ZEROS OF SMALLEST MODULUS OF FUNCTIONS RESEMBLING exp ( z )

To determine (in various senses) the zeros of the Laplace transform of a signed mass distribution is of great importance for many problems in classical analysis and number theory. For example, if the mass consists of finitely many atoms, the transform is an exponential polynomial. This survey studies what is known when the distribution is a probability density function of small variance, and examines in what sense the zeros must have large moduli. In particular, classical results on Bessel function zeros, of Szego on zeros of partial sums of the exponential, of I. J. Schoenberg on k-times positive functions, and a result stemming from Graeffe's method, are all presented from a unified probabilistic point of view.

where g(t) is a probability density function of mean I and variance .When o 0 the g(t) becomes the Dirac "delta function" at I, and f(z) is then exp z.Thus, when all the mass is concentrated at i, there are no zeros; when the mass spreads out from i the zeros move in from and may come closer and closer to z 0, although they cannot attain it (nor any other point on the real axis).
In probability theory f(iz) is known as the characteristic function of g(t) [LU 1,2].We note that if g(t) consists of a finite number of atoms, then (i.i) becomes an exponential polynomial.
For an example, say g(t) ga(t) has two atoms of mass 1/2 located at i a and I + a respectively.Then Z f(z) e cosh az, (1.2) and as the parameter a increases the zeros become more and more dense along the imaginary axis, with the one of smallest modulus approaching 0.
For another example, note that as the parameter b increases from 0 to I, the modulus of the zeros of f(z) b + (l-b)cosh z (1.3) nearest to the origin tends to ; in fact, it can be shown to do so monotonically.
Here we have two atoms of mass (l-b)/2 at +i and one atom of mass b at 0, so the mean is 0 in this case.
The following informal assertion seems reasonable.
The -Hypothesis.The distance from the origin to the nearest zero tends to vary inversely with the variance Our goal in this paper is to determine to what extent this hypothesis is true.
To do this we assemble results scattered throughout the literature.However, the theorems of 3 and 4 may be new.
To achieve an appropriate level of generality, we shall study the hypothesis in the class of all functions of the form f(z) I eWZd (w)   (1.4) where (w) is a probability measure on the complex plane, not necessarily of corn- pact support, with "mean i" and "variance o," i.e.,

o2
For example, when (w) is purely atomic, symmetric about the real axis, and con- fined to Re(w) where J(z) J(z;g) is an entire function such that J(O) I.If J(z) is neither an exponential nor a constant, let z be its zero of smallest modulus.

O
Then the 6G-hypothesis for f(z) has in this case the very simple form O since G r.For (1.7) it is obvious that increasing r brings the nearest zero closer; for (1.6) it is mearly plausible that increasing one of the %j may have this effect.
REMARK i.For a non-trivlal characterization of exp z, via its lack of zeros, in a function class much larger than E, see [HA, pp. 66-67].
REMARK 2. It may be of interest to investigate the angular distribution of the zeros of J(z).If g( 8) is considerably larger for lel < /2 than for 181 > /2 (say -<_ 8 <_ ), will J have more zeros in the left half plane than in the right?The results of P61ya [PO 4] on Mittag-Leffler's function suggest that a nearly uniform g(8) will have its zeros distributed almost uniformly with respect to angle.
2. COUNTEREXAMPLES TO THE O-HYPOTHESIS.When" (w) is chosen so that f(z) has the form (1.6), one can sometimes create a larger zero free region about the origin by dispersing some of the mass arbitrarily far away from w i.We exhibit such an example.
Let D be the closed disc about 0 of radius n, and set f(z;e,%) --k[cos z + cos %z], (2.1) where k is chosen so dl(w) 1. (2.2) Fix a positive integer n.For I%1 <_ 5n, the functions of (2.1) converge equlunl- formly on D to cos z as e / 0. Hence for small they have only two zeros in D, one near w/2 and the other near -w/2.Since f is even, we need only con- slder the zero z near w/2.For % i this z is w/2.Thus for e(n) sufficiently small, the zeros of f(z;e,4n) nearest the origin are more distant than the nearest zero of f(z;e,l).
For further examples of a similar nature (involving cosine sums on the real line) see [NU].One might doubt, however, the existence of far more extreme counterexamples.For t >_ 0 let t(w) B(w;t) be a family of mass distributions compact support, continuous in t, such that for t o < t I the distribution (w;t I) is obtained from (w;t 0) by moving some of the mass of (w;t 0) further away from i in the radial direction.We shall even make the assumption (A): all the t are centrally summetric about i, and 0 has no mass in some neighborhood U of i.Let  z If 0(w) is the mas distribution corresponding to e cos z, it seems safe to conjecture that Roo is rather less than 3n/2, and that any mass distribution corresponding to a nearly maximal R t will have a large proportion of its mass very near the points i + i. REMARK i.For mass distributions along the real axis, the answer can be nega- rive, no matter how large U is! See 7.
REMARK 2. In the example (2.1), the highest frequency was attached to the smallest mass.The distribution of the roots of b f(z) etZg(t)dt a where g(t) is continuous is sometimes more peculiar or more difficult to analyze when g(a) g(b) 0; see [CART, TI].When g(t) is monotone, or M f(z) [ a k cos kz, a N < aN+1 <--.< aM, k=N (2.8) the situation is far simpler; see [PO 2].
The simplest quantitative formulation of the 6-hypothesis is that o >> I.
(2.9)More precisely, this means that given a family G of probability measures the pro- duct 6o exceeds a positive constant depending only on G.We now show that this is not true unless G is somehow restricted.Let eZ[ee -az + ee az + i 2], a,e > 0. (2.10)

Here
(2.11) By solving a quadratic equation, we find there are roots as close as -I a ln[-e + 0(e 2)]I, and - (2.12) Since el/2n e 0 as e O, (2.13) the formulation (2.9) cannot be correct.In fact, if P(z)  2 n e -I >> P(c0), (2.16) c o a contradiction for an appropriate c o by (2.13).
3. THE 6O-HYPOTHESIS FOR SYMMETRIC PEAKED DISTRIBUTIONS.The 6c-hypothesis is true for functions f(z) of the form (1.1) when g(t) is a probability density function symmetric about I and "strongly" peaked near 1.The result (3.3) asserts that 6 is large if is very small.
Note that here and elsewhere we use "peaked" as an informal adjective rather than as a concept with a precise logical definition.and by expanding the hyperbolic cosine into an infinite series we easily see that Then the last integral on the right of (3.7) is bounded by {exp [-M(M-I) 6 2 / (M-l) 6. (3.9) (3.11) The result follows.
Note that if g were replaced by gl in (3.2), the proof would make no essen- tial use of the nonnegativity of g. 4. GRAEFFE'S METHOD.The first naive formulation of the 6o-hypothesis, (2.9), does seem to hold when the g(t) in (i.i) is very close to a smooth unimodal dis- tribution that is "somewhat" peaked at 1.For example, when g(t) is a rectangular pulse of width 2e centered at i (this can be approximated arbitrarily well by a  We now show that something like (2.9) holds when g(t) is suitably concen- trated near i and 6 is small.In this case the old root-squaring method (Graeffe's method) that has been popular in the past for determining minimum modulus zeros of polynomials is quite usable.In fact, this method has been used previously on certain transcendental entire functions; see [POI, 5] and [DIR].We shall require that the power moments of g(t) do not grow too rapidly; our condition (4.3) seems quite natural since it is satisfied for g(t) exp(-It-I I) and g(t) exp[-(t-l)2], and commonly occurs in the study of characteristic functions [LUI, pp.19ff.,27ff].
THEOREM 4.1.If the (t) of (i.I) satisfies (4.3) for positive constants A and K, and z 0 is any zero of f(z) less than I/K in modulus, then f eiZtg(t)dt.we find for the even function below on the left that iz(t-s) where R(z) is an infinite sum of double integrals.Since (t-l) (s-l) ]2mg(t)g(s)dtds Since G(z 0) O, the result follows by summing the series on the right of (4.9).
For example, if A K 1 and f(z) has a zero in zl < 1/4, inequality (4.4) shows that the variance of g(t) must exceed 15.
Note that the proofs of the above theorems use (4.6), but not the nonnegativity of g.We add that if the mass is not supported by any inteval smaller than [a,b], then (b-a)/W is the "linear density" of the zeros of f(z) see [TI].
5. PARTIAL SUMS OF THE EXPONENTIAL SERIES.By Hurwitz's Theorem, the partial sums N SN(Z) Z zk/k! k=0 (5.1) have no zeros in a disc of radius r N about 0, where r N / as N oo.This statement has been made very precise by Szeg [SZ] who has shown that if the zeros of s N(z) are divided by N, these normalized zeros, for large N, lie very near the subset H of the curve C defined by 1-z ze 1 (5.2) for which Re(z) < i.In fact, H is a loop that encircles the origin.The curve (5.2) roughly resembles the Greek letter "a".It intersects itself at right angles at z i, and to the right of z I it consists of two curves, mirror images in the z-axis of each other, with the one in the first quadrant being convex, and having an ordinate that grows exponentially with its abscissa.We speak of the parts of (5.2) to the left and right of z i as being the "head H" and the "tail T", respectively of C.
Curiously enough, Szeg also proved that the normalized zeros of the "tail" of the exponential series k TN(Z) k k=N+l (5.3) lie very near T! In particular, all the zeros are at least N in absolute value, and this is asymptotically best possible.We now show how this bears on the present investigations.
Consider the probability density given by (N+l)t N 0 _< t _< 1 g(t) 0 otherwise. (5.4) Then g(t) can be approximated arbitrarily well by sharply peaked C unimodal densities.A simple calculation yields i N+I 2 N+I tg(t)dt N--$' C 0 (N+2) 2 (N+3) (5.5) so for N large it is very nearly the sort of g(t) we considered in (i.i).Since (5.8)Thus >> i, and o i as N . (5.9) This provides another example where (2.9) is valid, and with a smaller constant than in (4.2) or (4.18).
REMARK.Szeg also determined the asymptotic angular distribution of the zeros.
His results (particularly in a neighborhood of z I) have been both greatly refined and also extended to entries of the Pad table for exp(z).See [BU,NEW,.
6. SMALL ZEROS OF BESSEL FUNCTIONS.The rule (2.9) seems to hold quite generally in the absence of the sort of "side lobes" that played a role in constructing the counterexamples of 2. This leads us to suspect it may be satisfied, or even "oversatisfied," when g(t) is convex aside from some small neighborhood of its maximum.
For an example of this we turn to the book of B. C. Carlson, Special Functions of Applied Mathematics [CAR], that emphasizes the utility of the fact that most of the commonly occurring special functions are expectations of elementary functions (sometimes exponentials) with respect to common probability measures.For example, the Bessel functions Jr(X) have the form (x/2) r Jr (x) F(r+l) S(r+i/2,r+I/2;ix,-ix) (6.1) where [CAR, pp. 93-96]  S(bl,b2;Zl,Z2) exp[wz I + (l-w)z2]d (w)   (6.2) Here B(x,y) is the usual beta-function.For a random variable X with the dis- Hence S (r+i/2, r+i/2; z/2 ,-z/2) (6,4) is a function f(z) of the form (i.I) with r(_r+l) J (z/2f) (6 5) (z/41) r r o 8(r+l) ]-i/2. (66)   Thus the rule (2.9) yields 6 >> r I/2 (6.7) for Jr(Z).But Watson's treatise on Bessel functions [WA] tells us that the small- est zero of J (z) is r + 0(rI/3) r so in this case 6 >> i - (6.8)   7. SOME IDEAS FROM PROBABILITY THEORY.The example of 6 leads us to ask if there are some further hypotheses that can be made on the shape of a unitary spike so that the nearest zero would have to be even further away for o small, with perhaps a rule such as 6Om >> 1 (7.1) where m is some positive integer.
If we think of g(t) as the distribution function of a random variable X, it is natural to also consider the distribution function for the average X I +'''+ Xn x (7 2) n where XI,... ,Xn are random variables with the same distribution as X itself.Let f(z;Y) denote the moment generating function for a given random variable Y, let O(Y) denote its variance, etc.Then f(z;X--) [f(;X) in, O(X--) O(X)/n, (7.3) and (X) n(X).( 7.4) Thus if we begin with a class of functions g(t) for which the rule (2.9) holds, and replace each g(t) by the density for the corresponding X, the rule holds with "much more room to spare."What (7.3) and (7.4) suggest, of course, is that for "nicely shaped" pulses, the appropriate rule to investigate is a rule we already stumbled upon in 6.
We next observe that for g(t) with infinite support, it may be that 0. is large, and f(z) is zero free.For example, let g(t) be the normal density Since g(t) exp(-z+z20.2/2) (7.7) there are no zeros!We can now construct densities of compact upport that are nearly as good.Simply multiply g(t) by a number 1 + e where e > 0 and rede- fine it to be zero outside of some large interval [-L,L], so the resulting function again has total mass one.Various densities other than the g(t) of (7.6) are also available; in fact, when an infinitely divisible distribution has an entire charac- teristic function f(z), then f(z) is zero free [LUI, p. 187].
The graph of the normal density is bell-shaped; this means that its nth derivative changes signs n times.If this holds merely for n < M we can say it is M-bell shaped.It is conceivable that some such subtle measure of the shape of the spike and not merely its peakedness may determine the extent to which (2.9) or (7.5) may be replaced by a sharper rule.The concept of bell-shaped will also occur below in 8.
We now return to the problem of 2. Let g(t) be a "normal density distri- bution" with 1 and 0. extremely large, but modified as above to be of compact support.Fix e > 0 and let r(t;) be a rectangular pulse distribution with i and support [I-B,I+] where i+n 1-rl go(t) dt < e" (7.8) First shift /2 of the mass of r(t;N) to each of the points i N and i + , and then shift the remaining mass of r(t;) outwards so it becomes the mass distri- bution of go(t).By choice of we can insure that before the second mass shift the nearest zero is at a distance of approximately / while after that shift it is far away as desired.Thus the problem of 2 has a negative answer in this case.
PROBLEM.Can the mass of the rectangular pulse be moved out to infinity con- tinuously with respect to time so that the distance of the nearest zero from the origin is monotonically increasing?
Scale g(t) so that it becomes a probability density.Then as a- its variance tends to 0, and all the mass becomes concentrated at its mean n/2.In terms of variance we have, just as for the Bessel functions of 6, that 6 >> 0-2.(8.13) REMARK i.I know of no paper or monograph that discusses infinitely divisible distributions and totally positive functions together, although they seem to be sim- ilar in some ways.Note that exp(-t 2) belongs to both of these function classes.
REMARK 2. The property of being k times positive is very delicate and can be destroyed by a small perturbation of g(t).However, the parameters 6 and o are at most slightly changed by such perturbations, so a law such as (8.13) is "robust." 9. SEVERAL COMPLEX VARIABLES.The nonvanishing of the exponential has an analogue in several complex variables, namely that [ exp z. =0 j--i 3 (9.1)cannot occur if the z.exhibit only "small deviations from the mean."To make 3 this precise, define the distance d(A,P) between any points A (a I a n) P (z I z n) (9.2) in complex n-space by d2(A,P) r..la.l-Z.i 12 (9,3) The diagonal of complex n-space is the set of complex numbers having all components identical.from the diagonal for which (9.1) holds.
The above result again comfirms the "60-hypothesis," though in a rather differ- ent setting.For the proof, and further comments on the relation of this result to various statistical estimates, see [ST].
i0. OSCILLATION THEORY.It is curious that the 6o-hypothesis does not seem to be directly addressed by any previous literature.However, if we think of the para- To illustrate its bearing on the 6o-hypothesis, consider y" qy 0 (10.4) where q q(z) is analytic in a neighborhood of z 0 and real for real z.Let f(z) be the solution with f(0) i, f'(O) 0. If d(w) is a measure on the real axis satisfying (1.4), then a simple calculation (note that f"(0) q(0)) shows that (1.5) is valid with 02 q(O) + 1. (i05) Now a theorem of Hille [HI2] asserts that for q q(z) analytic in zl < R and satisfying lq(z) < M there, we have 6 > min(R,W/2).
(10.6) Thus, in some loose sense, Hille's oscillation theorem gives the same lower bound as the 6 hypothesis for slowly varying q.Both are quantitative versions of the classical principle that the larger the potential the more rapidly the solution oscillates.
ii. REMARKS.'The location of minimum modulus zeros has been extensively investi- gated for polynomials [MA], but the literature for analytic functions does not seem terribly large, despite the importance of the related problem of location of minimal eigenvalues.The literature on zero-free half planes, at least, for exponential polynomials, is perhaps larger owing to its importance for stability theory.See [PON] and also [BE] for many further references.
For the asymptotic distribution of zeros of exponential polynomials and closely related functions see [BE, CART, DII- Also relevant here, as well as to 5, is [GAl.For the local distribution of zeros of exponential polynomials in the complex plane see [POOl, TIJ, VOI], ACKNOWLEDGEMENT.I thank the referee for a number of valuable suggestions, and also for removing many misprints and obscurities.
THEOREM 4.2.If the support of g(t) lies in the interval [a,b], then > I/(b-a).(4.10)PROOF.Modify the proof of Theorem 4.1 by estimating R(z) as follows: equal atoms at a and b has zeros at -+in/(b-a), Theorem 4.2 is within a factor of n of being best possible.For this function 60 follows from (5.6) and (5.7) that z f(z) (N+I)!e (_z) N+I TN(-Z).
are known to be totally positive.Schoenberg[SCH02] has proved the THEOREM.If g(x) is totally positive, but not the exponentl,al of a linear Thus the Laplace transform of such a function is zero free.Moreover, it is known that the totally positive functions are bell-shaped [SCHOI, HIRI-2]!But of greatest interest in connection with the Go-hypothesis isSchoenberg's theorem LE, oscillation theory of ordinary differential equations, starting with the comparison theorems of Sturm (see [HI2, pp.373-384; pp.576-644]), is possibly relevant.The literature in this direction is large; we mention merelyLE, then