## Abstract

We give analytical results pertaining to the distributions of zeros of a class of sums which involve complex powers of the distance to points in a two-dimensional square lattice and trigonometric functions of their angle. Let denote the product of the Riemann zeta function and the Catalan beta function, and let denote a particular set of angular sums. We then introduce a function that is the quotient of the angular lattice sums with , and use its properties to prove that obeys the Riemann hypothesis for any *m* if and only if obeys the Riemann hypothesis. We furthermore prove that if the Riemann hypothesis holds, then and have the same distribution of zeros on the critical line (in a sense made precise in the proof). We also show that if obeys the Riemann hypothesis and all its zeros on the critical line have multiplicity one, then all the zeros of every have multiplicity one. We give numerical results illustrating these and other results.

## 1. Introduction

In this paper, we will compare the distributions of zeros of a class of sums over two-dimensional lattices involving trigonometric functions of the angle to points in the lattice, and a complex power 2*s* (with *s*=*σ*+i*t*) of their distance from the lattice origin with those of a similar sum not involving trigonometric functions. The latter was evaluated analytically by Lorenz (1871) and Hardy (1920), and was shown to have the value 4*ζ*(*s*)*L*_{−4}(*s*), where *ζ*(*s*) denotes the Riemann zeta function and *L*_{−4}(*s*) denotes a particular Dirichlet *L* function called the Catalan beta function. We will give numerical results suggesting that key properties of the distributions of zeros are the same for all the members of the set of angular sums, denoted , and for the angular independent sum, denoted . We will also give proofs of certain of these key properties (which substantiate and go further than the numerical results):

— obeys the Riemann hypothesis for any

*m*if and only if obeys the Riemann hypothesis (theorem 4.3);— if the Riemann hypothesis holds for , then and have the same distribution of zeros on the critical line (in the sense that the formulae for the number of zeros of each up to a given ordinate

*t*on the critical line must agree in all terms which diverge as*t*goes to infinity) (theorem 4.5, corollary 4.6);— if obeys the Riemann hypothesis and all its zeros on the critical line have multiplicity one, then all the zeros of every have multiplicity one (theorem 4.8).

These results are interesting and important in that they relate to the generalized Riemann hypothesis (GRH), which asserts that, like the zeros of *ζ*(*s*), the non-trivial zeros of Dirichlet *L* functions associated with characters (Zucker & Robertson 1976) have their real part equal to 1/2. There have been extensive numerical investigations made into the distributions of zeros of the Riemann zeta function, and also of Dirichlet *L* functions associated with characters, so that the results proved here mean that the two-dimensional lattice sums may be considered to have had the Riemann hypothesis for them substantiated numerically to the same extent as the product *ζ*(*s*)*L*_{−4}(*s*). Note also that the proofs presented are not of great technical difficulty, and yet some of the results are striking. It thus seems that the study of angular lattice sums may offer a favourable context in which properties related to the Riemann hypothesis may be studied.

We begin by briefly describing the key properties of the angular sums , such as their functional equation and an exponentially convergent representation that enables their numerical evaluation everywhere in the complex *s* plane. More comprehensive accounts of these properties may be found in previous papers (McPhedran *et al.* 2004, 2008, 2010). We then give numerical data on the distributions of zeros, which motivated the analytical results derived in §4. These are obtained by studying the properties of the quotient function Δ_{4}(2,2*m*;*s*) of and , whereas our previous investigations (McPhedran *et al.* 2008) had focused on their product function Δ_{3}(2,2*m*;*s*). As we will see, the quotient function is well adapted to the comparison of the distributions of zeros of and .

## 2. Basic properties of the *𝒞*(1, 4*m*; *s*)

We recall the definition from McPhedran *et al.* (2008), hereafter referred to as (I), of two sets of angular lattice sums for the square array
2.1where , and the prime denotes the exclusion of the point at the origin. The sum independent of the angle *θ*_{p1,p2} was evaluated by Lorenz (1871) and Hardy (1920) in terms of the product of Dirichlet *L* functions:
2.2A useful account of the properties of Dirichlet *L* functions such as *L*_{−4}(*s*) has been given by Zucker & Robertson (1976).

It is convenient to use a subset of the angular sums (2.1) as a basis for numerical evaluations. We note that the sums are zero if *n* is odd. We have for the non-zero sums
2.3where *K*_{ν}(*z*) denotes the modified Bessel function of the second kind, or Macdonald function, with order *ν* and argument *z*. The general form (2.3) may be derived following Kober (1936), in the way described in McPhedran *et al.* (2010), hereafter referred to as II. It should be noted that the double sum in equation (2.3) is exponentially convergent. The representation (2.3) and finite combinations of it thus furnish absolutely convergent representations of trigonometric sums from the family (2.1) and close relatives, for any values of *s* with finite modulus. These representations are easily represented numerically in any computational system incorporating routines for the Riemann zeta function of complex argument, and Macdonald functions of complex order and real argument. (It should be noted however that the straightforward implementation may not be sufficiently computationally efficient for the case of *s* near the critical line and of large modulus. Further development of algorithms based on equation (2.3) or on other representations may be necessary for this case.)

The class of angular sum of particular interest in this paper can easily be expanded in terms of the ,
2.4or, in terms of the Chebyshev polynomial of the first kind (Abramowitz & Stegun (1972), ch. 22),
2.5As the coefficients of this Chebyshev polynomial are explicitly known, the representation (2.5) enables any sum to be expressed as a linear combination of sums with 0≤*n*≤2*m*.

The functional equation is known (see McPhedran *et al.* (2004), eqns (32) and (59)) for
2.6This equation also holds for *m*=0, where it gives the functional equation for the product *ζ*(*s*)*L*_{−4}(*s*). It is in fact the *m* dependence of the functional equation (2.6) that enables the derivation of many of the results in (I) and the present paper.

This *m* dependence in equation (2.6) is represented in two related functions
2.7where *ϕ*_{2m}(*s*) is in general complex. Note that is the ratio of two polynomials of degree 2*m*, with one obtained from the other by replacing *s* by 1−*s*
2.8All its zeros and poles lie on the real axis. We note that, for |*s*|>>4*m*^{2},
2.9

## 3. Numerical data on the phase distribution and zeros of Δ_{4}(2,2*m*;*s*)

We now give numerical results on the variation with *s* and the distribution of zeros of the quotient function
3.1for small values of the integer parameter *m*. These have been obtained using the equations (2.2)–(2.4). Proofs of non-evident properties will be given in the next section.

Figure 1 shows the phase distribution of this function for *t* in the range (0,40). The phase variation in the region below the real axis would be obtained by conjugation of Δ_{4}(2,2;*s*). The phase along the real axis is either zero or *π*, taking the latter value only between the first-order zero of Δ_{4}(2,2;*s*) at *s*=1 and its first-order pole at *s*=−1. The region *σ*>2 is composed of interspersed areas with phase lying in either the first or fourth quadrants. The same is true of the region *σ*<−1.5, with the region *σ*≪−1 having a phase lying entirely in the first quadrant.

On the critical line with *t*>4, the phase of Δ_{4}(2,2;*s*) lies either in the first or third quadrant. In some regions of figure 2*a* this is not evident, and so the more detailed plots have been included to clarify the phase variation. The zeros and poles of Δ_{4}(2,2;*s*) are all first order, with all four quadrants of phase coinciding at the zero or pole. The lines of constant phase equal to zero coming in from meet the critical line at a pole, the lower end of a fourth-quadrant region, or a zero, the upper end of a fourth quadrant region. There are also lines of phase zero delineating the fourth quadrant regions lying largely to the left of the critical line; these have a pole at their upper intersection with the critical line and a zero at their lower intersection. In either case, the number of zeros lying on these lines are equal to the number of poles on them. Of course, as figure 2 shows, there are more localized contours than those just described, but these also have equal numbers of zeros and poles lying on them.

We consider next the distribution of zeros and poles on the critical line of Δ_{4}(2,2*m*;*s*) for *m*=1,2,3 (table 1). The zeros are those of , while the poles are those of . Now, from Titchmarsh & Heath-Brown (1987), assuming the Riemann Hypothesis to hold, the asymptotic form of the distribution function for the zeros of the Riemann zeta function on the critical line is
3.2(The expression on the right-hand side of equation (3.2) gives the expectation value for the number of zeros of *ζ*(1/2+i*t*) in the interval (0,*t*] for *t*≫1.)

We complement this with the numerical estimate from McPhedran *et al.* (2007) for the asymptotic form of the distribution function of the zeros of *L*_{−4}(*s*) on the critical line
3.3Adding equations (3.2) and (3.3), we obtain the distribution function for the zeros of :
3.4The data in table 1 suggest the hypothesis that the distribution function of zeros of is the same as that of equation (3.4), to the number of terms quoted
3.5Table 1 also shows zero counts for and . Note that the numbers of zeros found for , and are virtually the same. This rules out any variation with increasing order similar to that of Dirichlet *L* functions, where increasing order results in significant increases in density of zeros (compare equations (3.3) and (3.2), or the second and third columns of table 1).

Comparing the data of table 1 with the discussion in Bogomolny & Leboeuf (1994), we can see that the split up of *N*(*t*) into averaged parts given by expressions like (3.2)–(3.5) and oscillating parts applies to and to the . However, it would be of value to extend the numerical investigations of table 1 to much higher values of *t*, to render the characterization of the oscillating term more accurate. Such an extension might require the development of an alternative algorithm to that based on equation (2.3), or at least its implementation in a form appropriate to numerically intensive computation.

In figure 3, we compare the distributions of the differences between zeros on the critical line for (*a*) and (*b*). The distributions are quite different, even with this modest dataset. Bogomolny & Leboeuf (1994) have studied the case of using 10 000 zeros after *t*=10^{5}, and contrast the distribution for *ζ*(*s*)*L*_{−4}(*s*) with that for each function separately. The separate factors in fact have distributions like that that for . The function compared with the histogram in the right of figure 3 corresponds to the Wigner surmise, which (Dietz & Zyczkowski 1991; Bogomolny & Leboeuf 1994) for the Gaussian unitary ensemble takes the form
3.6Here, the separation between zeros has been rescaled to have a mean of one. Bogomolny & Leboeuf (1994) comment that the left distribution is that of an uncorrelated superposition of two unitary ensemble sets. Note as one indicator of this that there is not the same pronounced tendency for the probability to go to zero with separation on the left as on the right, where the distribution clearly comes from a single ensemble. Recall that table 1 shows that the frequency distribution for zeros is the same for and . This makes the strong difference in the distributions of the gaps all the more interesting, and provides further motivation for extension of the data of table 1 and figure 3 to higher values of *t*.

## 4. Relation between distributions of zeros

We now investigate the relationship between the distributions of zeros of and , using their quotient function Δ_{4}(2,2*m*;*s*). We will now prove some of the important properties of this function. Note that the first two proofs do not assume the Riemann Hypothesis, while all subsequent proofs assume it for either or for a , for some *m*.

### Theorem 4.1

*The analytical function* *obeys the functional equation*
4.1
*It has first-order poles at s*=−(2*m*−1),−(2*m*−2),…,−1 *and a first-order zero at s*=1. *Its only essential singularity is at infinity. On the critical line in the asymptotic region t*≫4*m*^{2}, its argument lies in either the first or third quadrants, with its argument being related to the function *ϕ*_{2m}(1/2+i*t*)=*ϕ*_{2m,c}*(t) by*
4.2*As* *for any t, the argument of* Δ_{4} *tends to zero exponentially, while as* *for any t, the argument of* Δ_{4} *tends to zero algebraically.*

### Proof.

The functional equation (4.1) follows readily from the functional equations (2.6) for and . The poles on the negative real axis arise from the poles of , while the zero at *s*=1 arises from the first-order pole there of . The residues of Δ_{4}(2,2*m*;*s*) at the poles on the negative real axis have to be evaluated numerically. The zeros of *C*(0,1;*s*) for *s* a negative odd integer are due to *L*_{−4}(*s*) vanishing, and for *s* an even negative integer due to *ζ*(*s*) vanishing. A convenient formula for evaluating the residues is
4.3for *n*=−1,−2,…,1−2*m*. Some numerical values for the numerator in equation (4.3) are *m*=1, , *m*=2, , , , *m*=3, , , , , The first eight approximate values for the denominator are *n*=−1, −0.194374, *n*=−2, 0.0608969, *n*=−3, −0.051032, *n*=−4,0.079381, *n*=−5, −0.198916, *n*=−6,0.719771, *n*=−7, −3.55939, *n*=−8, 23.0358.

The function has an infinite set of zeros on the critical line which it inherits from *ζ*(*s*) (Titchmarsh & Heath-Brown 1987). This will make the point at infinity a limit point of poles of Δ_{4}(2,2*m*;*s*), and so an essential singularity of it. (Note from equation (2.3) that the sums can only have poles at the single location *s*=1 in the finite part of the plane; the as linear combinations of the with coefficients independent of *s* share this property. Thus the cannot have essential singularities in the finite part of the plane. Similar comments can be made for on the basis of its representation as a Macdonald function sum (Kober 1936).) The phase of Δ_{4}(2,2*m*;*s*) on the critical line as given by equation (4.2) follows from equation (4.1) when .

For *σ* positive and not small, we may expand and by direct summation:
4.4and
4.5Combining equations (4.4) and (4.5), we find for *m*=1,2,…
4.6and
4.7Thus, in either case, Δ_{4} tends to unity in exponential fashion as *σ* increases positively. The difference from unity in the case of order 4*m*−2 tends to zero as , and in the case of order 4*m* as , so that the phase of Δ_{4} tends to zero exponentially.

For *σ* negative and not small, we may take the phase of Δ_{4}(2,4*m*;1−*s*) to be zero, and write
4.8This quantity tends to zero algebraically as |*s*| increases. Its maximum value is 4*m*^{2}/(1/2−*σ*), which occurs when *t*=1/2−*σ*. ■

If necessary, it is not hard to construct uniform bounds on the remainder terms in series like that of equation (4.5) for in the region of absolute convergence *σ*>1. Indeed, suppose we break the series there into a part with , and a remainder term from :
4.9and |cos (4*mθ*_{p1,p2})|≤1 for every *m*, *p*_{1}, *p*_{2}, so that
4.10In turn,
4.11so bounding uniformly the truncated series, the remainders and the .

### Theorem 4.2

*The only lines of constant phase of the* Δ_{4}*(2,2m;s) which can attain* *are equally spaced, and have interspersed lines of constant modulus. All such lines of constant phase reach the critical line in the asymptotic region of t at a pole or a zero of* Δ_{4}(2,2*m;s).*

### Proof.

From equation (4.6), we have
4.12and so the leading order estimate gives the lines of phase zero for Δ_{4}(2,4*m*−2;*σ*+i*t*) as occurring at , for *n*=0,1,2,…. Halfway between these lines of phase zero are the lines on which the leading order estimate gives ∂Δ_{4}(2,4*m*−2;*σ*+i*t*)/∂*t*=0: these are lines of constant modulus, and in fact correspond to |Δ_{4}(2,4*m*−2;*σ*+i*t*)|=1, for . The same argument applies to Δ_{4}(2,4*m*;*σ*+i*t*), with replacing in the estimate for asymptotic placement of lines of phase zero and amplitude unity.

Next, consider where these lines of constant phase and amplitude can go as *σ* decreases towards 1/2. Along the line of unit amplitude, as *σ* decreases curves of constant phase emanate from the line at right angles and head above and below it towards smaller values of *σ*. We see that the phase of Δ_{4}(2,2*m*;*s*) must vary monotonically along the line of unit amplitude; were it to be otherwise, lines of constant phase would form closed loops about the line, constraining the phase to be everywhere constant within the closed loop—a contradiction. On either side of each line of phase zero, we have one line of unit amplitude whose phase increases monotonically as *σ* decreases, and another line of unit amplitude along which the phase decreases monotonically. A similar argument applies to the lines of phase zero, along which the amplitude of Δ_{4}(2,2*m*;*s*) either increases monotonically or decreases monotonically as *σ* decreases. The phase of Δ_{4}(2,2*m*;*s*) on the lines of unit amplitude moves away monotonically from zero, so that these lines can never intersect lines of phase zero in the finite part of the plane (although they could asymptote towards such lines at ). They cannot return to , since their phase cannot tend back towards zero. Thus, the lines of constant amplitude and phase coming from must all cut the critical line, or they could from a certain point on all commence to curve up towards .

Those lines of constant phase zero, which reach the critical line in the region where the phase of is accurately constrained by the asymptotic estimate (2.9) can only cut the line at a pole or zero of Δ_{4}(2,2*m*;*s*).

We finally consider the possibility of all lines of constant amplitude and phase of from a certain point curving up towards . Now in *t*≫|*σ*−1/2|, we find from equation (2.9) that
4.13Hence, from equation (2.9), the lines of constant phase and amplitude curving up towards in *σ*>1/2 must be accompanied by corresponding lines in *σ*<1/2.

Now, in *σ*<1/2, and , and these quantities vary algebraically, whereas Δ_{4}(2,2*m*;1−*s*) has an amplitude and phase which vary in an exponentially weak fashion for 1−*σ* positive and sufficiently large. Thus, subject to this constraint, the contribution from Δ_{4}(2,2*m*;1−*s*) in equation (4.1) to both |Δ_{4}(2,2*m*;1−*s*)| and will be negligible. This means that lines of constant phase and lines of constant modulus are restricted to the former being in the first quadrant and the latter exceeding unity. We then see that lines of constant |Δ_{4}(2,2*m*;1−*s*)| originating at high *t* in *σ*<1/2 with the constant modulus smaller than unity cannot proceed into the region of *σ*−1/2 strongly negative; their images via the functional equation do reach the strongly positive region. Similar remarks apply to lines of constant phase with that phase lying in other than the first quadrant.

We conclude that the topology of lines of constant amplitude and phase curving up towards as *σ* decreases towards 1/2 is incompatible with the topology of such lines in *σ*≪−1/2. ■

We now give one of the principal results of this section.

### Theorem 4.3

*Suppose* *obeys the Riemann hypothesis. Then* * obeys the Riemann hypothesis for any positive integer m. Conversely, if* * obeys the Riemann hypothesis for some m, then* *obeys the Riemann hypothesis.*

### Proof.

Consider lines *L*_{1} and *L*_{2} in *t*>0 along which the phase of Δ_{4}(2,2*m*;*s*) for a given *m* is zero. Join these lines with two lines to the right of the critical line along which *σ* is constant. Then the change of argument of Δ_{4}(2,2*m*;*s*) around the closed contour *C* so formed is zero, so by the Argument Principle, the number of poles inside the contour equals the number of zeros. Each non-trivial pole is formed by a zero of , so if there are no such zeros within *C* there can be no zeros of within *C*. We can repeat this procedure for all *m*.

Conversely, each non-trivial zero of Δ_{4}(2,2*m*;*s*) is formed by a zero of , so if there are no such zeros for some *m*, then there can be no poles, and hence no zeros of . These arguments prove the theorem in the region to the right of the critical line lying between lines of zero phase of Δ_{4}(2,2*m*;*s*), with the result to the left of the critical line then guaranteed by the functional equation (4.1).

To complete the proof we need to show that any point in the region *σ*>1/2, *t*>0 is enclosed between lines of phase zero of Δ_{4}(2,2*m*;*s*) coming from . We note that *t*=0 is one such line, and that for any *σ*>0 the infinite number of such constant phase lines cannot cluster into a finite interval of *t*, since that would indicate an essential singularity of Δ_{4}(2,2*m*;*s*) for that *σ*. ■

### Corollary 4.4

*Assuming the Riemann hypothesis applies to* *or* *apart from a small number of exceptions, lines of phase equal to* 0 *or π of* Δ_{4}(2,2*m*;*s*) *which leave the critical line going into σ*<1/2 *cannot reach the asymptotic region in which the phase estimate (4.8) is accurate. Instead, they must loop back to cut the critical line.*

### Proof.

If *t* is sufficiently large to ensure 4*m*^{2}*t*/(*σ*−1/2)^{2}+*t*^{2}<*π*, then no line with phase *π* can enter the asymptotic region. Lines with phase zero are also excluded. Hence, lines leaving *σ*=1/2 with *t* large and going left, which do not return to the critical line must either curve upwards and go to or curve downwards and cut the line *t*=0. If the first alternative applies, then the image of this line under in the functional equation is a line of variable phase running upwards in *σ*>1/2, which must cut lines of phase zero running left to intersect the critical line. Such intersection points would have to be zeros or poles of Δ_{4}(2,2*m*;*s*), contradicting our assumption. The second alternative means the line of constant phase must intersect the axis *t*=0 at one of the 2*m*−1 poles on the axis. Thus, the exceptional cases are limited to the region *t*<4*m*^{2} and to lines passing through the 2*m*−1 poles.

In fact, it may be numerically verified in particular cases whether such exceptions do in fact occur. They do not for the three cases we have investigated (*m*= 1,2,3). ■

We are now in a position to prove the second of the main results of this section.

### Theorem 4.5

*Assuming the Riemann hypothesis applies to* *or* *, then given any two lines of phase zero of* Δ_{4}*(2,2m;s) running from* *and intersecting the critical line, the number of zeros and poles of* Δ_{4}*(2,2m;s) counted according to multiplicity and lying properly between the lines must be the same.*

### Proof.

We consider a contour composed of the two lines of phase zero, the segment between them on the critical line and a segment between them in the region *σ*≫1. The total phase change around this contour is strictly zero, since the region *σ*≫1 has the phase of Δ_{4}(2,2*m*;*s*) constrained: −*π*≪arg[Δ_{4}(2,2*m*;*s*)]≪*π*. More particularly, if *P*_{u}=(1/2,*t*_{u}) lies at the upper end of the segment on the critical line, and *P*_{l}=(1/2,*t*_{l}) at the lower end, the total phase change between a point approaching *P*_{u} on the contour from the right and a point leaving *P*_{l} going right is zero. This phase change is made up of contributions from the changes of phase at the zero or pole *P*_{u}, from the zero or pole *P*_{l}, from the *N*_{z} zeros and *N*_{p} poles on the critical line between *P*_{u} and *P*_{l}, and from the phase change between the zeros and poles. In this list, the first change is *ϕ*_{2m,c}(*t*_{u}), the phase on the critical line just below *P*_{u}. (We could also have a phase *ϕ*_{2m,c}(*t*_{u})−*π*, but it will be easily seen that in this alternative case, the argument that follows will arrive at exactly the same conclusion.) The second change is −*ϕ*_{2m,c}(*t*_{l}), where *ϕ*_{2m,c}(*t*_{l}) is the phase just above *t*_{l}. Giving zero *n* a multiplicity *z*_{n}, and pole *n* a multiplicity *p*_{n}, the phase change at the former is −*πz*_{n} and the latter *πp*_{n}. The phase change between zeros and poles is *ϕ*_{2m,c}(*t*_{l})−*ϕ*_{2m,c}(*t*_{u}). Hence, the phase constraint is
4.14leading to
4.15as asserted. ■

### Corollary 4.6

*If all zeros and poles on the critical line have multiplicity one, the numbers of zeros and poles on the critical line between any pair of lines of phase zero of* Δ_{4}(2,2*m*;*s*) *coming from* *are the same. The distribution functions for zeros* *and* *of equation (3.5) then must agree in all terms which go to infinity with t*.

### Proof.

The first assertion is a simple consequence of theorem 4.5. The second follows from theorems 4.2 and 4.5: the number of zeros and poles between successive zero lines coming from match for all such pairs of lines, and there are only a finite number of exceptional poles and zeros, which may disturb the equality between numbers of zeros and poles. ■

We now return to figures 1 and 2, to discuss some aspects of the morphologies of lines of constant phase of Δ_{4}(2,2;*s*) near the critical line. Figure 1 gives at top left a diagram in which there are three lines of phase zero coming from . The first two give a loop going right that intersects the critical line at a pole, near *t*=6, and a zero above (between *t*=7 and 8. The third again intersects the critical line at a pole just above *t*=13, and ends what we will describe as a ‘cell’. By this we mean what is a repeat unit in the loose sense; above the third line we start another loop going right. Figure 2*a* shows a view at a structure for *t* in the range (40 : 50), in which the resolution of contour lines is insufficient to give a correct impression of the relationship between the lines of constant phase. The magnified views show that the apparent intersection of lines of phase 0 and *π* near *t*=49.75 is in fact an avoided crossing, where the lines behave in hyperbolic fashion. In figure 2*c*, we see a small line of phase *π*/2 going left away from *σ*=1/2, with an even smaller line element of phase 0 occurring in *σ*>1/2. Note that, denoting poles and zeros by prefixes *P* and *Z*, the values of *t* at which zeros and poles occur in the region of interest are: *P*45.6, *Z*45.9, *Z*46.9, *Z*47.71, *P*47.74, *P*48.0, *Z*49.2, *P*49.72, *P*49.77.

These data then show a succession of three zeros of uninterrupted by zeros of . We have encountered cases where the opposite situation occurs, but no sequences of four successive zeros or poles.

The graph at figure 2*d* shows the hyperbolic distribution of lines of constant phase occurring around a zero of Δ′(2,2;*s*) near *t*=49.75. The discussion following theorem 5.6 of II may be extended from Δ_{3}(2,2*m*;*s*) to Δ_{4}(2,2*m*;*s*), with the result that between successive zeros of Δ_{4}(2,2*m*;*s*) uninterrupted by a pole, there will be a hyperbolic point to the right of the critical line. Between successive poles of Δ_{4}(2,2*m*;*s*) uninterrupted by a zero, there will be a hyperbolic point to the left of the critical line. The graph at figure 2*d* shows the case of two poles, with a hyperbolic point in *σ*<0.5. An example of a hyperbolic point to the right of the critical line may be seen in figure 2*b*, between the successive zeros at *t*=45.9 and 46.9.

Table 2 gives the numbers of zeros and poles in successive cells, for the first 22 cells of Δ_{4}(2,2;*s*). These evidently satisfy the requirements of theorem 4.5. Note, the tendency of the numbers of zeros and poles to increase with cell number; this is a consequence of the cell length tending to be roughly constant, while the density of poles and zeros increases logarithmically, in keeping with equation (3.5). For the fourth column, the mean length is 9.01 and the standard deviation is 0.90. Note that the corresponding separation at is , or around 9.06, so that there is no significant change in average spacing with variation of *σ*. For Δ_{4}(2,4;*s*), table 3 gives data for the first 24 cells, with the mean cell length being 3.92 and its standard deviation 0.53. The separation at infinity here is , or around 3.90.

### Corollary 4.7

*If all zeros and poles on the critical line have multiplicity one, on any closed contour (including those closed at* *there must be an equal number of zeros and poles and they must alternate. i.e. following the contour in a clockwise direction, any pole will be followed by a zero and zero will be followed by a pole. Every cell begins with a pole on the critical line.*

### Proof.

We follow the line of zero phase coming in from , which forms the start of the cell. It must intersect the critical line, in accordance with theorem 4.2 and meet there a line of constant *π* phase going towards increasing *t*, which is also a branch cut discontinuity. The phase *ϕ*_{2m,c}(*t*) is decreasing as *t* increases along the critical line and above the intersection point (call it *t*_{1}) there is a change of *π* to the phase on the critical line, forcing the phase to drop to *ϕ*_{2m,c}−*π*, which lies between (−*π*,0). As phase is increasing in a clockwise direction around the point of intersection, that point must be a pole (i.e. every cell begins with a pole).

We continue to follow the branch cut as it loops back to the critical line in accordance with corollary 4.4. The phase below the next intersection point *t*_{2} is still negative as *ϕ*_{2m,c}(*t*) decreases along the critical line. The change of *π* to the phase at *t*_{2} must be a positive change, such that the phase above the point of intersection must be *ϕ*_{2m,c}(*t*_{2})>0 . The phase is increasing in an anti clockwise direction, which implies that the intersection point *t*_{2} is a zero. This argument is continued for successive poles and zeros until we reach the end of the cell in question. ■

### Remarks

The following consequences are clear from the arguments in corollary 4.7:

— Any closed contour intersecting the critical line at a finite set of points will have maximum and minimum intersections of opposite type, i.e. a zero and a pole.

— If there are two consecutive zeros or poles on the critical line they cannot lie on the same closed contour of zero phase.

— Any closed loop of zero phase going from

*σ*=1/2 into*σ*<1/2 must have a zero at its lowest intersection with*σ*=1/2.

We conclude with a result on the multiplicity of zeros and poles.

### Theorem 4.8

*Assume* *obeys the Riemann hypothesis. Then in the asymptotic region t*≫4*m*^{2}*,* *has no multiple zeros if and only if* * for an m*>0 *has no multiple zeros.*

### Proof.

If obeys the Riemann hypothesis, all zeros of it and for every *m*>1 are on the critical line. The sequence of zeros and poles on the critical line in *t*≫4*m*^{2} is constrained by the phase of Δ_{4}(2,2*m*;*s*) there, which must lie in either the first or third quadrant. All four quadrants of this phase must be represented around each pole or zero.

Consider a succession of poles or zeros starting when a line of phase zero comes in from *σ*≫1/2 and intersects the critical line. Every such line has phases in the fourth quadrant above it and in the first quadrant below it. Phases increase in the clockwise sense around the intersection point, which must be a pole.

Suppose every pole of Δ_{4}(2,2*m*;*s*) is first order. Then, the phase of Δ_{4}(2,2*m*;*s*) above the pole on the critical line is incremented from that below by *π*, and lies in the third quadrant. The functional equation for Δ_{4}(2,2*m*;*s*) guarantees that adjacent to this is a region with phase in the second quadrant. This makes the next zero or pole on the critical line in fact a zero. Were this a zero of even order, the phase on the critical line above the zero would again lie in the third quadrant, and we could deduce that the next zero or pole would be a zero. Only by reverting to a zero of odd order can we again encounter a pole on the critical line. The best we can then do is to have a succession of zeros followed by poles; we cannot compensate for the zero of multiple even order. For a zero of odd order greater than one, we follow it with a pole of single order, then a zero, etc., and again we have not compensated for the zero of multiple order. Thus, zeros of multiple order destroy the balance between multiplicities of zeros and poles required by theorem 4.5. This proves that having single zeros implies for any integer *m* cannot have multiple zeros in the asymptotic region, for a zero-phase contour coming in from *σ*≫1/2. The case of a zero-phase contour coming in from *σ*≪1/2 is dealt with by an analogous argument.

To prove the contrary, start from a zero of at the top of the phase loop, now of single order. It is evident that, if we work down the critical line, poles of multiple order cannot be compensated as required by theorem 4.5. ■

## Acknowledgements

The research of R.McP. on this project was supported by the Australian Research Council Discovery Grants Scheme.

- Received October 30, 2010.
- Accepted February 7, 2011.

- This journal is © 2011 The Royal Society