## Abstract

A description of the properties of *L*-series with complex characters is given. By using these, together with the more familiar *L*-series with real characters, it is shown how certain two-dimensional lattice sums, which previously could not be put into closed form, may now be expressed in this way.

## 1. Introduction

Lattice sums are expressions of the form(1.1)where the vectors ** l** range over a

*d*-dimensional lattice. The reduction of multiple sums to products of single sums, when accomplished, provides an elegant decrease in complexity. If this can be done, we shall refer to the multiple sums as having been solved ‘exactly’. There are many examples of this in the literature, where two- and higher dimensional lattice sums may be solved exactly in terms of Dirichlet

*L*-series (Glasser & Zucker 1980; Zucker & Robertson 1984; Zucker 1990), but in this communication our concern is with two-dimensional sums for which much more is known. The classic example in this case, often ascribed to Hardy (1920) but first given by Lorenz (1871), is(1.2)The first series on the r.h.s. of (1.2) represents the well-known Riemann zeta function for

*s*>1. The second series is also well known and often referred to as the Catalan beta function. Both of these are examples of series introduced by Dirichlet in order to prove a famous theorem in number theory, namely if

*k*and

*l*are relatively prime integers then the arithmetic progression

*kn*+

*l*contains an infinite number of primes. Here we first survey the properties of

*L*-series, which are relevant to lattice sums discussed below.

Elementary *L*-series (modulo *k*) are given by(1.3)where *k* will be referred to as the *period* or *order* of the *L*-series. In (1.3) *Χ*_{k} is called a *character*. It is a multiplicative function and periodic in *k* and is defined as follows. For *m*, *n* integers(1.4)It was shown by Dickson (1939) that *Χ*_{k} can assume values that are only the *ϕ*(*k*)'th roots of unity, where *ϕ*(*k*) is the Euler totient function that gives the number of positive integers less than *k*, which are relatively prime to *k*. Thus characters may be real or complex. The properties of *L*-series with real characters are well known and have been given elsewhere, e.g. Ayoub (1963) and Zucker & Robertson (1976). However, the properties of *L*-series with complex characters seem less widely known and their attributes are described here. First, for completeness, we summarize the properties of real character *L*-series.

## 2. Properties of *L*-series with real characters

All *L*-series are divided into two types according to whether *Χ*_{k}(*k*−1)=±1. If *Χ*_{k}(*k*−1)=+1, the series will be said to have positive parity. For such series, the *signs* of the characters will be mirrored exactly in the midpoint of the series. If *Χ*_{k}(*k*−1)=−1, the signs of the characters will be mirrored with opposite signs. For real characters, it may be shown that *Χ*_{k}=±1. Theorems apposite to such *Χ*_{k} and given in Ayoub (1963) show that the only possible characters are given by the *Legendre*–*Jacobi*–*Kronecker* symbol, *Χ*_{k}(*n*)=(*n*|*k*). The number of independent real character *L*-series is then found to be as follows. Let *P*=1 or *P* be a product of all *different* primes, i.e. *P* is odd and square-free, then

if

*k*=*P*there is just one primitive*L*-series,if

*k*=4*P*there is just one primitive*L*-series,if

*k*=8*P*there are two primitive*L*-series, andif

*k*=2*P*, 2^{α}*P*where*α*>3, or*P*is not square-free, there are*no*primitive*L*-series.

The definition of a primitive *L*-series is complicated. A good account is given by Ayoub (1963), and it is discussed fully by Zucker & Robertson (1976).

The parity of a real *L*-series is determined as follows.

If

*k*=*P*≡1 (mod4), the*L*-series has positive parity.If

*k*=*P*≡3 (mod4), the*L*-series has negative parity.If

*k*=4*P*with*P*≡3 (mod4), the*L*-series has positive parity.If

*k*=4*P*with*P*≡1 (mod4), the*L*-series has negative parity.If

*k*=8*P*, there is an*L*-series of each type.

The suffix *k* will be *signed* according to whether *Χ*_{k}(*k*−1)=±1. If *s* is a positive integer, then explicit formulae for *L*_{−k}(2*s*−1) and *L*_{k}(2*s*) may be established. They are(2.1)and(2.2)where *B*_{s}(*x*) are the Bernoulli polynomials. As both *n* and *k* are positive integers, and *B*_{s}(1−*n*/*k*) are rational numbers, then for *s* a positive integer(2.3)(2.4)where *R*(*k*) and *R*′(*k*) are rational numbers. It is also known that(2.5)where *h*(*k*) is the class number of the binary quadratic form of discriminant *k* and *ϵ*_{0} is the fundamental unit of the number field . Some examples of real character *L*-series, which are relevant to this report, are given below. It is pertinent here to introduce another notation for *L*-series, which emphasizes their periodicity. Let(2.6)then(2.7)The Riemann zeta function is thus expressed as

The Catalan *β* function is

Other real character *L*-series that will be used here are(2.8)In addition to these, there is always one further real *L*-series of period *k* formed by all the characters *Χ*_{k}(*n*) being equal to 1, but this can be shown to be equal to , where *p*_{n} are all the different prime factors of *k*. This is illustrated below for *k*=5 by making use of the expansion property of (*k*, *l*), thusSoBut (5, 5)=5^{−s}(1, 1) andSums of the form(2.9)have been found exactly using real *L*-series. Some examples are(2.10)In particular, many *Q*(1, 0, *λ*) were solved by expressing them as the Mellin transforms of the product of the Jacobian *θ*_{3}(*q*) functions with different arguments. Thus,(2.11)where(2.12)

By expressing *θ*_{3}(*q*)*θ*_{3}(*q*^{λ})−1 as Lambert series where possible, the integral in (2.11) is easily evaluated in terms of *L*-series. A large list of such results may be found in Glasser & Zucker (1980), supplemented by others in Zucker & Robertson (1984) and Zucker (1990). In a recent work (McPhedran *et al*. 2007) related to the Green's function connected with sums over the square lattice, displaced lattice sums of the form(2.13)were encountered. There were also associated phased lattice sums(2.14)with the displaced and phased lattice sums connected by the Poisson summation formula.

It was found possible to solve *S*(*p*, *r*, *j*) for all *j*=2–4 in terms of real *L*-series, but for *j*=5 no such similar solutions, apart from *S*(1, 2, 5), were found. This prompted us to look for solutions in terms of complex character *L*-series. Whereas the properties of real *L*-series are well known, complex *L*-series seem less well documented, and in the §3 some account of them is given.

## 3. Properties of *L*-series with complex characters

We first list all the possible *L*-series with both the real and complex characters for periods *k*=1–10 and for *k*=16. For *k*, a prime, these are found by taking each *ϕ*(*k*)th root of unity in turn and using the rules for characters given in (1.4) to produce a given series always starting with +1 for the first term. If *k* has factors, then these factors also have period *k* and will reduce the number of independent *L*-series of period *k*. This will become evident from the following:*k*=3, *ϕ*(3)=2; as 1 and 2 are relatively prime to 3, there are two possible order 3 *L*-series*k*=4, *ϕ*(4)=2; as 1 and 3 are relatively prime to 4, two possible order 4 *L*-series exist*k*=5, *ϕ*(5)=4 and the four roots of unity are ±1 and ±i, giving four order 5 *L*-series(3.1)*k*=6, *ϕ*(6)=2; 1 and 5 are relatively prime to 6. There are just two possible *L*-series, both real and both of order lower than 6(3.2)*k*=7, *ϕ*(7)=6; 1, 2, 3, 4, 5, 6 are relatively prime to 7. The sixth roots of unity are 1, −1, *ω*, −*ω*, *ω*^{2}, −*ω*^{2} where *ω*=exp (i*π*/3). The six possible *L*-series are(3.3)*ϕ*(8)=4; 1, 3, 5, 7 are relatively prime to 8. The four order 8 *L*-series are all real(3.4)*ϕ*(9)=6; 1, 2, 4, 5, 7, 8 are relatively prime to 9. There are six order 9 *L*-series(3.5)*ϕ*(10)=4; 1, 3, 7, 9 are relatively prime to 10. The four possible *L*-series are all of order lower than 10(3.6)*ϕ*(16)=8; are 1, 3, 5, 7, 9, 11, 13, 15 are relatively prime to 16, and there are eight *L*-series. As *ϕ*(*k*) gets larger, the depiction of *L*-series in (*k*, *l*) symbols can become unwieldy. It is thus convenient here to introduce a slightly modified notation, namely a *signed* (*k*, *l*) symbol defined by

This enables us to halve the number of symbols required to show an *L*-series. It will be seen that positive parity series are described entirely by (*k*, *l*)_{+} symbols and negative parity series by (*k*, *l*)_{−} symbols. Thus, for *k*=16 we have(3.7)

These results illustrate most of the properties that have been observed with complex *L*-series and allow us to make the following conjectures regarding them.

The number of positive parity series always equals the number of negative parity series.

The number of complex character positive parity

*L*-series is even and they divide into pairs of complex conjugates. The same is true for complex character*L*-series with negative parity.The first and last terms of all

*L*-series are always real.Knowing the parity of the series and the first non-real term, then, using the properties of characters given in (1.4), the coefficients of all the other terms may be established. It allows us to create a concise notation for complex

*L*-series. The subscript gives the parity and period, while the superscript gives the coefficient of the first non-real term, and this specifies the given series completely.The number of (

*k*,*l*) symbols needed to specify a given*L*-series equals the number of*L*-series for a given*k*. Thus, every (*k*,*l*) is expressible as a linear combination of*L*-series of period*k*.

We note from the above that for *k*=6, 10 the inclusion of complex *L*-series does not yield any *L*-series with these periods, and this has been found for *k*=14 and 18. So part of the statement made in §2 regarding real *L*-series seems to hold for complex *L*-series, namely if *k* is equal to twice an odd number no *L*-series of such a period exists. It would appear that this is the result of the fact that *ϕ*(2*n*)=*ϕ*(*n*) if *n* is an odd number. However, whereas for real *L*-series if *k* is a perfect square greater than 4 no such period *L*-series are found, it is seen here that for such *k* complex *L*-series of that period do exist.

It is necessary to point out here that all the statements made about real *L*-series made in §2 have been proved, whereas the conjectures (i–v) made in this section about complex *L*-series have not been proved.

## 4. Expressions for *S*(*p*, *r*, *j*) in closed form for *j*=2–10

It is easily seen that *S*(*p*, *r*, *j*)=*S*(*j*−*p*, *j*−*r*, *j*), so all the independent *S* for a given *j* will be found by allowing both *p* and *r* to take on all values up to and including *j*/2. Then the displaced lattice sums may be expressed as the following Mellin transform:(4.1)The exponential sums are disjoint and each can be evaluated separately. Then letting e^{−t}=*q* and writing(4.2)we have(4.3)For small *j*, *θ*(*j*, *p*) may be expressed in terms of *θ*_{3}-functions (Zucker 1990), and we list the results for *j*=2, 3, 4, 6(4.4)

(4.5)

For these *j* values, all the independent *S*(*p*, *r*, *j*) may be expressed in terms of *Q*(1, 0, *λ*) using (2.11), thus(4.6a)(4.6b)

Of the six different *Q*(*a*, *b*, *c*) that appear in the preceding displaced sums, the values of four have been found in terms of Dirichlet series with real characters. They are(4.7)*Q*(1, 0, 36) and *Q*(4, 0, 9) cannot be individually found.1 However, it will be seen that ±[*Q*(1, 0, 36)+*Q*(4, 0, 9)] appear together in three of the cases considered, and it may be shown via the theory of which numbers the binary quadratic forms (*m*^{2}+36*n*^{2}) and (4*m*^{2}+9*n*^{2}) represent that(4.8)So we are able to express 8 of the 10 possible *S*(*p*, *r*, 6) in terms of known Dirichlet series. All the results for *j*=2–4 and *j*=6 are listed below.(4.9)

It may also be seen that the sum of the two unknown members may also be expressed in terms of Dirichlet series, thusWhether or not *Q*(1, 0, 36) or *Q*(4, 0, 9) may be individually expressed in terms of Dirichlet series with complex characters is still an open question.

For *j*=5, the previous method can be used to establish one factorization (Zucker 1990)(4.10)The other results for *j*=5 all involve Dirichlet series with complex characters and have been obtained (McPhedran *et al*. 2007) by evaluating coefficients *c*_{n} in the expansion for a sufficient set of *c*_{n}. These may be compared with the corresponding expansions generated from appropriate combinations of Dirichlet functions, where, if a product with a pair of complex characters occurs, this must be accompanied by the product with complex conjugated characters, to ensure a real result for Im (*s*)=0. The following two complex *L*-functions of order 20 were required:(4.11)

Then definingwe have the solutions(4.12)

Note that in the expansion of *S*(*p*, *r*, *j*) as a sum over factors 1/*n*^{s}, all terms with non-zero coefficients have . For *j*=5, each modulus value 1, 2, 3, 4, 5 occurs only once.

Following the method described above for *j*=5, it has been possible to find similar results for larger values of *j*. For *j*=7, there are nine independent lattice sums. Of these three on their own were solved. Pairs of the other six sums could also be evaluated in terms of real and complex *L*-series. New complex *L*-series of order 28 were required and these wereLet(4.13)ThenThese three sums that have been solved have expansions with terms with values 6, 5, 3 (mod7), respectively,The three sets of pair relations contain terms with values of 1, 2, 4 (mod7), respectively.

For *j*=8, let

The following three individual sums have been found:

The six remaining independent sums occur as three pairs

For *j*=9 where there are 12 independent terms, the following *L*-series of order 36 were required:Let

The 12 sums divide into six pairs giving

These results as displayed are associated with terms having values 8, 2, 5, 1, 7, 4 (mod9), respectively. The similarity between these results and those of *j*=7 is noteworthy.

For *j*=10 there are 13 independent terms, and the complex *L*-series of order 20 given for *j*=5 were required. LetOf the 13 independent terms, seven could be solved exactly

Of the remaining six forms, the following three pairs have solutions:It appears to us that there is no apparent reason why similar results cannot be found for larger values of *j*. However, at this stage we have no rules for determining which particular *S*(*p*, *r*, *j*) or combination of such terms can be put into closed form and some criteria are desirable in order to go further.

## 5. Exact solutions of a lattice sum involving an indefinite quadratic form

Efforts to solve *Q*(*a*, *b*, *c*) in terms of *L*-series have concentrated on the case when the binary quadratic form is positive definite, i.e. *a*>0 and the discriminant . Following Lorenz (1871), who found an exact form for *T*(1, 0, −1) defined in (5.1) below, Zucker & Robertson (1984) attempted to solve some lattice sums involving indefinite quadratic forms. They investigated(5.1)and in particular found an expression for *T*(1, 0, −*r*^{2}) in terms of (*k*, *l*) symbols defined in (2.6). This was(5.2)

For *r*=1–6 it was found possible to find solutions in terms of *squares* of positive parity real *L*-series, but for larger values of *r* no such solutions could be found. Now it was noted in §3 that every (*k*, *l*) is expressible as a linear combination of *L*-series of period *k* if the complex *L*-series are included. Thus, in principle, (5.2) may be written in *L*-series for *every r*. To illustrate this, closed forms for *r*=1–13 have been evaluated and the outcome displayed in table 1. In the table, we have(5.3)and(5.4)

These results show that, in addition to squares of positive parity *L*-series with real characters, products of pairs of positive parity complex *L*-series will in general be required to solve *T*(1, 0, −*r*^{2}). It is apparent that with sufficient labour there is no limit to which one may go, but until now no clear pattern has yet emerged. We have established numerically, but not analytically, the following functional equation:(5.5)It may also be shown that *T* has the following expansion near its second-order pole at *s*=1:(5.6)*γ*_{1} is the first Stieltjes constant and *C*(*r*) is a constant depending on *r*. The first three are

The expansion near *s*=0 is then(5.7)

## 6. Conclusion

It has been found that certain two-dimensional lattice sums, which hitherto were not expressible in closed form, may now be written exactly if *L*-series with complex characters are employed. This increases considerably the number of two-dimensional lattice sums that can be expressed in closed form. An obvious question is can complex character *L*-series play a role in higher dimensional sums, and this may be an interesting path to pursue.

## Acknowledgments

R.C.M. acknowledges the support of the Australian Research Council, and valuable discussions with Prof. Lindsay Botten and Dr Nicolae Nicorovici.

## Footnotes

↵We are grateful to Mark Watkins, University of Bristol, for obtaining this result for us.

- Received August 15, 2007.
- Accepted January 28, 2008.

- © 2008 The Royal Society