## Abstract

We consider the Hurwitz zeta function *ζ*(*s*, *a*) and develop asymptotic results for *a*=*p*/*q*, with *q* large, and, in particular, for *p*/*q* tending to 1/2. We also study the properties of lines along which the symmetrized parts of *ζ*(*s*, *a*), *ζ*_{+}(*s*, *a*) and *ζ*_{−}(*s*, *a*) are zero. We find that these lines may be grouped into four families, with the start and end points for each family being simply characterized. At values of *a*=1/2, 2/3 and 3/4, the curves pass through points which may also be characterized, in terms of zeros of the Riemann zeta function, or the Dirichlet functions *L*_{−3}(*s*) and *L*_{−4}(*s*), or of simple trigonometric functions. Consideration of these trajectories enables us to relate the densities of zeros of *L*_{−3}(*s*) and *L*_{−4}(*s*) to that of *ζ*(*s*) on the critical line.

## 1. Introduction

In an accompanying paper (McPhedran *et al*. in press), hereafter denoted I, we considered the properties of the Hurwitz zeta function, *ζ*(*s*,*a*), and showed how it could be decomposed into four parts, each obeying functional equations of the type associated with Dirichlet *L* functions. We then considered this decomposition for zeta functions, with *a*=*p*/*q* rational, *q* ranging from 1 to 10. The results became steadily more complicated with increasing *q*.

Here, we will consider first the asymptotics of the decomposition for large *q*. We will show how these asymptotics may be developed as *p*/*q* approaches values such as 1/2, where the form of *ζ*(*s*,*a*) is analytically known. We will consider the properties of the zeros of the four constituent functions in the asymptotic domain and, in particular, will construct sequences of simple zeros on Re(*s*)=1/2, whose limit is a zero of the Riemann zeta function.

We will also consider the properties of lines along which the two functions *ζ*_{+}(*s*,*a*) and *ζ*_{−}(*s*,*a*) are zero. For the former, these lines start either at *a*=1/2 with Re(*s*)=σ=0 and end at *a*=3/4 with *σ*→−∞ or *a*=1/2 with *σ*=1/2 and end at *a*=1 with *σ* and *t*=Im(*s*), both tending to zero as a reciprocal logarithm of *a*. For *ζ*_{−}(*s*,*a*), the null lines start either at *σ*=−1 or −1/2 and end as *a*→1, with *σ* approaching −1 and *t* tending to zero. We use the properties of these null lines to relate the densities of zeros of *L*_{−3}(*s*) and *L*_{−4}(*s*) to that of *L*_{1}(*s*) on the critical line.

Our interest in the properties of zeros of *ζ*(*s*,*a*) and its symmetrized parts *ζ*_{+}(*s*,*a*) and *ζ*_{−}(*s*,*a*) relates to their occurrence in the lattice sums, which occur in diffraction by gratings (Nicorovici & McPhedran 1994; Nicorovici *et al*. 1994; McPhedran *et al*. 2004). Poles of lattice sums correspond to light lines, a case of extreme-field interaction between the constituent parts of a periodic structure, while zeros correspond to the absence of such interaction. To view all figures in this article reproduced in colour, please see electronic supplementary material.

## 2. Asymptotics for large *q*

We have seen that for arbitrary *q*, we can construct four sets of basis functions, generalizing the Dirichlet functions of positive and negative order. The analysis of their properties becomes more complicated for increasing *q*, so it is of interest to investigate the asymptotic case, where *q*≫|*s*|, in search of simple results. We now follow this approach.

We note that as *q* increases, the number of basis functions of the four types increases. However, all basis functions of a given type obey the same functional equation, as does any linear combination with coefficients independent of *s*. Hence, we can write for the functions *ζ*_{+}(*s*,*a*) and *ζ*_{−}(*s*,*a*), with *a* equal to any rational *p*/*q*, that(2.1)

Using (4.11) and (4.12) from I, we find that(2.2)so that we can solve (2.1) and (2.2) to find a way of calculating the new basis functions(2.3)and(2.4)

The two functions generalizing the Dirichlet functions of negative order are(2.5)and(2.6)

One interesting way to use the expressions (2.3)–(2.6) is to let the ratio *p*/*q* approach a value for which *ζ*(*s*,*a*) is analytically known, e.g. one of the particular values such as 1/2, 1/3, 1/4, 1/5, 2/5 and 1/6 as mentioned previously. This generates expansions for *ζ*_{+}(*s*,*p*/*q*) and *ζ*_{−}(*s*,*p*/*q*) involving lower-order *L* functions, with the expansion parameter being *s*/*n*. We illustrate the procedure in the simplest case, *a*=1/2.

We use (5.6) and (5.7) from I with *a*′=(*n*+2/2*n*). Then(2.7)and(2.8)

We find(2.9)and(2.10)

In figure 1, we compare the logarithms of the magnitudes of the left-hand side terms in (2.9) with the leading terms of the right-hand side. Note that the leading terms of the right-hand side give the zeros of the left-hand side terms accurately in the region |*s*|≪2*n*, which here translates to |*s*|≲20. Note also the large number of prefactor zeros evident, coming from the term in square brackets in (2.9).

Let us consider some properties of these prefactor zeros. They satisfy the equations(2.11)and if *s* is a root, then by construction, so is 1−*s*. It is obvious that the roots of these two equations cannot coincide. Note that and have zeros on the real axis: two in the former case at −*Δ*_{n} and 1+*Δ*_{n}, and three in the latter case at , 1/2 and , where *Δ*_{n} and tend to zero as *n*→∞. In both cases, the prefactors have an infinite number of zeros on Re(*s*)=1/2, with their density increasing with *n*.

These properties are more evident if we use the transformations (4.21) and (4.23) defined in I. We find(2.12)where(2.13)

Note that the first term on the right-hand side of (2.12) is real on Re(*s*)=1/2, by virtue of the reflection formula for *ζ*(*s*). We find that the prefactor zeros of are given in terms of *s*=1/2+*δσ*+i*t* by the two equations(2.14)and(2.15)

It is easy to see that this pair of equations has no solutions if *δσ*≪1. If *δσ*=0, only (2.14) applies, so that the prefactor zeros on Re(*s*)=1/2+i*t* are given in the asymptotic region by the transcendental equation(2.16)while those of satisfy(2.17)

There will be for large *n* on average one zero of (2.16) or (2.17) in each interval of length 2*π*/log *n*.

Note that if we consider an interval around each zero *ζ*_{p}=1/2+i*t*_{p} of *ζ*(*s*) on *σ*=1/2, in which tan[(*t*/2)log 2] does not change sign, then zeros of (2.14) and (2.15) alternate. Of course, zeros in smaller and smaller neighbourhoods of *t*_{p} correspond to values of *n* for which the two functions(2.18)are close to integer values, for some integer *m*. This would be a difficult algorithm to implement numerically, since construction of prefactor zeros ever closer to *t*_{p} would require evermore accurate values for *t*_{p}. However, the fact that there is one zero of each of (2.16) and (2.17) in regular intervals of length 2*mπ*/log *n* guarantees that we can construct sequences of these simple zeros tending towards any *ζ*_{p} on Re(*s*)=1/2. The sequences interlace, as we have noted previously.

For the functions of negative order,(2.19)where(2.20)

We may rewrite (2.7) and (2.8) in terms of the functions in (2.12) and (2.19) as(2.21)and(2.22)

Note that one needs to use the reflection equation for *ζ*(*s*) (Abramowitz & Stegun 1972, eqn (23.2.6)),(2.23)and the properties of the gamma function to compare (2.22) and (2.8).

## 3. Null lines

Let us consider the construction of lines along which *ζ*_{+}(*s*,*a*)=0 and *ζ*_{−}(*s*,*a*)=0. Such contours are described by the partial differential equation(3.1)

Such lines can then be constructed using (3.1) through points for which the denominator is non-zero. (If the numerator is zero, a higher-order treatment is necessary.)

Any zero of *ζ*_{±}(*s*,*a*) for which ∂*ζ*_{±}(*s*,*a*)/∂*s*≠0 must lie on a null trajectory. It will lie either on one trajectory, if it is a zero of one but not the other, or on two trajectories, if it is a zero of both (assuming the order of the zeros to be unity).

Note that *ζ*_{+}(*s*,*a*) is symmetric about *a*=1/2, while *ζ*_{−}(*s*,*a*) is antisymmetric about *a*=1/2. This means that the null trajectories for both are identical for *a*≤1/2 and *a*≥1/2, and may be regarded as starting at *a*=1/2 and continuing towards higher values of *a*. As we shall see, they either ‘end’ with *σ*→−∞ or with *t* tending to zero as *σ*→0 or −1. For the second and third of these alternatives, they continue into the lower half-plane, arriving at *a*=1/2 with *s* given by the complex conjugate of its value at *a*=1/2 in the upper half-plane.

We illustrate the properties of the null trajectories, starting with *ζ*_{+}(*s*,*a*). From (5.2) in I,(3.2)so that the null trajectories of *ζ*_{+} commence either at a zero of *ζ*(*s*) or at *s*=2*mπ*i/log 2, for *m* an integer. The trajectories are points in the space of the three real variables *a*, *σ* and *t*, so we use a technique (Trott 2004) for their visualization which combines the trajectory with its projections on the *σ*−*t*, *a*−*σ* and *a*−*t* planes. All the five trajectories shown in figures 2–4 start at *a*=1/2, *σ*=1/2 and move towards increasing *σ*. For a trajectory starting at a zero *s*_{0} of *ζ*(*s*), the first-order estimate of the initial step on the trajectory is(3.3)

We plot *δσ* in figure 5 for the first hundred zeros of *ζ*(*s*); it is always positive.

As *a* increases towards 1, the trajectories pass through special points (see (6.3), (7.2) and (9.3) in I) at(3.4)(3.5)and(3.6)

Finally, all trajectories end with *s* tending to zero as a series in 1/log *a*, for *a*→1. The general trend along trajectories is that *t* decreases as *a* increases, and the trajectories become more highly oscillatory as the order of the zero of *ζ*(*s*) increases.

To ascertain the form of the null trajectories near *a*=1, we let *a*=1−*δ* and find(3.7)or(3.8)

Solving for the zeros of (3.8), we find the following estimate for the null trajectory near *a*=1:(3.9)

Thus, *t* tends to zero as a series with first term going as 1/log *δ*, while *σ* tends to zero as 1/(log *δ*)^{3}. The limiting trajectories are numbered by even integer orders 2*m*.

Next, consider the trajectories starting at *s*=2*mπ*i/log 2. The first four of these are shown in figures 6 and 7.

The trajectories now have a first step estimate given by(3.10)

As shown in figure 5, this can be either positive or negative. Figure 7, for *m*=4, shows an example of a trajectory where this is negative. The trajectories pass through a special point given by (3.4) before *a*→3/4, *t*→(2*m*−1)*π*i/log 2 and *σ*→−∞.

For the trajectories of *ζ*_{−} (figures 8–10), we see from (2.8) that they start at(3.11)

Their special point values are given by (see (6.5), (7.3) and (9.10) in I)(3.12)(3.13)and(3.14)

These trajectories then provide a 1 : 1 mapping between the zeros of *L*_{−3}(*s*) and *L*_{−4}(*s*). An interesting deduction is that both *L*_{−3}(*s*) and *L*_{−4}(*s*) must have at least one zero in each interval 2*π*(*m*−1)/log 2<*t*<2*πm*/log 2, for *m* a positive integer. In figure 11, we show the histograms of the number of zeros of the functions *ζ*(*s*), *L*_{−3}(*s*) and *L*_{−4}(*s*) in each such interval, for the range in which *ζ*(*s*) has 750, *L*_{−3}(*s*) has 946 and *L*_{−4}(*s*) has 997 zeros on the critical line. The zeros used to compile these and the following figures are taken from the dataset of M. Rubinstein.1

As *a* approaches 1, we have(3.15)or(3.16)where *γ* is the Euler–Mascheroni constant and *γ*_{1} is the Stieltjes constant.2 The null trajectories near *a*=1 are then(3.17)

Thus, as *a* approaches unity, *σ* tends to −1, while *t* tends to zero as a series with first term 2*mπ*/log *δ*.

From equation (3.1) for the null trajectories and the Cauchy–Riemann equations, it follows that if there are two or more null trajectories at a particular point, they must be tangent there. Thus, even in the presence of zeros of multiple order, we can still count null trajectories. We can also argue from the study of the *ζ*_{−} trajectories that the multiplicity of any zero of *L*_{−3}(*s*) is shared with that of the following zero of *L*_{−4}(*s*) on the null trajectory.

Note that the asymptotic results (3.9) and (3.17) establish that *s*=−1 and 0 are special points, where an infinite number of null trajectories are tangent, before fanning out in the upper and lower half-planes. The other special point is at *s*=−∞, where the *ζ*_{+} trajectories starting on *σ*=0 end.

We can use the form of the null trajectories to establish recurrence relations involving the number of zeros of the functions *L*_{1}(*s*), *L*_{−3}(*s*) and *L*_{−4}(*s*) in successive intervals (2*m*−2)*π*/log 2<Im(*s*)<2*mπ*/log 2. Let these be denoted by *n*_{1}(*m*), *n*_{−3}(*m*) and *n*_{−4}(*m*), respectively, and let *l*_{3}(*m*) denote the number of occurrences of integer multiples of 2*π*/log 3 in the same interval. This last quantity is either one or two, as shown in figure 12.

Consider first the simpler case: the null trajectories of *ζ*_{−}(*s*). In the *m*th interval, *n*_{1}(*m*)+1 trajectories start at *a*=1/2. At *a*=2/3, *n*_{−3}(*m*) points in the interval are available for trajectories. This leaves an excess of *n*_{−3}(*m*)−*n*_{1}(*m*)−1 points which lie on trajectories starting in intervals above the *m*th interval. Hence, if we introduce an excess function for *ζ*_{−} null trajectories, we see that(3.18)

Similarly, at *a*=3/4, *n*_{−4}(*m*) points are supplied, so that(3.19)

Finally, at *a*=5/6, *n*_{−3}(*m*)+1 points are supplied, so(3.20)

Note that and *n*_{1}(1)=0. Thus, and , given that *n*_{−3}(1)=*n*_{−4}(1)=1 (as required by (3.18) and (3.19)). These functions are shown in figure 13; both and decrease only in two instances, in the range shown.

Turning to *ζ*_{+}(*s*), the number of trajectories starting on *a*=1/2 is *n*_{1}(*m*)+1, of which one ends at *a*=3/4 with *σ*→−∞. At *a*=2/3, the number of points on trajectories is *n*_{1}(*m*)+*l*_{3}(*m*) and the number of points required by the starting trajectories is *n*_{1}(*m*)+1, so that(3.21)

At *a*=3/4, the number of points on trajectories is again *n*_{1}(*m*)+1 and the number required by starting trajectories is now *n*_{1}(*m*), so that(3.22)

Again, at *a*=5/6, the number of points on trajectories is *l*_{3}(*m*)+*n*_{1}(*m*)+1, so that(3.23)

The initial conditions are , and , so that , and . The solution of this second recurrence relation is of course . The behaviour of is shown in figure 14. Note that this function has only two types of behaviour: it remains constant or increases by 1.

The mean value of the second term in (3.22) is 〈*l*_{3}(*m*)−1〉=log 3/log 2−1≃0.584963, so that the trend line for is(3.24)

As can be seen from figure 14, this trend line fits the numerical data very well. It is interesting to note that the binned data in figures 13 and 14 are much smoother than those in figure 8. It is also highly significant that the same trend lines we have deduced for the *ζ*_{+} excess functions fit very well the *ζ*_{−} data. In the case of figure 13*b*, the comparison line plotted is just . Also, comparison between (3.21) and (3.23) shows that(3.25)

A similar relationship exists for ,(3.26)

Assuming the validity of the fitted lines in figures 13 and 14, we can hypothesize the relationships between the mean densities of zeros of *L*_{−3}(*s*) and *L*_{−4}(*s*) with that of *L*_{1}(*s*) on the critical line. We find that(3.27)and(3.28)

Histograms of the distributions of *n*_{−3}(*m*)−*n*_{1}(*m*) and *n*_{−4}(*m*)−*n*_{1}(*m*) are given in figure 15. The datasets shown in figure 15, 〈*n*_{−3}(*m*)−*n*_{1}(*m*)〉=1.58065 and 〈*n*_{−4}(*m*)−*n*_{1}(*m*)〉=1.99194, are in good agreement with the estimates in (3.27) and (3.28) (1.58496 and 2, respectively). If we increase the number of zeros of *L*_{−3}(*s*) and *L*_{−4}(*s*) to around 100 000, then the estimates get closer to the analytical values 1.58493 and 1.99986, respectively.

Note that we have also checked the difference *n*_{5}(*m*)−*n*_{1}(*m*) for the function *L*_{5}(*s*). For a sample containing 1115 zeros, generated using Ersek's algorithm (Ersek 2005), we find an estimate of 2.31061, compared with log(5)/log(2)≃2.32193. This suggests the possibility of a general formula(3.29)

This formula would be in accord with the result (4.16) from I,(3.30)and embodies the assumption that Dirichlet *L* functions, and possibly the related functions defined in I, all have a density of zeros related to that of *L*_{1}(*s*) by their order. Note further that (3.29) is structurally satisfied by the forms of function from I: (4.17) and (4.18) but not (4.19) and (4.20). However, the functions (4.19) and (4.20) from I are convenient but not essential in the construction of bases of solutions of the functional equations, since(3.31)while(3.32)is a difference of two solutions of the requisite functional equation, each satisfying (3.29).

We have examined the other functions investigated in I for *q*=5. For the two functions and shown in fig. 4 of I, they are of the form (3.30), and thus satisfy (3.29) automatically. However, it is already evident from fig. 3 of I that and have different densities of zeros on *s*=1/2+i*t*, for *t*≤30. This continues for larger *t*: has 1045 zeros for *t*<1200, while has 742 zeros. This choice of basis functions does not obey (3.29). However, the choice of functions *P*(*s*) and *Q*(*s*) of (8.5) and (8.6) from I, based on series with imaginary characters, gives a different result. In fact, *P*(*s*) has exactly the same number of zeros (1118) for *t*<1200 as does *L*_{5}. Hence, it gives exactly the same estimate (2.31061) for *n*_{5}(*m*)−*n*_{1}(*m*) as does *L*_{5}. For *Q*(*s*), the number of zeros is 1120 for the same interval, giving an estimate of 2.31818, which is slightly closer to the analytical value. Thus, grouping basis functions according to the distribution of zeros rather than the form of the functional equation would suggest for *q*=5 the set rather than .

Taking into account the above-mentioned remarks in reference to (3.31) and (3.32), it is evident that a basis of solutions of the functional equations exists for *q*=6. For *q*=8, in addition to functions satisfying (3.29) by virtue of their form, there are two Dirichlet *L* functions, *L*_{±8}(*s*). Each has 1209 zeros for *t*<1200, and these give rise to the numerical estimate of 2.99242 for *n*_{8}(*m*)−*n*_{1}(*m*), in comparison with the analytical value of 3. Clearly, further analytical and numerical investigation of the hypothesis (3.29) for higher values of *q* is warranted.

As a final comment, it is interesting to contrast the relative smoothness of the histograms in figures 13 and 14 with the much less regular behaviour in figure 8. This suggests that there is a common origin for the irregularity in the distribution of zeros of *L*_{1}(*s*) and of higher-order functions *L*_{±q}(*s*), and this irregularity is suppressed when 〈*n*_{±q}(*m*)−*n*_{1}(*m*)〉 is studied.

## 4. Conclusions

The results we have established here suggest that there are interesting connections between the Dirichlet functions and the Riemann zeta function which remain to be discovered, and that geometrical tools may well be useful in this quest. We have confined our attention to *ζ*_{±}(*s*,*a*) for *a*=1/2, 2/3, 3/4, 5/6 and 1; it may also be the case that the exact solutions in I for other values of *a* can also be exploited. In addition, it will be of value to use these results in the investigation of double sums and their possible factorizations.

## Acknowledgments

This work was supported by the Australian Research Council. Numerical, analytical and graphical results were obtained with Mathematica v. 5.0. This work benefited from the online tabulations of zeros of M. Rubinstein.

## Footnotes

The electronic supplementary material is available at http://dx.doi.org/10.1098/rspa.2006.1763 or via http://www.journals.royalsoc.ac.uk.

↵http://pmmac03.math.uwaterloo.ca/∼mrubinst/l_function_public/ZEROS.

- Received March 27, 2006.
- Accepted July 14, 2006.

- © 2006 The Royal Society