3(mod4), there is an Lq. If q=4P and P≡1(mod4), there is an Lq, whereas if P≡3(mod4), there is an Lq. Finally, if q=8P, there are both an Lq and an Lq.

For each value of q, we seek a basis of solutions, i.e. q independent functions satisfying the reflection equations. As we have seen that there are ⌊(q−1)/2⌋ such functions of type (−), there will be q−⌊(q−1)/2⌋ functions of type (+).

Consider next the identity (3.7) for the functions ζ+. For q even, these functions will contain elements from systems with lower values of q, already solved for. Thus, we concentrate on the case q odd, and replace q by Q=2q+1. Then equation (3.7) can be written asEmbedded Image(4.25)or, using equation (3.9),Embedded Image(4.26)

We can use equation (3.14) in (4.26) to obtainEmbedded Image(4.27)

Introduce the q×q matrix, Embedded Image, whereEmbedded Image(4.28)

Here, an elementary argument shows that with q=(Q−1)/2,Embedded Image(4.29)so thatEmbedded Image(4.30)for all p=1,2, …, q. Thus, Embedded Image has an eigenvalue Embedded Image, which corresponds to an eigenvector v1 whose q entries are all unity.

We denote the eigenvalues of the real symmetric matrix Embedded Image by λ12, …, λq, and the orthogonal basis of eigenvectors by v1,v2, …, vq. Then equation (4.27) becomes, expanding the solutions as linear combinations of eigenvectors with coefficients ci(s),Embedded Image(4.31)

We can project equation (4.31) onto the basis of eigenvectors. The equation for i=1 isEmbedded Image(4.32)which has the solutionEmbedded Image(4.33)

For i≠1, the equation (4.31) givesEmbedded Image(4.34)which, using Embedded Image, shows that λi=±1 for all i=2,3, …, q. Hence,Embedded Image(4.35)

Note that all eigenvectors vi with i>1 are orthogonal to v1, and thus the sum of their elements is always 0. These properties, together with the fact that as s→1, the pre-factor in equation (4.33) tends to unity, show that the term c1(s)v1 contains the entire contribution of ζ+(s,1) to Embedded Image. Note further that this contribution may be written in terms of the functions introduced in equation (4.15),Embedded Image(4.36)

The trace of the matrix Embedded Image isEmbedded Image(4.37)

This tells us that there is one eigenvalue, Embedded Image, and the other eigenvalues appear in pairs ±1 for q odd, while there is one −1 more than the number of +1's if q is even. Bearing in mind the sign reversal in equation (4.35), this says that the numbers of functions Embedded Image and Embedded Image needed apart from Embedded Image and Embedded Image either are both equal to (q−1)/2 if q is odd, or the number of functions Embedded Image exceeds the number of Embedded Image by 1, with the latter being ⌊(q−1)/2⌋. Denote these by p(Q) and n(Q), respectively.

Using these results, we can write a formal expansion for the set of sums Embedded Image. We renumber the eigenvectors of Embedded Image, so that those corresponding to eigenvalues −1,+1 are separated. Then,Embedded Image(4.38)

In equation (4.38), Embedded Image and Embedded Image denote appropriate linear combinations of solutions of equations (4.11) and (4.12), with the coefficients being independent of s.

5. The cases q=1,2

For q=1, there is only one full-range sum and Dirichlet function,Embedded Image(5.1)

For q=2, ⌊(q−1)/2⌋=0, andEmbedded Image(5.2)

We can re-express these in terms of the following symmetrized functions:Embedded Image(5.3)so thatEmbedded Image(5.4)andEmbedded Image(5.5)

Here, Embedded Image and Embedded Image obey the reflection equations (4.11) and (4.12) with q=2.

Note that the function Embedded Image has its zeros for Embedded Image, in addition to the zeros of L1(s). The symmetrized functions Embedded Image and Embedded Image have their pre-factor zeros at Embedded Image and Embedded Image, respectively.

The expansions (2.10) and (2.11) are particularly simple about a=1/2, and using equation (5.2), they becomeEmbedded Image(5.6)andEmbedded Image(5.7)

We now introduce a vector notation for the basis functions we have found. We note that the series for Embedded Image and Embedded Image, for Re(s) greater than 1 have coefficients of 1/ns, which are periodic with a period of 2. Thus, we can specify these two functions in terms of the first two coefficients in such series,Embedded Image(5.8)

We note that these two vectors are orthogonal, and that in their terms, equation (5.4) becomes ζ+(s, 1)=2s[0, 2]=2[0, 1], while equation (5.5) becomes ζ+(s, 1/2)=2s+1[1, 0].

6. The case q=3

Here, ⌊(q−1)/2⌋=1. Using equation (3.9), the identity (3.7) becomesEmbedded Image(6.1)

The second equation in (6.1) enables us to solve for ζ+(s,(1/3))Embedded Image(6.2)orEmbedded Image(6.3)

The identity (4.1) is a scalar equation, since ⌊(q−1)/2⌋=1,Embedded Image(6.4)

This is exactly the equation satisfied by 3sL−3(s),Embedded Image(6.5)

Combining equations (6.3) and (6.5),Embedded Image(6.6)Embedded Image(6.7)

Both ζ(s, 1/3) and ζ(s, 2/3) have poles at s=1, with unit residue.

We can re-express these results in terms of three symmetrized functions, obeying equations (4.11)–(4.14),Embedded Image(6.8)andEmbedded Image(6.9)

For example,Embedded Image(6.10)

In vector notation,Embedded Image(6.11)

These are orthogonal vectors. Additionally, in this notation, ζ+(s, 1/3)=3s[1, 1, 0] and ζ(s, 1/3)=3s[1, −1, 0].

7. The case q=4

Again ⌊(q−1)/2⌋=1, which means we seek three basis functions of positive type and one of negative type. The result p=1 from equation (3.7) gives usEmbedded Image(7.1)

This can be solved to giveEmbedded Image(7.2)

The single equation from equation (4.1) givesEmbedded Image(7.3)which is the reflection equation for 4sL−4(s). Indeed,Embedded Image(7.4)

The symmetrized functions of positive type areEmbedded Image(7.5)

These satisfy equations (4.11) and (4.12) with q=4. The single symmetrized function of negative type is Embedded Image, which satisfies equation (4.13). In terms of these,Embedded Image(7.6)Embedded Image(7.7)andEmbedded Image(7.8)

Using equation (7.6), we findEmbedded Image(7.9)andEmbedded Image(7.10)

Using vector notation, the basis functions areEmbedded Image(7.11)

The vectors in equation (7.11) are orthogonal.

Note that of the three functions in equation (7.5), the first and the third have their pre-factor zeros on Re(s)=1/2. However, for the second,Embedded Image(7.12)and so the zeros of its two pre-factors lie on Re(s)=0 and 1, respectively. However, note that in equation (7.6), the function of positive type isEmbedded Image(7.13)which has its zeros at Embedded Image, on the critical line. In equation (7.7), Embedded Image has its zeros on Re(s)=1/2.

8. The case q=5

We discuss q=5 in some detail, as it is the first instance where the Dirichlet L functions need supplementation to generate q independent solutions of the appropriate reflection equations. The Dirichlet series of order 5 with real characters areEmbedded Image(8.1)andEmbedded Image(8.2)

Hence, these equations determine the even full-range sumsEmbedded Image(8.3)andEmbedded Image(8.4)

The odd full-range sums occur in two negative parity L-series with imaginary charactersEmbedded Image(8.5)andEmbedded Image(8.6)

Let us now derive the reflection equation applying to ζ(s,(1/5)) and ζ(s,(2/5)). The matrix Embedded Imageof equation (4.5) isEmbedded Image(8.7)which has trace 0 and determinant −1, consistent with its two eigenvalues, −1 and 1. Its Jordan decomposition has J, the diagonal matrix with entries, −1 and 1. The matrix S has its inverse given byEmbedded Image(8.8)

Hence, our two reflection equations areEmbedded Image(8.9)andEmbedded Image(8.10)

The equations (8.9) and (8.10) involve two functions which can be multiplied by the scaling factors. We choose to make the scaling condition, the requirement that each tends to unity for large Re(s). Then, they becomeEmbedded Image(8.11)andEmbedded Image(8.12)

They satisfy the reflection equations (4.13) and (4.14) with q=5.

We can re-express the even full-range sums (8.3) and (8.4) in terms of three basis functionsEmbedded Image(8.13)withEmbedded Image(8.14)andEmbedded Image(8.15)

For the third basis function, we haveEmbedded Image(8.16)

Here, the two functions, Embedded Image and Embedded Image, satisfy the reflection equation (4.11) with q=5, while Embedded Image satisfies equation (4.12).

We show the behaviour of the functions Embedded Image and Embedded Image on the critical line s=1/2+i t in figure 3. The function Embedded Image is zero for s=1/2, since the pre-factor in equation (4.14) is equal to unity there. Note that Embedded Image oscillates more rapidly than Embedded Image; it has 10 zeros in the range 0–30 as against 5 (and in the range of t from 30 to 60, this continues with 18 versus 11 zeros). In table 1, we show the zeros from figure 3.

Figure 3

The absolute value of the two functions Embedded Image (solid line) and Embedded Image (dashed line) on the critical line: s=1/2+it.

View this table:
Table 1

The zeros of Embedded Image and Embedded Image on the critical line: s=1/2+it in figure 3.

In figure 4, we show the behaviour of Embedded Image and Embedded Image on the critical line. The former has zeros at t=(2)/log(5), and the latter at t=[(2n+1)π]/log(5), in addition to the shared zeros, which are those of L1(1/2+i t).

Figure 4

The absolute value of the two functions Embedded Image (solid) and Embedded Image (dashed) on the critical line: s=1/2+it.

Using vector notation, the basis functions areEmbedded Image(8.17)andEmbedded Image(8.18)whereEmbedded Image(8.19)

Written in terms of these vectors,Embedded Image(8.20)andEmbedded Image(8.21)

The ζ expressions areEmbedded Image(8.22)andEmbedded Image(8.23)

Note that the functions P(s) and Q(s) defined from characters including imaginary elements satisfy the reflection equations somewhat more complicated than equations (8.9) and (8.10),Embedded Image(8.24)andEmbedded Image(8.25)

The function Embedded Image is of some interest, as it is used by Titchmarsh (1986, pp. 282–286), who exhibits a proof that this function, which is denoted as f (s), has an infinity of zeros in the half-plane σ=Re(s)>1, with those zeros occurring just to the right of σ=1. The proof compares f (s) with the functionEmbedded Image(8.26)and asserts that, given ϵ>0 and δ>0, there is a τ, such thatEmbedded Image(8.27)

The proof then establishes that f(s) and N(s) each have at least one zero in close proximity. It would be of interest to have a numerical study of zeros close to Re(s)=1.

9. Conclusions

We have shown that the Hurwitz zeta function may be symmetrized, being divided into four components each having properties akin to those of Dirichlet L functions. In particular, they obey appropriate reflection equations, but are not necessarily associated with the sets of characters. We have concentrated on the symmetrization of ζ(s, p/q), for p=1,2, …, q/2, and exemplified the results obtained for q ranging up to 10. The results show that for each value of q, there are q basis functions, of which only one diverges at s=1. All the others have the last component in their representative vectors equal to 0, as is the sum of the elements of the representative vector. These properties, together with the form of the reflection equation, are like those of Dirichlet L functions, and the example for q=5 shows that it is in fact impossible to combine these properties with compatibility with a set of characters.

Of course, the results become steadily more complicated with increasing q, and so it is of value to study the large q asymptotics of the symmetrized functions. We will present such an analysis, together with a study of the properties of lines in the (s, a) space along which the symmetrized zeta functions vanish, in subsequent work.


This work was supported by the Australian Research Council. Numerical, analytic and graphical results were obtained with Mathematica v. 5.0. Helpful discussions with Prof. M. L. Glasser and Prof. D. R. Heath-Brown are acknowledged.



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