Further Results on the Riemann Hypothesis for Angular Lattice Sums
Ross C. McPhedran Lindsay C. Botten, Nicolae-Alexandru P. Nicorovici
aa r X i v : . [ m a t h - ph ] N ov Further Results on the RiemannHypothesis for Angular Lattice Sums
By Ross C. McPhedran ,Lindsay C. Botten and Nicolae-Alexandru P. Nicorovici CUDOS, School of Physics, University of Sydney, NSW 2006, Australia, School of Mathematical Sciences, University of Technology, Sydney, N.S.W. 2007Australia
We present further results on a class of sums which involve complex powers of thedistance to points in a two-dimensional square lattice and trigonometric functionsof their angle, supplementing those in a previous paper (McPhedran et al , 2008).We give a general expression which permits numerical evaluation of members of theclass of sums to arbitrary order. We use this to illustrate numerically the propertiesof trajectories along which the real and imaginary parts of the sums are zero, andwe show results for the first two of a particular set of angular sums which indicatetheir density of zeros on the critical line of the complex exponent is the same asthat for the product of the Riemann zeta function and the Catalan beta function.
Keywords: Lattice sums, Dirichlet L functions, Riemann hypothesis
1. Introduction
This paper adds to results in McPhedran et al (2008) (hereafter referred to as I)on the properties of a class of sums over two-dimensional lattices involving trigono-metric functions of the angle to points in the lattice, and a complex power 2 s oftheir distance from the lattice origin. There, it was shown that certain of theseangular sums had zeros on the critical line Re( s ) = 1 /
2, but could not have zerosin a neighbourhood of it.We derive a general expression which is exponentially convergent and permitsthe rapid and accurate evaluation of the angular sums irrespective of the valueof the complex parameter s . We demonstrate the high-order convergence of thisformula by using it to illustrate a limiting formula for a particular set of angularlattice sums. We go on to consider the properties of trajectories along which thereal and imaginary parts of a class of angular sums are zero, and in particular weestablish accurate approximations for these trajectories when Re( s ) lies well outsidethe critical strip 0 < Re( s ) <
1. We give preliminary results on the distribution ofzeros on the critical line Re( s ) = 1 / Article submitted to Royal Society
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R.C. McPhedran and others sums and the location of their zeros will be given here, while others will be presentedin a companion paper.There are two principal motivations for the study presented here. The firstis that the very general expression for the class of angular lattice sums derived inSection 2, and their connections with other angular lattice sums shown in AppendixA, enables them to be used in physical applications requiring regularization ofsums over a two dimensional square lattice. The class of summands which canbe addressed is wide, as it consists of any function which has a Taylor series ininteger powers of trigonometric functions of the lattice point angle in the plane,and any complex power of its distance from the origin. For example, any summandof the type often encountered in solid state physics combining a Bloch-type phasefactor and a function of distance having a Taylor series could be so represented.The second is that we show the angular lattice sums to be connected with theproduct of the Riemann zeta function and the Catalan beta function in a verynatural way- for example, it seems that the densities of their zeros on the criticalline are the same to leading orders. In this way, these angular sums may providea new way to forge a link between the Riemann hypothesis and its generalizationto other Dirichlet L functions, as well as providing a wide class of functions, inwhich identified members obey a priori the hypothesis, and others do not. Thereare interesting parallels between this work and that of S. Gonek (2007), althoughwe deal with double sums and Gonek with single sums.
2. An absolutely-convergent expression for angular latticesums
We recall the definition from (I) of two sets of angular lattice sums for the squarearray: C ( n, m ; s ) = X p ,p ′ cos n ( mθ p ,p )( p + p ) s , S ( n, m ; s ) = X p ,p ′ sin n ( mθ p ,p )( p + p ) s , (2.1)where θ p ,p = arg( p + i p ), and the prime denotes the exclusion of the point atthe origin. The sum independent of the angle θ p ,p was evaluated by Lorenz (1871)and Hardy(1920) in terms of the product of Dirichlet L functions: C (0 , m ; s ) = S (0 , m ; s ) ≡ C (0 , s ) = 4 L ( s ) L − ( s ) = 4 ζ ( s ) L − ( s ) . (2.2)Here L ( s ) is more commonly referred to as the Riemann zeta function, and L − ( s )as the Catalan beta function. A useful account of the properties of Dirichlet L functions has been given by Zucker & Robertson (1976).It is convenient to use a subset of the angular sums (2.1) as a basis for numericalevaluations. We note that the sums C ( n, s ) are zero if n is odd. We next derivethe following relationship for the non-zero sums C (2 n, s ): X ( p ,p ) ′ p n ( p + p ) s + n = C (2 n, s ) = 2 √ π Γ( s + n − / ζ (2 s − s + n )+ 8 π s Γ( s + n ) ∞ X p =1 ∞ X p =1 (cid:18) p p (cid:19) s − / p n p n π n K s + n − / (2 πp p ) , (2.3) Article submitted to Royal Society ew Properties of Angular Lattice Sums K ν ( z ) denotes the modified Bessel function of the second kind, or Macdonaldfunction, with order ν and argument z . The general form (2.3) may be derivedfollowing Kober (1936) in the usual way: a Mellin transform is used to give X ( p ,p ) ′ p n ( p + p ) s + n = X ( p ,p ) ′ p n Γ( s + n ) Z ∞ t s + n − e − t ( p + p ) d t. (2.4)The Poisson summation formula is then used to transform the sum over p , giving X ( p ,p ) ′ p n ( p + p ) s + n = X ( p ,p ) ′ p n Γ( s + n ) Z ∞ t s + n − e − tp r πt e − π p /t d t. (2.5)We then separate the axial contribution, which for n = 0 comes from p = 0 alone,and use Hobson’s integral Z ∞ t s − e − pt − q/t d t = 2 (cid:18) qp (cid:19) s/ K s (2 √ qp ) (2.6)on the remaining double sum. This leads directly to (2.3).It should be noted that the double sum in (2.3) is exponentially convergent.Indeed, from relation 9.7.2 in Abramowitz and Stegun (1972), the large argumentapproximation for the Macdonald function of order ν is K ν ( z ) ∼ r π z e − z . (2.7)This means that the double sum starts to converge rapidly as soon as the argument2 πp p everywhere exceeds the modulus of the order s + n − /
2. In practice, accu-rate answers are achieved when sums are carried out over p and p from 1 to P ,where P ∼ | s + n − / | /π (the precise value of P required being fixed by studiesof the effect of increasing P on the stability of the result). The representation (2.3)and finite combinations of it thus furnish absolutely convergent representations oftrigonometric sums from the family (2.1) and close relatives, for any values of s with finite modulus. These representations are easily represented numerically inany computational system incorporating routines for the Riemann zeta function ofcomplex argument, and Macdonald functions of complex order and real argument.(In practice, the Macdonald function evaluations are most time-expensive in the re-gion of ( p , p ) values where p p has comparable magnitude to | s + n − / | / (2 π ).Thus, it is efficient to create a table of these values for p varying with p = 1, run-ning up to an argument where the Macdonald function is less than an appropriatetolerance times its value for p = 1.)As an example of the numerical efficacy of (2.1), we consider its use in illustratinga limiting property of the sums C (2 m, s ).We have C (2 m, s ) = X ( p ,p ) ′ cos m θ p ,p ( p + p ) s , (2.8)and as m → ∞ we require | cos θ p ,p | = 1 for a contribution, i.e.lim m →∞ C (2 m, s ) = 2 ζ (2 s ) . (2.9) Article submitted to Royal Society
R.C. McPhedran and others
Figure 1. The modulus of C (2 m, s ) (red) and 2 ζ (2 s ) (green) as a function of s = 1 / t , for t ∈ [0 , m = 10 (left) and m = 100 (right). The relationship (2.9) is illustrated numerically in Fig. 1. For the right-hand side of(2.9) to be accurate, the required order m increases with t , although convergenceis also slow for t near zero.Another angular sum of great importance in this paper can easily be expandedin terms of the C (2 n, s ): C (1 , m ; s ) = X p ,p ′ cos(4 mθ p ,p )( p + p ) s = C (2 , m ; s ) − S (2 , m ; s ) , (2.10)or, in terms of the Chebyshev polynomial of the first kind (Abramowitz & Stegun(1972), Chapter 22), C (1 , m ; s ) = X p ,p ′ T m (cos θ p ,p )( p + p ) s . (2.11)As the coefficients of this Chebyshev polynomial are explicitly known, the represen-tation (2.11) enables any sum C (1 , m ; s ) to be expressed as a linear combinationof sums C (2 n, s ) with 0 ≤ n ≤ m .The connections between various angular lattice sums grouped in systems withorder up to 10 are explored in Appendix A.
3. Some properties of trigonometric lattice sums
The functional equation is known (see McPhedran et al. (2004), eqs. 32 and 59) for C (1 , m ; s ): G m ( s ) = C (1 , m ; s ) Γ( s + 2 m ) π s = G m (1 − s ) . (3.1)This equation also holds for m = 0, where it gives the functional equation forthe product ζ ( s ) L − ( s ). It is in fact the m dependence of the functional equation(3.1) which enables the derivation of many of the results in (I) and the presentpaper. As the derivation in McPhedran et al (2004) uses different notation to thatin subsequent papers and here, we give a brief discussion of the argument leadingto (3.1) in Appendix B.This m dependence in (3.1) is represented in two related functions: F m ( s ) = Γ(1 − s + 2 m )Γ( s )Γ(1 − s )Γ( s + 2 m ) = exp(2i φ m ( s )) , (3.2) Article submitted to Royal Society ew Properties of Angular Lattice Sums φ m ( s ) is in general complex. Note that F m ( s ) is the ratio of two polynomialsof degree 2 m , with one obtained from the other by replacing s by 1 − s : F m ( s ) = (2 m − s )(2 m − − s ) . . . (1 − s )[2 m − (1 − s )][2 m − − (1 − s )] . . . [1 − (1 − s )] . (3.3)We then introduce two sets of rescaled lattice sums: e C (2 , m ; s ) = Γ( s )2 π s p F m ( s ) [ C (0 , s ) + C (1 , m ; s )] , (3.4)and e S (2 , m ; s ) = Γ( s )2 π s p F m ( s ) [ C (0 , s ) − C (1 , m ; s )] . (3.5)Note that this definition means e C and e S have branch cuts where F m ( s ) is real andnegative. For example, for F ( s ), the branch cut includes a circle in the ( σ, t ) plane( s = σ + i t ), with centre (1 / ,
0) and radius √ / (2 , m ; s ) = e C (2 , m ; s ) − e S (2 , m ; s ) = Γ( s ) π s F m ( s ) C (0 , s ) C (1 , m ; s ) . (3.6)It obeys the functional equation∆ (2 , m ; s ) = F m (1 − s )∆ (2 , m ; 1 − s ) , (3.7)and on the critical line s = 1 / t ,∆ (2 , m ; s ) = [1 − i tan( φ m ( s ))][ | e C (2 , m ; s ) | − | e S (2 , m ; s ) | ] , (3.8)and Im[ e C (2 , m ; 1 / t )] ≡ Im[ e S (2 , m ; 1 / t )] . (3.9)We note from (3.8) thatIm ∆ (2 , m ; 12 + i t ) = − tan( φ m,c t ) Re ∆ (2 , m ; 12 + i t ) , (3.10)using the notation φ m (1 / t ) = φ m,c ( t ), a real-valued function. We can takethe derivative with respect to t of (3.10), to obtain: ∂∂t Im ∆ (2 , m ; 12 + i t ) + tan φ m,c ( t ) ∂∂t Re ∆ (2 , m ; 12 + i t )= − φ ′ m,c ( t )cos ( φ m,c ( t )) Re ∆ (2 , m ; 12 + i t ) . (3.11)We use the Cauchy-Riemann equations in (3.11), to obtain ∂∂σ Re ∆ (2 , m ; 12 + i t ) + tan φ m,c ( t ) ∂∂t Re ∆ (2 , m ; 12 + i t )= − φ ′ m,c ( t )cos ( φ m,c ( t )) Re ∆ (2 , m ; 12 + i t ) , (3.12) Article submitted to Royal Society
R.C. McPhedran and others and ∂∂t
Im ∆ (2 , m ; 12 + i t ) − tan φ m,c ( t ) ∂∂σ Im ∆ (2 , m ; 12 + i t )= − φ ′ m,c ( t )cos ( φ m,c ( t )) Re ∆ (2 , m ; 12 + i t ) . (3.13)The equation (3.12) indicates that, at points where contours of Re ∆ (2 , m ; σ +i t ) = 0 intersect the critical line, their tangent vector is given by (1 , tan φ m,c ( t )),provided ∂ ∆ (2 , m ; + i t ) /∂t = 0. (This requirement is that the left-hand sideof (3.12) can be interpreted as the scalar product of the tangent vector and thegradient of Re ∆ , with the latter having a well-defined direction.) The equation(3.13) indicates that the tangent vectors for the contours of Im ∆ = 0 at the criticalline are given by ( − tan φ m,c ( t ) , | s | >>
1, or, more strictly, for | s | >> m , F m ( s ) ≃ − m s − / m ( s − / , φ m ( s ) ≃ m is − / im (8 m − s − / . (3.14)Thus, for | t | >>
1, (3.12) takes the approximate form ∂∂σ
Re ∆ (2 , m ; 12 + i t ) + 2 m t ∂∂t Re ∆ (2 , m ; 12 + i t ) = 2 m t Re ∆ (2 , m ; 12 + i t ) . (3.15)This shows that, as t increases, the contours of Re ∆ (2 , m ; σ + i t ) = 0 strike thecritical line at ever flatter angles (although the angle increases as m increases). Thedirection of the intersection with the critical line is unique for every point where ∂ ∆ (2 , m ; + i t ) /∂t = 0. Similar remarks apply to Im ∆ (2 , m ; σ + i t ) = 0, wherethe equivalue contours cut the critical line at a direction given by the tangent vector( − m /t, t .
4. Equivalue contours of
Re ∆ and Im ∆ In Figs. 2-4 we show contours on which the real and imaginary parts of ∆ (2 , s )are zero. Fig. 2 gives some detail of the region near σ = 0 . t ∈ [0 . , F ( s ) is real). Fig. 3 gives a more global view of null contours for t ranging up to 20. Fig. 4 gives the detail of the null contours for t near two values atwhich contours nearly touch. We introduce the notation 14 for a zero on σ = 1 / C (1 , s ), +1 for a zero of ζ ( s ) and − L − ( s ). Then the zeros evidentin Fig.3 are categorized as: − , , , − , , − , +1 , , − , , − , , σ axis, which decreases as t increases, while the contours for the imaginarypart strike the critical line almost vertically. Secondly, the contours for the real andimaginary parts intersect the critical line simultaneously, except for one point. Thisis at (1 / , √ / φ (1 / i p /
2) = π/
2, which permitsRe ∆ (2 , s ) to be zero, while Im ∆ (2 , s ) is non-zero (see (3.8)). Article submitted to Royal Society ew Properties of Angular Lattice Sums - - Figure 2. Null contours of the real part (red) and imaginary part (blue) of ∆ (2 , σ + i t ),with (left) σ ∈ [ − . , . t ∈ [0 , t ∈ [0 . , . F ( s ) isreal and negative. We can see four null contours for the real part and five null contours for theimaginary part intersecting the axis t = 0. If we use the expression∆ (2 , s ) = Γ( s ) s ( s + 1)(2 − s )(1 − s ) π s [ C (2 , s ) − S (2 , s )] , (4.1)we find that ∆ (2 , s ) has a second order pole at s = 1, and first order poles at s = 0 and s = 2. Numerical investigations show that near these points∆ (2 ,
2; 1 + δ ) ≃ − . δ , (4.2)and ∆ (2 , δ ) ≃ − . δ , ∆ (2 ,
2; 2 + δ ) ≃ . δ . (4.3)We note that Re ∆ (2 , s ) has zeros at the two first-order poles, where the tra-jectories approach the poles broadside (i.e., parallel to the t axis), and two nulltrajectories approaching the second-order pole at 45 ◦ and 135 ◦ to the σ axis.Im ∆ (2 , s ) has a zero at the second-order pole, where the null trajectory ap-proaches broadside. The other null trajectory crossings of the real axis occur at0.29782, 1.67735, -2.65568 and 4.21422. The last two of these are associated withminima of Re ∆ (2 , σ ).We show in Fig. 5 the behaviour of the null contours for σ large enough ( σ > C (2 , s ) − S (2 , s ) ≃ s + 365 s +2 ) , (4.4)so that ∆ (2 , s ) ≃ (cid:20) − s )Γ( s + 2)Γ( s )Γ(3 − s ) π s (cid:21) (1 + 14 s + 365 s +2 ) . (4.5) Article submitted to Royal Society
R.C. McPhedran and others
Figure 3. Null contours of the real part (red) and imaginary part (blue) of∆ (2 , σ + i t ), with σ ∈ [ − . , . t ∈ [0 . , For σ beyond 4.21422, we see from Fig. 5 that neither the null trajectories of thereal part or the imaginary part of ∆ (2 , s ) can attain the real axis. In Fig. 6,we show for comparison the null trajectories associated with the prefactor term insquare brackets in (4.5). It is evident from a comparison of Figs. 5 and 6 that thenull trajectories are given accurately by the prefactor in the range of σ shown. UsingStirling’s formula (Abramowitz and Stegun (1972)), we can place the prefactor inexponential form for | s | large, and find the constraint for null trajectories to beIm[2 s log( sπ ) − s − log( s ) + 256 s + 2 s ] = ( n + 12 ) π, or nπ, (4.6)where on the right-hand side the first value is for real-part zeros, and the secondfor imaginary-part zeros ( n being an integer). From (4.6) we find that the equiv-alue contours alternate for σ large, with null contours for the real part sandwichedbetween those for the imaginary part, and vice versa (with each trajectory corre-sponding to a first-order zero). Each contour tends to zero as 1 / log( σ/π ) as σ → ∞ . Article submitted to Royal Society ew Properties of Angular Lattice Sums - - Figure 4. Detail of the null contours of the real part (red) and imaginary part (blue) of∆ (2 , σ + i t ), for t near 10 (left) and near 18.5 (right). Figure 5. Equivalue contours of the real part (red) and imaginary part (blue) of the form(4.5) for ∆ (2 , σ + i t ), with σ ∈ [3 . , . t ∈ [0 . , Note that we may use the equation (3.7) to deduce relations governing the phaseof ∆ (2 , m ; s ). Dividing (3.7) by its conjugated form, we obtainarg(∆ (2 , m ; s )) + arg(∆ (2 , m ; 1 − ¯ s )) = − arg( F m ( s )) . (4.7)This agrees with (3.8) and (3.10) when Re( s ) = 1 /
2. Furthermore, on the right-handside of Fig. 3 we know that arg(∆ (2 , s ) increases monotonically with t , from itsvalue of zero on the real axis. This enables us to assign values of the constant phaseof the null contours there: π/ π , − π/
2, 0, π/
2, etc. Using (4.7) and (3.14), we seethat the monotonic increase of arg(∆ (2 , s ) with t on the right in Fig. 3) forcesa monotonic decrease of arg(∆ (2 , s ) with t on the left. We can then assign the Article submitted to Royal Society R.C. McPhedran and others
Figure 6. Null contours of the real part (red) and imaginary part (blue) of the prefactorterm in the expression (4.5) for ∆ (2 , σ + i t ), with σ ∈ [3 . , . t ∈ [0 . , phase values of the null contours there, again starting from zero on the real axis: − π/ − π , π/
2, 0, − π/
2, etc.We can use this information to understand the behaviour of the null contoursshown in Fig. 3. As the null contours of Re(∆ (2 , m ; s )) pass through a simplezero, their phase must change by π , from π/ − π/
2, or vice versa. However, wenote that the relevant contours on the left in Fig. 3 which are almost symmetricto those on the right have opposite phase values. This means the null contours ofRe(∆ (2 , m ; s )) vary smoothly as they cross the critical line. For the null contoursof Im(∆ (2 , m ; s )) the situation is different: the phase change of π forces themto avoid their almost symmetric counterpart on the left and ”jump up” a contouras they pass through a zero on the critical line. These remarks are in accord withthe derivative estimates at the critical line (see equation (3.15) and subsequentdiscussion).
5. Some properties of zeros
The properties of the angular sums given in the preceding sections enable simpleproofs to be given of some properties concerning the location of zeros of angularlattice sums. These properties form an interesting counterpoint to those of thezeros of Epstein zeta functions (see Bogomolny and Lebouef (1994) and references[26-30] of that paper). The Epstein zeta functions are characterized as having aninfinite number of zeros lying on the critical line, but with many zeros lying offthat line and with almost all zeros lying on the critical line or in its immediateneighbourhood. We have already seen in the example of Section 2 that this thirdproperty of Epstein zeta function zeros is not shared by those of the trigonometricdouble sum C (2 m, s ) for large m . Theorem 5.1.
The trigonometric sums e C (2 , m ; s ) and e S (2 , m ; s ) have no zerosfor s on the critical line in the asymptotic region t >> m which are not zeros ofboth, and these will then be simultaneous zeros of C (0 , s ) and C (1 , m ; s ) . Article submitted to Royal Society ew Properties of Angular Lattice Sums Proof.
We combine equations (3.4) and (3.5), to obtain e C (2 , m ; s ) + e S (2 , m ; s ) = 1 p F m ( s ) (cid:20) Γ( s ) π s C (0 , s ) (cid:21) . (5.1)As the term in square brackets is real on the critical line, from the functionalequation (3.1), we see thatarg[ e C (2 , m ; 12 + it ) + e S (2 , m ; 12 + it )] = − φ m ; c ( t ) + (cid:20) π (cid:21) . (5.2)The second term on the right-hand side of (5.2) is chosen according to the sign ofthe term in square brackets in (5.1). We can also use (3.9) to deduce thatarg[ e C (2 , m ; 12 + it ) − e S (2 , m ; 12 + it )] = (cid:20) π (cid:21) . (5.3)Note that the term on the right-hand side in (5.3) incorporates the phase of thereal function on the left-hand side, and is not necessarily the same as the secondterm on the right-hand side of (5.2).If we suppose that e S (2 , m ; + it ) = 0, we require from (5.2) and (5.3) that φ m ; c ( t ) = nπ for some integer n . We know from (3.14) that this cannot occur.Exactly the same argument applies if e C (2 , m ; + it ) = 0, so neither function canbe separately zero on the critical line in the asymptotic region. If both are zerothen, trivially, C (0 , s ) = 0 and C (1 , m ; s ) = 0.We now consider the phases of e S (2 , m ; + it ) and e C (2 , m ; + it ), particularlyin the neighbourhood of zeros C (0 , s ) or C (1 , m ; s ). For brevity, we will adoptthe following notations: | e C (2 , m ; 12 + it ) | = | e C | , arg e C (2 , m ; 12 + it ) = Θ c , (5.4)and | e S (2 , m ; 12 + it ) | = | e S | , arg e S (2 , m ; 12 + it ) = Θ s . (5.5)Then on the critical line, we have for e C and e S that e C (2 , m ; 12 + it ) = | e C | (cos Θ c + i sin Θ c ) , e S (2 , m ; 12 + it ) = | e S | (cos Θ s + i sin Θ s ) , (5.6)and | e C | sin Θ c = | e S | sin Θ s . (5.7)Using (5.6) and (5.7), we can rewrite (5.2) as2 sin Θ c sin Θ s sin(Θ c + Θ s ) = − tan φ m,c ( t ) . (5.8)With a prime denoting t derivatives, we obtain by differentiating (5.8)(2 cos Θ c sin Θ s )Θ ′ c + (2 sin Θ c cos Θ s )Θ ′ s = − φ ′ m,c ( t )cos φ m,c ( t ) sin(Θ s + Θ c ) − tan φ m,c ( t ) cos(Θ s + Θ c )(Θ ′ s + Θ ′ c ) . (5.9) Article submitted to Royal Society R.C. McPhedran and others
12 14 16 18 20 - - - - - - - - - Figure 7. Variation of the quantities φ ,c ( t ) (red), Θ c ( t ) (blue) and Θ s ( t ) (green) with t ∈ [9 . ,
20] (left) and [20 , .
5] (right), where those angles pertain to e C (2 , s ) and e C (2 , s ) . We now consider the values of Θ s , Θ c and their t derivatives in the neighbour-hood of zeros of C (1 , m ; + it ) or C (0 , + it ), which are not zeros of bothfunctions. Close to a zero of C (1 , m ; + it ), we require | e C | cos Θ c − | e S | cos Θ s → , | e C | → | e S | , Θ c → Θ s , (5.10)where | e C | 6 = 0, | e S | 6 = 0 at the zero by Theorem 5.1. Using the last of (5.10) in (5.8)and (5.9), we find at the zero of C (1 , m ; + it ) thatΘ s = Θ c = − φ m,c ( t ) + (cid:20) π (cid:21) , Θ ′ c + Θ ′ s = − φ ′ m,c ( t ) . (5.11)Close to a zero of C (0 , + it ), we require | e C | cos Θ c + | e S | cos Θ s → , | e C | sin Θ c + | e S | sin Θ s → , | e C | → | e S | . (5.12)From (5.12), at the zeroΘ c = π + Θ s = (cid:20) π (cid:21) , Θ ′ s = − Θ ′ c . (5.13)The relationships are exemplified in Fig. 7, where the variations with t of thefunctions cot(Θ c ) and cot(Θ s ) are shown for the case m = 1. Note that, in terms ofthese, the relationship (5.8) controlling the variation of ∆ (2 , m ; s ) on the criticalline is cot(Θ c ) + cot(Θ s ) = − φ m,c ( t ) . (5.14)Then, from (5.11), the zeros of C (1 , m ; + it ) are evident where the curves givingcot(Θ c ) and cot(Θ s ) intersect, which they do on the line giving the variation ofcot( φ m,c ( t ). The curves of cot(Θ c ) and cot(Θ s ) also go off to infinity at zerosof C (0 , + it ), with their movement being in opposite directions in accord with(5.14). In Fig. 7 we see examples of there being none, one or two zeros C (1 , m ; + it )between successive zeros of C (0 , + it ).The next result relates to the independence of the functions ∆ (2 , m ; s ) on thecritical line, again in the asymptotic region. Article submitted to Royal Society ew Properties of Angular Lattice Sums Theorem 5.2. If s is on the critical line in the asymptotic region t >> l and α l , β are non-zero reals, then α l C (1 , l ; s ) + βC (0 , s ) = 0 , unless C (1 , l ; s ) = 0 = C (0 , s ) . (5.15) If s is on the critical line in the asymptotic region t >> l , m ) and α l , α m are non-zero reals with l = m , then α l C (1 , l ; s ) + α m C (1 , m ; s ) = 0 unless C (1 , l ; s ) = 0 = C (1 , m ; s ) . (5.16) Proof.
We suppose α l C (1 , l ; s ) + βC (0 , s ) = 0 with C (0 , s ) = 0 . (5.17)Then α l C (1 , l ; s ) C (0 , s ) = − β, arg (cid:20) C (1 , l ; s ) C (0 , s ) (cid:21) = mπ, (5.18)for m an integer.We now use the functional equations (3.1) for C (0 , s ) and C (1 , m ; s ), whichon the critical line with 1 − s = s enable their arguments to be deduced:arg C (0 , s ) = − [arg π s + arg Γ( s )] , arg C (1 , m ; s ) = − [arg π s + arg Γ( s + 2 m )] . (5.19)Using (5.19) we arrive at the estimate for s on the critical line in the asymptoticregion arg C (0 , s ) − arg C (1 , m ; s ) = mπ − m t + O ( 1 t ) . (5.20)Using (5.20), (5.18) requires2 l t + O ( 1 t ) − lπ = mπ, (5.21)which is not possible. This proves (5.15).Consider next (5.16), and suppose α l C (1 , l ; s ) + α m C (1 , m ; s ) = 0 with C (1 , m ; s ) = 0 . (5.22)Then this requires arg (cid:20) C (1 , l ; s ) C (1 , m ; s ) (cid:21) = pπ, (5.23)for an integer p . However, from (5.20),arg C (1 , l ; s ) − arg C (1 , m ; s ) = ( m − l ) π + 2( l − m ) t + O ( 1 t ) . (5.24)Hence (5.22) requires 2( l − m ) t + O ( 1 t ) = ( p + l − m ) π, (5.25)which is not possible if l = m . This proves (5.16). Article submitted to Royal Society R.C. McPhedran and others
Corollary 5.3.
The sums C (2 , , s ) , S (2 , , s ) and C (1 , s ) have no zeros s on thecritical line in the asymptotic region t >> which are not zeros of both C (0 , s ) and C (1 , s ) .Proof. We apply Theorem 5.2 and equation (5.15). Now the sums C (2 , , s ), S (2 , , s )and C (1 , s ) belong to the system of order 4 discussed in Appendix A, which is gen-erated by linear combinations of C (0 , s ) and C (1 , s ). All such combinations areof the form (5.15), and are thus zero only if both C (0 , s ) = 0 and C (1 , s ) = 0.We can also simply establish a corresponding result for the derivative ∆ ′ (2 , m ; s )for s on the critical line. Theorem 5.4.
The derivative function ∆ ′ (2 , m ; s ) has no zeros for s = 1 / it in the asymptotic region t >> m which are not zeros of ∆ (2 , m ; s ) of multipleorder.Proof. We recall the result (3.8) for s = 1 / it :∆ (2 , m ; 1 / it ) = [1 − i tan( φ m,c ( t ))][ | e C (2 , m ; 1 / it ) | − | e S (2 , m ; 1 / it ) | ] , (5.26)where φ m,c ( t )) is a real-valued function. Taking the derivative of ∆ (2 , m ; 1 / it )with respect to 1 / it and separating into real and imaginary parts, we find∆ ′ (2 , m ; 1 / it ) = − (cid:26) φ ′ m,c ( t )cos φ m,c ( t ) [ | e C (2 , m ; 1 / it ) | − | e S (2 , m ; 1 / it ) | ]+ tan φ m,c ( t ) ddt [ | e C (2 , m ; 1 / it ) | − | e S (2 , m ; 1 / it ) | ] (cid:27) − i ddt [ | e C (2 , m ; 1 / it ) | − | e S (2 , m ; 1 / it ) | ] . (5.27)For ∆ ′ (2 , m ; 1 / it ) to be zero, we find from the imaginary part of (5.27) that ddt [ | e C (2 , m ; 1 / it ) | − | e S (2 , m ; 1 / it ) | ] = 0 . (5.28)We use (5.28) in the real part of (5.27), and, assuming tan φ m,c ( t ) is non-singularand φ ′ m,c ( t ) = 0, we complement (5.28) with[ | e C (2 , m ; 1 / it ) | − | e S (2 , m ; 1 / it ) | ] = 0 . (5.29)Since both the assumptions we have just mentioned are true in the asymptoticregion, we have proved the theorem.We provide plots of ∆ ′ (2 ,
2; 1 / t ) in Fig. 8. These show that the derivativeis non-zero in the non-asymptotic region as well as in the asymptotic region. Thenumerical value of ∆ ′ (2 ,
2; 1 /
2) is 0.918604.We next consider the properties of trajectories of constant phase which followfrom the assumption that the Riemann Hypothesis holds for ∆ (2 , m ; s ). As canbe seen from Fig. 9, lines of constant phase between zeros of ∆ (2 , s ) do not ingeneral cross the critical line, but asymptote towards it, and their configurationis arranged about a zero of the derivative ∂ | ∆ (2 , σ + it ) | /∂t or equivalently of ∂ arg ∆ (2 , σ + it ) /∂σ . Article submitted to Royal Society ew Properties of Angular Lattice Sums - - - - Figure 8. The real part (red), imaginary part (blue) and modulus (green) of∆ ′ (2 ,
2; 1 / t ), with t ∈ [0 . ,
20] (left) and [20 ,
40] (right).
Figure 9. Contours of constant phase for ∆ (2 , s ) in the region around its zero at s = 0 . i . s = 0 . i . ∂ log | ∆ (2 , σ + it ) | /∂t . Theorem 5.5.
Given the Riemann hypothesis holds for ∆ (2 , m ; s ) , then linesof constant phase coming from σ = −∞ can only cut the critical line at a zero of ∆ (2 , m ; s ) or of ∂ arg ∆ (2 , σ + it ) /∂σ . Those lines of constant phase comingfrom σ = ∞ cutting the critical line at a point which is not a zero of ∆ (2 , m ; s ) orof ∂ arg ∆ (2 , σ + it ) /∂σ must curve back and pass through a zero of ∆ (2 , m ; s ) .Proof. The assumption of the Riemann hypothesis holding enables us to say thatlines of constant phase coming from σ = −∞ do not intersect before the criticalline. Their phase monotonically decreases as t increases in σ < /
2, while thephase of ∆(2 , m ; 1 / it ) monotonically increases with t . Thus, groups of constantphase lines coming in from σ = −∞ cannot cut the critical line, except at a zeroof ∆(2 , m ; s ). Isolated trajectories passing through points where ∂ arg ∆(2 , σ + it ) /∂σ = 0 are allowed. Such special trajectories separate lines of constant phasewhich curve up as σ → / , m ; s ), fromtrajectories which curve down as σ → / σ = 1 / t , but if they were able to cross the line σ = 1 / Article submitted to Royal Society R.C. McPhedran and others progress towards σ = −∞ it would have to decrease with t . Thus they must turn andrun alongside σ = 1 /
2, with lines above the special trajectory curving up towardsa zero of ∆(2 , m ; s ), and those below curving down towards a zero. Alternatively,they can cut the critical line at a point which is not a zero of ∆ (2 , m ; s ) or of ∂ arg ∆(2 , σ + it ) /∂σ . Such trajectories supply the lines of constant argumentrequired for generic points on the critical line, and must return back to cut thecritical line at a zero of ∆(2 , m ; s ). The region where they cross into σ < / (2 , m ; s )with an adjacent zero of ∂ arg ∆(2 , σ + it ) /∂σ .Note that the region where zeros of ∂ arg ∆(2 , σ + it ) /∂σ exist is confinedto the neighbourhood of the critical line, since this partial derivative being zerocorresponds to a horizontal segment on a line of constant phase (see, for example,Fig. 3). However, we may use the line of constant phase passing through the pointon the critical line where ∂ arg ∆(2 , σ + it ) /∂σ = 0 as the separator between linesof constant phase going to the zero above this line from those going to the zerobelow (given the Riemann hypothesis is assumed to hold). Theorem 5.6.
Given the Riemann hypothesis holds for ∆ (2 , m ; s ) , then thereexists one and only one zero of ∂ arg ∆ (2 , σ + it ) /∂σ on the critical line betweentwo successive zeros of ∆ (2 , m ; s ) .Proof. We consider the analytic functionlog ∆ (2 , m ; s ) = log | ∆ (2 , m ; s ) | + i arg ∆ (2 , m ; s ) . (5.30)The real part of this function goes to −∞ at any zero of ∆ (2 , m ; s ), andincreases away from these logarithmic singularities. It must have at least one turningpoint between successive zeros. By the Cauchy-Riemann equations, such a turningpoint is a zero of ∂ arg ∆ (2 , σ + it ) /∂σ .Next, suppose there two or more zeros of ∂ arg ∆(2 , σ + it ) /∂σ between suc-cessive zeros s = 1 / it and s = 1 / it . Denote the upper two of suchderivative zeros by s ∗ = 1 / it ∗ and s ∗∗ = 1 / it ∗∗ . As we have seen, each ofthese has constant phase lines coming from σ = −∞ and passing through it, aroundwhich constant phase trajectories reverse their course. Those above s ∗ curve up to s as they approach the critical line, while those below it curve down. They cannotcross the constant phase line passing through s ∗∗ , nor can they cross the criticalline. They must then head back to σ = −∞ , where they will breach the mono-tonic nature of the variation of arg ∆ (2 , m ; s ) with t . Thus, this situation cannotarise.We conclude this section with an investigation of the structure of lines of con-stant phase which cut the critical line and turn back to σ = ∞ thereafter. We startwith the expansion of arg ∆ (2 , σ + it ) around a point s ∗ = σ ∗ + it ∗ = 1 / it ∗ in the asymptotic region where ∂ arg ∆ (2 , σ + it ) /∂σ = 0, which is of the form:arg ∆ (2 , σ + it ) = arg ∆ (2 ,
2; 1 / it ∗ ) + 2 m t ∗ ( σ − / − m t ∗ ( t − t ∗ ) + 2 m t ∗ ( t − t ∗ ) − (cid:20) ∂ ∂t log | ∆ (2 , σ + it ) | (cid:21) s = s ∗ ( σ − σ ∗ )( t − t ∗ ) + . . . . (5.31) Article submitted to Royal Society ew Properties of Angular Lattice Sums Figure 10. Contours of constant phase for ∆ (2 , s ) in the region around its zero at s = 0 . i . ∂ arg ∆ (2 , σ + it ) /∂t is zero.(a)The fine lines correspond to the phases 3.04,3.04076, 3.0410, 3.0412, 3.0414, 3.0416,3.0418, 3.0419, 3.04198, 3.042, 3.0425, 3.043, 3.044, 3.045, 3.046, 3.047 and 3.048. (b) Thethick dashed line is the contour on which ∂ log | ∆ (2 , σ + it ) | /∂t is zero. The contoursare -0.1090, -0.1080, -0.1072, -0.1068, -0.1064, -0.1060, -0.1056, -0.1055, -0.1054, -0.1053,-0.1052, -0.1048, -0.104, -0.103, -0.102, -0.1015 and -0.10. Here we have employed the asymptotic estimates for arg ∆ (2 ,
2; 1 / it ) basedon (3.14). The corresponding trajectories of constant phase may be shown to berectangular hyperbolae, with their centre at σ − / (cid:2) ∂ ∂t log | ∆ (2 , σ + it ) | (cid:3) s = s ∗ + m t ∗ (cid:18) m t ∗ (cid:19) (cid:20) ∂ ∂t log | ∆ (2 , σ + it ) | (cid:21) s = s ∗ ,t − t ∗ = 1 (cid:2) ∂ ∂t log | ∆ (2 , σ + it ) | (cid:3) s = s ∗ + m t ∗ (cid:18) m t ∗ (cid:19) . (5.32)The second derivative factor in (5.32) is always negative, and tends to be muchlarger than the terms involving powers of 1 /t ∗ . Thus, we see that the centre of thehyperbolic trajectories of constant phase will always lie to the left of the criticalline, with its ordinate very close to t ∗ . This displacement of the centre into σ < / σ = 1 / σ > / (2 , s ). Note that the centre corresponds to apoint at which two lines of constant phase intersect; thus, it must have derivativesof phase along two independent lines which are zero. It therefore is a point atwhich both ∂ arg ∆ (2 , σ + it ) /∂t and ∂ arg ∆ (2 , σ + it ) /∂σ are zero. From theCauchy-Riemann equations, it also is a point at which ∂ log | ∆ (2 , σ + it ) | /∂t and ∂ log | ∆ (2 , σ + it ) | /∂σ are zero- i.e., it is a point of extremum for both amplitudeand phase.These hyperbolic centre points are locations at which ∆ (2 , s ) ′ = 0. The factthat they lie to the left of ℜ ( s ) = 1 / ζ ( s ). Article submitted to Royal Society R.C. McPhedran and others
The curves of constant phase of ∆ (2 , s ) given in Fig. 10 illustrate some moreof their general characteristics in the neighbourhood of zeros. For Fig. 10 (a), atthe top of the figure, we see the centre of the hyperbolic phase curves; through thiscentre passes a curve on which ∂ arg ∆ (2 , σ + it ) /∂t = 0. This curve connectsthe hyperbolic centre to the zero of ∆ (2 , s ) below it, and also continues to thezero above; at each zero it is tangent to the t axis. The curve marks points wherethe constant phase lines have vertical slope. The lowest line of constant phase onthe right has vertical slope when it arrives at the zero of ∆ (2 , s ); it correspondsto 3.04076, which is the value given by (3.8) for the phase on the critical linejust above the zero. A second important line has the phase 3.04198, which is thephase corresponding to the centre of the hyperbola. This line again connects thehyperbolic centre to the zero. All lines whose phase lies between these values comein from the right, cross the zero line of ∂ arg ∆ (2 , σ + it ) /∂t , and curve back topass through the zero of ∆ (2 , s ). Where they lie to the left of the zero line of ∂ arg ∆ (2 , σ + it ) /∂t , their phase increases as t decreases; where they lie to theright, it increases as t increases. Curves coming in from the left all pass throughthe zero without crossing the zero derivative line; their phase always increases as t decreases.For Fig. 10 (b), we show the phase contours below the zero, in the region downto the next zero. The phase at the centre of the hyperbolae is -0.1055, while thephase just above the bottom zero and just below the upper zero are respectively-0.107604 and -0.101134. Note the dashed line passing through the centre, alongwhich ∂ log | ∆ (2 , σ + it ) | /∂t = 0. From the Cauchy-Riemann equations, thisis perpendicular to the solid line defined by ∂ arg ∆ (2 , σ + it ) /∂t = 0 (a factdisguised by the different scales on the horizontal and vertical axes).
6. Distributions of zeros
We return to the left-hand side of expression (4.6), in which we replace s by σ + i t ,and expand assuming | t | >> σ , with σ large enough to ensure accuracy of (4.6).The result is 2 t log( t ) − t (log π + 1) + π ( σ −
12 ) + σt (1 − σ ) . (6.1)As for each increment of π of this expression we get one null line of the real partof ∆ and one of the imaginary part, and these intersect at σ = 1 / (2 , m ; s )), we can divide(6.1) by π , and regard the result as a distribution function for zeros of ∆ : N ∆ ( σ, t ) = 2 tπ log( t ) − tπ (1 + log π ) + σ −
12 + σπt (1 − σ ) . (6.2)Now, from Titchmarsh and Heath-Brown (1987), the distribution function forthe zeros of the Riemann zeta function on the critical line is N ζ ( 12 , t ) = t π log( t ) − t π (1 + log(2 π )) + O (log t ) . (6.3)We complement this with the numerical estimate from McPhedran et al (2007) forthe distribution function of the zeros of L − ( s ): N − ( 12 , t ) = t π log( t ) − t π (1 + log( π/ O (log t ) . (6.4) Article submitted to Royal Society ew Properties of Angular Lattice Sums Figure 11. The distributions of the differences between successive zeros of C (0 , s )(left)and C (1 , s ) (right) on s = 1 / it , for t < Adding (6.3) and (6.4) we obtain the distribution function for the zeros of C (0 , s )(see (3.6)): N C , ( 12 , t ) = tπ log( t ) − tπ (1 + log( π )) + O (log t ) . (6.5)When we compare this with (6.2), and use the equation N ∆ ( 12 , t ) = N C , ( 12 , t ) + N C , ( 12 , t ) , (6.6)it suggests the hypothesis that the distribution function of zeros of C (1 , s ) is thesame as that of (6.5), to the number of terms quoted: N C , ( 12 , t ) = N C , ( 12 , t ) = tπ log( t ) − tπ (1 + log( π )) + O (log t ) . (6.7)Strong numerical evidence supporting this is given in Table 1, which also showszero counts for C (1 , s ) and C (1 , s ). Note that the numbers of zeros found for C (1 , s ), C (1 , s ) and C (1 , s ) are virtually the same. This rules out any variationwith increasing order similar to that of Dirichlet L functions, where increasing orderresults in significant increases in density of zeros (compare (6.4) and (6.3), or thesecond and third columns of Table 1).Comparing the data of Table 1 with the discussion in Bogomolny and Lebouef(1994), we can see that the split up of N ( t ) into averaged parts given by expres-sions like (6.3-6.7) and oscillating parts applies to C (0 , s ) and to the C (1 , m ; s ).However, it would be value to extend the numerical investigations of Table 1 tomuch higher values of t , to render the characterization of the oscillating term moreaccurate. Such an extension may require the development of an alternative algo-rithm to that based on (2.3), which will probably become unwieldy for values of t of order 10 − .In Fig. 11 we compare the distributions of the differences between zeros onthe critical line for C (0 , s )(left) and C (1 , s ) (right). The distributions are quitedifferent, even with this modest data set. Bogomolny and Lebouef (1994) havestudied the case of C (0 , s ) using 10,000 zeros after t = 10 , and contrast thedistribution for ζ ( s ) L − ( s ) with that for each function separately. The separatefactors in fact have distributions like that that for C (1 , s ). The function comparedwith the histogram in the right of Fig. 11 corresponds to the Wigner surmise, Article submitted to Royal Society R.C. McPhedran and others which (Bogomolny and Lebouef, 1994, Dietz and Zyczkowski, 1991) for the unitaryensemble takes the form P ( S ) = 9 S π exp (cid:18) − S π (cid:19) . (6.8)Here the separation between zeros has been rescaled to have a mean of one. Bo-gomolny and Lebouef (1994) comment that the left distribution is that of an un-correlated superposition of two unitary ensemble sets. Note as one indicator of thisthat there is not the same pronounced tendency for the probability to go to zerowith separation on the left as on the right, where the distribution clearly comesfrom a single ensemble. Recall that Table 1 shows that the frequency distributionfor zeros is the same for C (0 , s ) and C (1 , s ). This makes the strong differencein the distributions of the gaps all the more interesting. The research of R.McP. on this project was supported by the Australian Research CouncilDiscovery Grants Scheme.
References
Abramowitz, M. & Stegun, I. A. 1972
Handbook of Mathematical Functions with Formulas,Graphs and Mathematical Tables . New York: Dover.Bogomolny, E. & Lebouef, P. 1994 Statistical properties of the zeros of zeta functions-beyond the Riemann case.
Nonlinearity Z. Phys. B- Condensed Matter Finite Euler Products and the Riemann Hypothesis arXiv:0704.3448Hardy, G. H. 1920 On some definite integral considered by Mellin.
Mess.Math. , 86-91.Kober, H. 1936 Transformation formula of certain Bessel series, with reference to zetafunctions Math. Zeitschrift , 609-624Lorenz, L. 1871 Bidrag til talienes theori, Tidsskrift for Math. , 97-114.McPhedran, R. C., Smith, G. H., Nicorovici, N. A. & Botten, L. C. 2004 Distributive andanalytic properties of lattice sums. J. Math. Phys. , 2560-2578.McPhedran, R. C., Botten, L. C. & Nicorovici, N. A. 2007 Null trajectories for the sym-metrized Hurwitz Zeta Function Proc. Roy. Soc. A
Proc. Roy. Soc. A , , 3327-3352.Speiser, A., 1934 Geometrisches zur Riemannschen Zetafunktion. Math. Ann. , 514-521.Titchmarsh, E. C. & Heath-Brown, D. R. 1987
The theory of the Riemann zeta function ,Oxford: Science Publications.Zucker, I.J. & Robertson, M.M.. 1976 Some properties of Dirichlet L series. J. Phys. A.Math. Gen. Article submitted to Royal Society ew Properties of Angular Lattice Sums Table 1.
Numbers of zeros of ζ (1 / t ) , L − (1 / t ) , C (1 ,
4; 1 / t ) , C (1 ,
8; 1 / t ) and C (1 ,
12; 1 / t ) in successive intervals of t . t n ζ n − n C n ζ + n C n C n − + n C Appendix A. Supplementary Notes on Systems of AngularSums
We give here results linking trigonometric lattice sums of order up to ten, whichshow that they may be generated from three independent sums, C (0 , s ), C (1 , s ), Article submitted to Royal Society R.C. McPhedran and others and C (1 , s ). We also give expressions in terms of sums of Macdonald functions ofthe Kober-type from which these independent sums may be calculated.A basic result we use relies on the symmetry of the square lattice: C (1 , m − s ) = X p ,p ′ cos(4 m − θ p ,p ( p + p ) s = 0 = C (2 , m − s ) − S (2 , m − s ) , (A 1)from which we obtain C (2 , m − s ) = S (2 , m − s ) = 12 C (0 , s ) , (A 2)for m any positive integer. We can also expand cos(4 m − θ p ,p = T m − (cos θ p ,p )in terms of powers of cos θ p ,p , using the expressions for the Chebyshev polynomialsin Table 22.3 of Abramowitz & Stegun (1972). This enables us to express C (4 m − , s ) in terms of C (4 n, s ) with 4 n < m −
2, and n being a positive integer. Inthis way, we can inductively arrive at the results below.( a ) Order 0
The only sum of this type is C (0 , s ), given by equation (2.2). It is given by amodified form of (2.3), since there are contributions from both axes p = 0, p = 0rather than just p = 0: C (0 , s ) = 2 ζ (2 s ) + 2 √ π Γ( s − / s ) ζ (2 s − π s Γ( s ) ∞ X p =1 ∞ X p =1 (cid:18) p p (cid:19) s − / K s − / (2 πp p ) . (A 3)( b ) Order 2
The single sum of order 2 is X ( p ,p ) ′ p ( p + p ) s +1 = C (2 , s ) = 12 C (0 , s )= 2 √ π Γ( s + 1 / ζ (2 s − s + 1) + 8 π s Γ( s + 1) ∞ X p =1 ∞ X p =1 (cid:18) p p (cid:19) s − / p p πK s +1 / (2 πp p ) . (A 4)( c ) Order 4
We generate this system from C (4 , s ), obtained from (2.3). In terms of this, C (1 , s ) = 8 C (4 , s ) − C (0 , s ) , (A 5)and C (2 , s ) = 4 C (4 , s ) − C (0 , s ) , S (2 , s ) = − C (4 , s ) + 2 C (0 , s ) . (A 6) Article submitted to Royal Society ew Properties of Angular Lattice Sums d ) Order 6
The result of expanding (A 1) for m = 2 is C (6 , s ) = 32 C (4 , s ) − C (0 , s ) , (A 7)while from (A 2) C (2 , s ) = S (2 , s ) = 12 C (0 , s ) . (A 8)Also, X ( p ,p ) ′ p p ( p + p ) s +3 = X ( p ,p ) ′ p p ( p + p ) s +3 = 12 X ( p ,p ) ′ p p ( p + p ) s +2 , (A 9)so X ( p ,p ) ′ cos ( θ p ,p ) sin ( θ p ,p )( p + p ) s = 18 S (2 , s ) . (A 10)Note that all sums mentioned so far can be generated from just two, say C (0 , s )and C (1 , s ). C (6 , s ) is given by (2.3).( e ) Order 8
The new sum we use here is C (8 , s ), obtained from (2.3). In terms of this, C (1 , s ) = 128 C (8 , s ) − C (4 , s ) + 49 C (0 , s ) , (A 11) C (2 , s ) = 64 C (8 , s ) − C (4 , s ) + 25 C (0 , s ) , (A 12)and S (2 , s ) = − C (8 , s ) + 112 C (4 , s ) − C (0 , s ) . (A 13)Two other sums in this system are X ( p ,p ) ′ p p ( p + p ) s +4 = −C (8 , s ) + 32 C (4 , s ) − C (0 , s ) , (A 14)and X ( p ,p ) ′ p p ( p + p ) s +4 = C (8 , s ) − C (4 , s ) + 12 C (0 , s ) . (A 15)Using the result (A 11), we have calculated curves showing the modulus of C (1 , s ) as a function of s = 1 / t on the critical line. These are given inFigs. 12-13, and the distributions of zeros they show are given in Table 1. Article submitted to Royal Society R.C. McPhedran and others
Figure 12. The modulus of C (1 , s ) (red) and C (1 , s ) (blue) as a function of s = 1 / t , for t ∈ [0 ,
20] (left) and t ∈ [20 ,
40] (right).
45 50 55 6051015202530 65 70 75 805101520253035
Figure 13. The modulus of C (1 , s ) (red) and C (1 , s ) (blue) as a function of s = 1 / t , for t ∈ [40 ,
60] (left) and t ∈ [60 ,
80] (right). ( f ) Recurrence Relations
We can generalize the above procedure by establishing recurrence relations forthe trigonometric sums. We consider X ( p ,p ) ′ p n ( p + p ) s + n = C (2 n, s ) = X ( p ,p ) =(0 , p + p ) s (cid:20) − p p + p (cid:21) n . (A 16)Expanding using the Binomial Theorem, we obtain C (2 n, s ) = n X l =0 n C l ( − l C (2 l, s ) . (A 17)If n is even, (A 17) gives an identity: n − X l =0 2 n C l ( − l C (2 l, s ) = 0 , (A 18)while for n odd we obtain an expression for C (4 n − , s ) in terms of lower ordersums: C (4 n − , s ) = 12 n − X l =0 2 n − C l ( − l C (2 l, s ) . (A 19) Article submitted to Royal Society ew Properties of Angular Lattice Sums Figure 14. Null contours of the real part (red) and imaginary part (blue) of∆ (2 , σ + i t ), with σ ∈ [ − . , . t ∈ [0 . , It may be checked that the two relations (A 18) and (A 19) give equivalent resultsin the cases given above.In Fig. 14 we give null contours of the real part and imaginary part of ∆ (2 , s ),for comparison with those of ∆ (2 , s ) in Fig. 3. It will be noted that there arenow three lines giving the contours on which Im ∆ (2 , s ) = 0, and these are notcircles (as was the single exemplar in Fig. 3). There is a corresponding increasein the number of null contours starting and finishing on the real axis, and in thevalue of σ at which the last one reaches the real axis. There are two examplesin Fig. 14 of zeros on the critical line of the real part, but not the imaginary part(tan ( φ m,c (1 / t ) = ∞ ), and one of a zero of the imaginary part but not the realpart (cot ( φ m,c (1 / t ) = ∞ ). The null contours starting at σ = ∞ and endingat σ = −∞ settle down to an asymptotic behaviour for somewhat larger values of t than in Fig. 3, but are otherwise similar to the lower order null contours. Article submitted to Royal Society R.C. McPhedran and others ( g ) Order 10
The new sum we use from (2.3) is C (10 , s ), which, from (A 19), is C (10 , s ) = 12 C (0 , s ) − C (4 , s ) + 52 C (8 , s ) . (A 20)Other sums in the system are: X ( p ,p ) ′ p p ( p + p ) s +5 = C (8 , s ) − C (10 , s ) , (A 21)and X ( p ,p ) ′ p p ( p + p ) s +5 = 12 X ( p ,p ) ′ p p ( p + p ) s +4 . (A 22) Appendix B. Derivation of the Functional Equation for C (1 , m ; s ) We start with the equation (30) from McPhedran et al (2004). This equation usesthe Poisson summation formula to derive a connection between a sum over a directlattice of points R p ≡ R p ,p = d ( p , p ) with polar coordinates ( R p , φ p ), and acorresponding sum over the reciprocal lattice, where the lattice points are labelled K h ≡ K h ,h . Considering the case of a square lattice with period d , the reciprocallattice points are K h = (2 π/d )( h , h ). The sum in the direct lattice incorporatesa phase term of the Bloch type, with wave vector k , and the result of the Poissonformula is2 s − Γ( l s ) X ( p ,p ) ′ e i k · R p e i lφ p R sp = 2 π i l d Γ( l − s ) e i lθ k − s X ( h ,h ) ′ e i lθ h Q − sh . (B 1)Here the sum over the reciprocal lattice in fact runs over a set of displaced vectors: Q h = k + K h = ( 2 πh d + k x , πh d + k y ) = ( Q h , θ h ) , (B 2)where in (B 2) the second and third expressions are in rectangular and polar coor-dinates. Note that in (B 1) and (B 2), p , p and h , h run over all integer values,and θ gives the direction of k .We express the relation (B 1) in non-dimensionalized form, taking out a factor d s on the left-hand side, and a factor (2 π/d ) − s on the right-hand side. We put k = (2 π/d ) κ , and obtainΓ( l s ) X ( p ,p ) ′ e π i( κ x p + κ y p ) e i lφ p ( p + p ) s = i l π s − Γ( l − s ) × e i lθ κ − s + X ( h ,h ) ′ e i lθ h (( κ x + h ) + ( κ y + h ) ) − s . (B 3) Article submitted to Royal Society ew Properties of Angular Lattice Sums κ →
0, set l = 4 m and take Re( s ) >
1, we obtain the desired result:Γ(2 m + s ) π s C (1 , m ; s ) = Γ(2 m + 1 − s ) π − s C (1 , m ; 1 − s ) . (B 4)(B 4)