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 and , each of which obeys a functional equation of the Dirichlet L type, with a multiplicative factor of −1 for the functions . 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 and have some of the key properties of Dirichlet L functions.
The Hurwitz zeta function (Whittaker & Watson 2004, p. 265),(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)(1.2)
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,(1.3)and(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.
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 form(1.6)with p1,p2,r integers and p1≤r/2, p2≤r/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 (http://functions.wolfram.com), 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 (http://functions.wolfram.com/LerchPhi),(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,(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; then(2.3)
The half-range Hurwitz zeta function is (http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta2)(2.4)
We define two full-range zeta functions, differing in the way the negative k terms are folded into the positive k terms,(2.5)and(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 equation(2.7)
The function ζ−(s,a) is odd about a=1/2, differs from one unit interval to the next according to(2.8)and satisfies the partial differential equation(2.9)
Hence, we can form Taylor series(2.10)and(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 relation(2.12)and with the Hurwitz zeta expressed in the form (Whittaker & Watson 2004, p. 267)(2.13)for m, a non-positive integer, we have(2.14)so that(2.15)
Here, Bm(a) denotes the Bernoulli polynomial.
Note that ζ(s, a) has a pole at s=1,(2.16)(http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta2).
Hence, ζ+(s, a) also has a pole at s=1 with residue 2,(2.17)while ζ−(s, a) is regular, there(2.18)
We also have(2.19)
Note that the case a=1 is exceptional,(2.20)
3. Reflection formulae for ζ+ and ζ−
The Lerch Phi function and the Hurwitz zeta function are closely related (http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/LerchPhi),(3.1)
Hence, with p=q=1,(3.2)
This may be written as(3.3)
The Lerch Phi function also satisfies the relation (http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/LerchPhi)(3.4)where p≥0 and q>0 are integers.
We now replace the left-hand side of equation (3.5) with one half of the original sum and the sum with replaced by , 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 is(3.6)
Similarly in equation (3.8),(3.10)
Further, from equation (3.13),(3.14)
Consider now the behaviour of ζ+(s, a) and ζ−(s, a) in the neighbourhood of s=1 and 0. We use (http://functions.wolfram.com/Zeta)(3.16)so that(3.17)
We combine the particular expressions (Whittaker & Watson 2004, p. 271)(3.18)and(3.19)to obtain the Maclaurin series(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.(3.21)
4. Eigenfunctions of the reflection formulae
Consider the reduced form of equation (3.8), taking into account (3.10). The number of independent functions is given by the floor function ⌊(q−1)/2⌋, and . Hence, equation (3.8) becomes(4.1)for p=1,2, …, ⌊(q−1)/2⌋. Here,(4.2)so that(4.3)
We next arrange our functions with in a vector [ζ−(s)], and write equation (4.1) in matrix form as(4.4)
Here, the matrix is independent of s,(4.5)
From equation (4.4),(4.6)so that(4.7)
The matrix thus has eigenvalues, which are either ±1. The number of positive and negative eigenvalues is controlled by the trace(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 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,(4.9)or(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 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 and . 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 and −1 for , and all are normalized, thus their leading coefficient is unity for Re(s)→∞. The reflection equations they will obey are(4.11)(4.12)(4.13)and(4.14)
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 and satisfy equations (4.11) and (4.12), respectively, then we can construct solutions for equation (4.11) of the form(4.16)
We can also construct solutions for equation (4.12) of the form(4.17)
Given that two solutions of equation (4.11) are(4.18)then another solution is(4.19)
A similar argument leads to another solution of equation (4.12),(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 or , with variously .
Note that the reflection equations (4.11)–(4.14) admit a simplification. We define(4.21)and then the reflection equations for these functions are simply(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 are(4.23)with(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 L−q. If q=4P and P≡1(mod4), there is an L−q, whereas if P≡3(mod4), there is an Lq. Finally, if q=8P, there are both an Lq and an L−q.
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 as(4.25)or, using equation (3.9),(4.26)
Introduce the q×q matrix, , where(4.28)
Here, an elementary argument shows that with q=(Q−1)/2,(4.29)so that(4.30)for all p=1,2, …, q. Thus, has an eigenvalue , which corresponds to an eigenvector v1 whose q entries are all unity.
We denote the eigenvalues of the real symmetric matrix by λ1,λ2, …, 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),(4.31)
We can project equation (4.31) onto the basis of eigenvectors. The equation for i=1 is(4.32)which has the solution(4.33)
For i≠1, the equation (4.31) gives(4.34)which, using , shows that λi=±1 for all i=2,3, …, q. Hence,(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 . Note further that this contribution may be written in terms of the functions introduced in equation (4.15),(4.36)
The trace of the matrix is(4.37)
This tells us that there is one eigenvalue, , 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 and needed apart from and either are both equal to (q−1)/2 if q is odd, or the number of functions exceeds the number of 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 . We renumber the eigenvectors of , so that those corresponding to eigenvalues −1,+1 are separated. Then,(4.38)
5. The cases q=1,2
For q=1, there is only one full-range sum and Dirichlet function,(5.1)
For q=2, ⌊(q−1)/2⌋=0, and(5.2)
We can re-express these in terms of the following symmetrized functions:(5.3)so that(5.4)and(5.5)
Note that the function has its zeros for , in addition to the zeros of L1(s). The symmetrized functions and have their pre-factor zeros at and , respectively.
We now introduce a vector notation for the basis functions we have found. We note that the series for and , 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,(5.8)
6. The case q=3
The second equation in (6.1) enables us to solve for ζ+(s,(1/3))(6.2)or(6.3)
The identity (4.1) is a scalar equation, since ⌊(q−1)/2⌋=1,(6.4)
This is exactly the equation satisfied by 3sL−3(s),(6.5)
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),(6.8)and(6.9)
In vector notation,(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 us(7.1)
This can be solved to give(7.2)
The single equation from equation (4.1) gives(7.3)which is the reflection equation for 4sL−4(s). Indeed,(7.4)
The symmetrized functions of positive type are(7.5)
Using equation (7.6), we find(7.9)and(7.10)
Using vector notation, the basis functions are(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,(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 is(7.13)which has its zeros at , on the critical line. In equation (7.7), 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 are(8.1)and(8.2)
Hence, these equations determine the even full-range sums(8.3)and(8.4)
The odd full-range sums occur in two negative parity L-series with imaginary characters(8.5)and(8.6)
Let us now derive the reflection equation applying to ζ−(s,(1/5)) and ζ−(s,(2/5)). The matrix of equation (4.5) is(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 by(8.8)
Hence, our two reflection equations are(8.9)and(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 become(8.11)and(8.12)
For the third basis function, we have(8.16)
We show the behaviour of the functions and on the critical line s=1/2+i t in figure 3. The function is zero for s=1/2, since the pre-factor in equation (4.14) is equal to unity there. Note that oscillates more rapidly than ; 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.
In figure 4, we show the behaviour of and on the critical line. The former has zeros at t=(2nπ)/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).
Using vector notation, the basis functions are(8.17)and(8.18)where(8.19)
Written in terms of these vectors,(8.20)and(8.21)
The ζ− expressions are(8.22)and(8.23)
The function 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 function(8.26)and asserts that, given ϵ>0 and δ>0, there is a τ, such that(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.
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.