A note on the zeroes of the Fredholm series
AA NOTE ON THE ZEROES OF THE FREDHOLM SERIES
UMBERTO ZANNIER
With an appendix by F. Veneziano
Dedicated to the memory of Edoardo Vesentini
Abstract . The issue had been raised whether the so-called
Fredholm series z + z + · · · + z n + · · · has infinitely many zeroes in the unit disk. We provide an affirmative answer, provingthat in fact every complex number occurs as a value infinitely many times.1. Introduction
Let D = { z ∈ C : | z | < } denote the complex unit disk and let(1) f ( z ) = z + z + z + · · · + z n + · · · , z ∈ D. This function is proposed in several books as a simple instance of a holomorphic function on D which cannot be holomorphically extended to any connected open domain of C containing D strictly (see e.g. [13], p. 159, or [14], p. 98). It is often called the Fredholm series , thoughapparently Fredholm had considered a different series; the terminology was introduced by A. J.Kempner in the belief that Fredholm studied it, and then this was followed by several authors:see J. Shallit’s paper [12], especially pp. 161–162, for an accurate description. (Fredholm is alsomentioned in R. Remmert’s book [10], p. 254 but concerning indeed a different series sharingthe said property.) We shall not change here this convention, though perhaps the functionshould be called
Kempner series .It satisfies the functional equation f ( z ) = z + f ( z ) .The function became better known also through the work of K. Mahler [5], who consideredits values at (nonzero) algebraic points in D and proved their transcendency (together withsimilar results for a much wider class of functions). In fact, concerning the function itself, thetranscendency of the values at points /n , n ∈ N + , had been already established by Kempner[4]; these values are interesting and related to iterated paperfolding (see [12], pp. 162–163).All of this gave rise to many-sided generalisations and several works, both of arithmeticaland functional nature, by a number of authors. See also D. Masser’s book [8] for an accountof Mahler’s method in particular, and for some references, and see [6] p. 132 for Mahler’sreminiscences on this topic.In the book [8] the issue is raised (see Ex. 3.15) whether the function has infinitely manyzeroes in D , and it is remarked that Mahler found 16 zeroes (apparently using a pocket cal-culator). Actually already Mahler enquired about the location of the zeroes in the paper [7],commenting that there are probably zeroes “ in every neighborhood of the unit circle ”. It iseasy to locate the (unique) real nontrivial zero − . ... on looking at signs, but otherwisethe location of zeroes seems not obvious to detect, especially due to the fairly complicatedoscillatory nature of f ( z ) near the boundary of D (which is related to exponential sums withwidely growing arguments - see [7] and see Remarks 3.1, 3.2 below). Of course in any disk rD of fixed radius r < there are only finitely many zeroes of f , so Mahler’s expectation literallyamounts to the existence of an infinity of zeroes.Note that several ‘general’ methods for studying the zeroes (and the values) of holomorphicfunctions (e.g. Picard Theorems, Nevanlinna Theory) usually apply to functions with singu-larities, or defined on the whole complex plane, and do not seem to be immediately helpfulhere since f ( z ) is regular on the whole D and has the unit circle as a natural boundary. (See Umberto Zannier, Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, [email protected]. a r X i v : . [ m a t h . C V ] J un UMBERTO ZANNIER however W. Hayman’s book [2], especially Chs. 5 and 6, and also W. Rudin’s book [11], Chs.15, 17, for some results about the distribution of zeroes of functions holomorphic in D . SeeRemark 3.1 below for some comments on this.)It is the main purpose of this note to prove that indeed there are infinitely many zeroes of f ( z ) in D , confirming the expectation of Mahler. Actually we shall prove a sharpening of thisresult, namely that the function attains every complex number as a value infinitely often, andeven arbitrarily near .By the results of Mahler any of the zeroes, apart from , is a transcendental number. Masserasked if for instance one can prove sharper results, e.g. of algebraic independence, or if one canstudy more accurately the distribution of zeroes. We have no answer to the former issue; thelatter maybe admits some kind of answer through the present method, but we haven’t carriedout any analysis in this direction (see the final remarks for a few comments).This note contains also an appendix by Francesco Veneziano, who kindly followed somerequests of mine, on performing several computations concerning this problem; in the Appendixhe lists some approximate zeroes of the function and locates a zero of a related function whichappears in the proof. Also, he represented in Figure 1 below some zeroes of a truncated series,rescaled with respect to the Poincaré distance in D , from the origin.Before stating our results, we introduce some notation. We denote as usual e ( w ) = exp(2 πiw ) and we let H = { w ∈ C : (cid:61) w > } denote the upper-half plane, with which we shall work inplace of D . Indeed, for w ∈ H we set(2) F ( w ) := f ( e ( w )) = e ( w ) + e (2 w ) + e (4 w ) + . . . , w ∈ H . This function is holomorphic in H and satisfies the functional equations(3) F ( w + 1) = F ( w ) , F ( w ) = e ( w ) + F (2 w ) . Consider now the function(4) G ( w ) := ∞ (cid:88) l =1 e (cid:18) − l (cid:19) (cid:16) e (cid:16) w l (cid:17) − (cid:17) , w ∈ H . Note that for w in a given compact set K ⊂ C we have | e ( w/ l ) − | (cid:28) K − l , hence theseries defining G converges uniformly and boundedly on compact sets of C , so in fact G is inparticular holomorphic on the whole C .We further define(5) S ( w ) = F ( w ) + G ( w ) , w ∈ H . We have
Proposition 1.1.
Let V = S ( H ) ⊂ C be the image of S on H . Then, for every open disk U centered at and for every v ∈ V , the Fredholm function f ( z ) assumes on D ∩ U infinitelymany times the value v . The following result also shows that the set V of Proposition 1.1 is stable by certain trans-lations, thus is in a sense ‘large’. Proposition 1.2.
Let V = S ( H ) ⊂ C be as above the image of S on H . Then V + Z = V andmoreover for each m ∈ Z we have V + c m ⊂ V , where c m is as in (12) below. A study of the numbers c m carried out in §3.2 will in fact imply the following sharpeningof the proposition (which essentially then becomes a lemma): Theorem 1.3.
We have V = C , that is, the function S ( w ) attains every complex value. In particular, combining these results we obtain the following consequence, which containsas a corollary the main motivation for this note:
Theorem 1.4.
For every open disk U centered at , the Fredholm series attains on D ∩ U every complex number as a value infinitely many times. NOTE ON THE ZEROES OF THE FREDHOLM SERIES 3
So in particular f ( z ) has infinitely many zeroes in D and even in D ∩ U . It may be that themethods allow to replace U with any disk centered on the unit circle but we have not verifiedthis. Acknowledgements . I thank David Masser for raising several issues and for remarks andreferences; similarly I thank Pietro Corvaja. I thank Francesco Veneziano for some computercalculations, especially helpful when writing the paper, and for providing the Appendix. I fur-ther thank Jeffrey Shallit for pointing out to me a correct historical account of the terminology,and some references. 2.
Some formulae
We now let θ ∈ R be real and n > be a positive integer, and seek an approximationformula for F ( w − n + θ ) . On iterating (3) we have F (cid:16) w n + θ (cid:17) = e (cid:16) w n + θ (cid:17) + e (cid:16) w n − + 2 θ (cid:17) + · · · + e (cid:16) w n − θ (cid:17) + F ( w + 2 n θ )= e (cid:16) w n (cid:17) e ( θ ) + e (cid:16) w n − (cid:17) e (2 θ ) + · · · + e (cid:16) w (cid:17) e (2 n − θ ) + F ( w + 2 n θ ) . (6)We shall use this formula, with suitable choices for the involved quantities, to approximate F with convenient functions.2.0.1. A Ramanujan sum and related ones.
As remarked by Mahler in [7] the behaviour of F ( w ) (for rational (cid:60) w ) is linked with certain Gaussian periods. Here we adopt a related butdifferent viewpoint, constructing vanishing Ramanujan sums .We let q = 3 k , where k ≥ is an integer. Note the Euler function value φ ( q ) = 2 · k − .Since is a primitive root modulo and since (cid:54)≡ , it follows easily (e.g. byinduction on k ) that is a primitive root modulo q , namely the powers m , ≤ m ≤ φ ( q ) − form a complete set of residues modulo q and coprime to q . Since q is not squarefree, it follows(see e.g. [3], §16.6, especially Thm. 271) that, for any integer s coprime with q , the Ramanujansum , i.e. of the primitive q -th roots of unity, vanishes, which amounts to:(7) φ ( q ) − (cid:88) m =0 e (cid:18) m sq (cid:19) = 0 , gcd( s, q ) = 1 , the same equation actually holding when the summation for m runs through a complete set ofresidues modulo φ ( q ) .We now choose an integer a ≥ and k = 2 a , a power of . We put θ = 1 q , n = φ ( q ) = 2 · a − . We also note that q ≡ a +2 ) , as follows easily by induction on a ≥ . Hence for l ≤ a + 2 the ratio (1 − q ) / l is an integer. Also, we have for any integer n ≥ l , n − l θ ≡ n (1 − q )2 l θ (mod Z ) , ≤ l ≤ a. In particular, let n = n , so n ≡ q ) . Then for l ≤ a ≤ n , on writing n =1 + q · integer, this congruence yields(8) n − l θ ≡ (1 − q )2 l q = θ l − l (mod Z ) , ≤ l ≤ a. In what follows this choice is immaterial and there is wide possibility of changing it.
UMBERTO ZANNIER
Preparing an approximate formula.
Setting s = 1 into (7) we obtain n − (cid:88) l =0 e (2 l θ ) = 0 , whence, setting θ = θ , n = n into (6) we derive, taking into account that n θ ≡ θ (mod Z ) and that F is Z -periodic, F (cid:16) w n + θ (cid:17) = n − (cid:88) l =0 e (2 l θ ) (cid:16) e (cid:16) w n − l (cid:17) − (cid:17) + F ( w + θ ) . Further, on writing n − l in place of l in the sum and using (8) for ≤ l ≤ a , this becomes(9) F (cid:16) w n + θ (cid:17) = a (cid:88) l =1 e (cid:18) θ − l (cid:19) (cid:16) e (cid:16) w l (cid:17) − (cid:17) + n (cid:88) l = a +1 e (2 n − l θ ) (cid:16) e (cid:16) w l (cid:17) − (cid:17) + F ( w + θ ) . Proofs of main assertions
Proof of Proposition 1.1.
Let K ⊂ H be a given compact subset of H , to which we shall referby a subscript to indicate dependence of the implicit constants which shall appear. For w ∈ K we have | F ( w + θ ) − F ( w ) | (cid:28) K q , (cid:12)(cid:12)(cid:12) e (cid:16) w l (cid:17) − (cid:12)(cid:12)(cid:12) (cid:28) K l , (cid:12)(cid:12)(cid:12)(cid:12) e (cid:18) θ − l (cid:19) − e (cid:18) − l (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:28) q ( l ≥ , whence, by (9), again for w ∈ K ,(10) (cid:12)(cid:12)(cid:12) F (cid:16) w n + θ (cid:17) − S ( w ) (cid:12)(cid:12)(cid:12) (cid:28) K ∞ (cid:88) l = a +1 (cid:12)(cid:12)(cid:12) e (cid:16) w l (cid:17) − (cid:12)(cid:12)(cid:12) + 1 q (cid:28) K a . The proof is now easily completed by a quite standard argument. Let v ∈ V be any valueof S on H , say v = S ( w ) for some w ∈ H . Let C be a circle inside H and centred at w suchthat S ( w ) (cid:54) = v for w ∈ C , and choose K to be a closed disk inside H , centred at w , and ofradius larger than that of C .The function S ( w ) − v is holomorphic in K and never zero on C , hence its absolute valueattains on C a minimum δ > .Put now, for any integer a ≥ , F a ( w ) = F (2 − n w + θ ) , where n , θ are as above, so by(10), | F a ( w ) − S ( w ) | ≤ c K − a for w ∈ K , where c K is a constant, i.e. independent of a . Nowsuppose that a is so large that c K < δ a . Then on C we have | ( F a ( w ) − v ) − ( S ( w ) − v ) | < δ < min w ∈ C | S ( w ) − v | . By Rouché’s theorem (see e.g. [13]) the functions S ( w ) − v , F a ( w ) − v have the same numberof zeroes inside C , so since the former vanishes at w (which is the centre of C ) the latter hasat least a zero ζ a inside C .This means that F (2 − n ζ a + θ ) = v . Now, the points − n ζ a + θ ∈ H , for varying a , tendto (because ζ a ∈ K ), so the corresponding images through e ( w ) tend to (but are differentfrom as they lie in D ). This proves that, if U ⊂ C is any neighbourhood of , f ( z ) attainsthe value v at infinitely many points of D ∩ U , concluding the argument. (cid:3) About the function S ( w ) . In this subsection we develop some properties of the function S ( w ) , in particular of its image on H , above denoted V .It will be equivalent and notationally convenient to work with the function S ( w ) := S ( w +1) .By definition we have(11) S ( w ) = F ( w + 1) + G ( w + 1) = F ( w ) + ∞ (cid:88) l =1 (cid:18) e (cid:16) w l (cid:17) − e (cid:18) − l (cid:19)(cid:19) . NOTE ON THE ZEROES OF THE FREDHOLM SERIES 5
For integers m ∈ Z , also define constants c m by(12) c m := ∞ (cid:88) l =1 (cid:16) e (cid:16) m l (cid:17) − (cid:17) . Hence, setting H ( w ) := (cid:80) l ≥ (cid:0) e (cid:0) w/ l (cid:1) − (cid:1) , we may rewrite (11) as(13) S ( w ) = F ( w ) + ∞ (cid:88) l =1 (cid:16) e (cid:16) w l (cid:17) − (cid:17) + ∞ (cid:88) l =1 (cid:18) − e (cid:18) − l (cid:19)(cid:19) = F ( w ) + H ( w ) − c − . Note that H is holomorphic in the whole C and c m = H ( m ) .As Masser pointed out to me, the function F ( w ) + H ( w ) appears already in S. Ramanujan’swork (see [1], p. 39); it also appears (as a function on D ) in Exercise 3.11, p. 44 of Masser’sbook [8] and in the previous article of lectures by Masser in [9] (see Lecture 3). It is observedthat it satisfies a functional equation when (in the present notation) w is changed into w .Indeed, we easily find that(14) S (2 w ) = S ( w ) − , This already shows that V + Z = V . Note also that this yields (Masser’s observation) thatthe function S ( w ) + (log w/ log 2) (which is well-defined in H on agreeing for instance on thevalue log i = π/ ) is holomorphic on H and invariant under w → w , as pointed out in theabove quoted sources.Let us now seek other functional equations.Setting w + m s in place of w , for integers m ∈ Z , s ≥ , we obtain S ( w + m s ) = S ( w ) + ∞ (cid:88) l = s +1 e (cid:16) w l (cid:17) (cid:16) e (cid:16) m l − s (cid:17) − (cid:17) = S ( w ) + ∞ (cid:88) l =1 (cid:16) e (cid:16) w l + s (cid:17) − (cid:17) (cid:16) e (cid:16) m l (cid:17) − (cid:17) + c m = S ( w ) + ∆ m,s ( w ) + c m , (15)say. Note that for w in a compact subset K of H , we have(16) ∆ m,s ( w ) := ∞ (cid:88) l =1 (cid:16) e (cid:16) w l + s (cid:17) − (cid:17) (cid:16) e (cid:16) m l (cid:17) − (cid:17) = O K (cid:18) s (cid:19) . Proof of Proposition 1.2.
We have already noted above that, in view of (14), the image of S is invariant under translation by integers, which yields the first claim of the proposition.Let now K be any compact disk inside H and consider formula (15). Note that by (16),for s → ∞ (and fixed m for instance) the functions ∆ m,s converge uniformly and boundedlyto for w ∈ K . Then, exactly by the same argument as in the proof of Proposition 1.1, if v = S ( w ) is a value of S at a point w in the interior of K , then for large s (and fixed m )the function S ( w + m s ) − c m assumes as well that value. Hence S (and therefore S ) assumesthe value v + c m , proving the contention. (cid:3) Before going ahead, we pause to give a proof of the special case of Theorem 1.4 in which C is replaced by R as a set of possible values. This proof is a bit simpler than the general caseand does not require the considerations of §3.2. Proof of Theorem 1.4 for real values.
By Proposition 1.1 it suffices to prove that R ⊂ V (i.e.that S attains all real numbers as values on H ). Also, by Proposition 1.2 applied with m = − ,it suffices to show that R − c − ∈ V ; namely, using (13), it suffices to show that F ( w ) + H ( w ) assumes any real value in H . Let us look at the values for purely imaginary w = it , t ∈ R .These values are real and by the formula S (2 w ) = S ( w ) − this set of real values is invariantby any integer translation. Thus it suffices to prove that it contains an interval of length ; onthe other hand this is clear by the same formula S (2 w ) = S ( w ) − , since the imaginary lineis connected. (cid:3) UMBERTO ZANNIER
About the constants c m . Note that c m = c m and that c m = c − m . As remarked above, c m is just the value of H ( z ) at z = m . These numbers seem interesting on their own and wewonder whether they have irrationality of transcendency properties. We cannot prove any neatassertion of this type, but some small information in this direction will appear below.Proposition 1.2 says that V is stable under translation by the semigroup generated by Z and the c m . Let σ m := c m + c − m noting it is real, and in particular we find that V is stableunder translation by the real semigroup generated by Z and the σ m . We shall study a bit thissemigroup, denoted here Γ .Now, it turns out that a related sequence has better approximation properties, useful forour purposes.Define k l := e (1 / l ) − . We have k = 0 , k = − , k = i − , k = ( √ − i √ / and k l = k l +1 ( k l +1 + 2) . Setting µ l := | k l | we also have µ l = 2 − π/ l − ) and µ l +1 =2 − √ − µ l .Further, e ( m/ l ) = ( k l + 1) m and c m = (cid:80) ∞ l =1 (( k l + 1) m − . For m ∈ Z we now set(17) c ( m ) = ∞ (cid:88) l =1 k ml . Note that the k l tend to exponentially so this formula yields good approximations of the c ( m ) (by linear recurrence sequences) on taking truncations of the series.These numbers are linear combinations of the c m with integer coefficients and conversely.In fact c ( m ) = (cid:80) ∞ l =1 ( e ( m/ l ) − (cid:0) m (cid:1) e (( m − / l ) + ... ) = c m − (cid:0) m (cid:1) c m − + . . . + ( − m − c .Similarly, c m = (cid:80) ∞ l =1 ( k ml + (cid:0) m (cid:1) k m − l + . . . + (cid:0) m (cid:1) k l ) = c ( m ) + (cid:0) m (cid:1) c ( m −
1) + . . . + (cid:0) m (cid:1) c (1) .For the sake of curiosity, we also note the recurrence c ( m ) = (cid:80) ∞ l =0 k ml +1 ( k l +1 + 2) m = c (2 m ) + 2 (cid:0) m (cid:1) c (2 m −
1) + 4 (cid:0) m (cid:1) c (2 m −
2) + . . . + 2 m − (cid:0) m (cid:1) c ( m + 1) + 2 m c ( m ) , whence (1 − m ) c ( m ) = m (cid:88) h =1 m − h (cid:18) mh (cid:19) c ( m + h ) , so the c ( m ) satisfy several linear relations with rational coefficients. These correspond in fact tothe more evident relations c m = c m . We wonder whether there are other such linear relationsholding among them.Now, for our purposes it is more useful to consider the numbers σ ( m ) := ∞ (cid:88) l =1 ( k l + ¯ k l ) m = ∞ (cid:88) l =1 (cid:18) e (cid:18) l (cid:19) + e (cid:18) − l (cid:19) − (cid:19) m . Observe that, for h (cid:54) = m , k hl ¯ k m − hl + k m − hl ¯ k hl equals an integer linear combination of terms e ( s/ l ) + e ( − s/ l ) − σ s : this is seen on expanding k hl = ( e (1 / l ) − h with the binomialtheorem (and similarly for the complex conjugate). The same holds for a term k hl ¯ k hl in case h = m . On summing over l we deduce that σ ( m ) is an integer linear combination of σ , . . . , σ m .(One can also write down an explicit formula in terms of Chebyshev polynomials.)Now, ( − m σ ( m ) is of the shape m + 2 m + O ((2 − √ m ) . We conclude that for growing m ,either one among σ , . . . , σ m is irrational, or the least common denominator of these numbersgrows to infinity. In fact, if all σ j would be rationals with a bounded denominator, the saidrelation would imply the same for σ ( m ) , whence, since < − √ < , for large m we wouldhave σ ( m ) = ( k + ¯ k ) m + ( k + ¯ k ) m , which is plainly false.We in turn deduce that the semigroup Γ generated by Z and the σ m is dense in R . In fact,if some σ m is irrational this follows from well-known (easy) theorems, and otherwise it followsfrom the deduction just obtained.3.3. Concluding arguments.
We can now use the results just obtained to complete the proofof Theorems 1.3 and 1.4.
Proof of Theorem 1.3 and Theorem 1.4.
In view of Proposition 1.1, it suffices to prove Theo-rem 1.3. Let z ∈ V and let A be an open disk with centre z , entirely contained in V (whichexists since S ( w ) is holomorphic). Recall that V is stable under translation by the semigroup Γ ⊂ R , so in particular A + Γ ⊂ V . Since Γ is dense in R we deduce that V contains a whole NOTE ON THE ZEROES OF THE FREDHOLM SERIES 7 horizontal strip around the line (cid:61) w = (cid:61) z . In particular, V contains the line (cid:61) w = (cid:61) z itself.Hence V contains any line (cid:61) w = t as soon as it contains a point in such a line. But since V is invariant under translation by c ± (cid:54) = 0 , thus by any multiple sc ± (any integer s > , anysign), the imaginary parts of elements of V are unbounded both from above and from below.But the set of such imaginary parts is connected, hence is the whole R . Combining this withthe above deduction, we conclude that V = C , as required. (cid:3) Remark 3.1.
Concerning the distribution of zeroes of f ( z ) , one could also think for instance ofusing criteria such as Theorem 15.23 of [11], which would predict a convergent sum (cid:80) (1 − | α i | ) for the zeros α i , provided (cid:82) log + | f ( re ( θ )) | d θ remained bounded as r → − .However one may show that this last condition does not hold for the present function, thatis, f does not belong to the class denoted N (Nevanlinna) in [11]. We give a very brief sketchfor this claim. One first may expand f ( re ( θ )) for r = e − c/ n ( / < c ≤ ), using the first n terms and bounding the remainder by a constant (see e.g. (6) above). Approximating in turn r m = 1 + O ( m/ n ) , m = 1 , , . . . , n − , for each of these n terms, one is reduced to estimatethe exponential sums s ( θ ) = s n ( θ ) = e ( θ ) + e (2 θ ) + . . . + e (2 n − θ ) in absolute value. (See also[7] for such approximate formulae.)In conclusion, if the said integral would be bounded then it is easy to see that, in particular,the measure of the set A = A n = { θ ∈ [0 ,
1] : | s ( θ ) | > √ n/ } would tend to , hence would be < (cid:15) , for any prescribed (cid:15) > and large enough n . Now, however, we easily find the moments (cid:82) | s ( θ ) | d θ = n , (cid:82) | s ( θ ) | d θ = 2 n − n . But on the one hand (cid:82) A | s | ≥ n − ( √ n/ = 3 n/ ,on the other hand ( (cid:82) A | s | ) ≤ (cid:15) (cid:82) A | s | ≤ (cid:15)n . This is false for (cid:15) < / , proving the claim.We do not know if nevertheless the series (cid:80) (1 − | α i | ) converges (though we are inclined tobelieve it does not). Remark 3.2.
Quite possibly there are other methods to prove the results of the paper (andmaybe more). For instance, one could apply Ahlfors’ theory (see especially [2], Ch. 5). Also,on approximating f ( z ) as indicated in the previous remark (or see [7]), one should obtain thatfor r = e − c/ n , ( / < c ≤ ), f ( re ( θ )) is approximated by s n ( θ ) up to a bounded function.In turn, we have s n ( θ ) = s n ( θ ) + s n (2 n θ ) , and on changing θ to θ a := θ + ( a/ n ) for anarbitrary integer a , we get s n ( θ a ) = s n ( θ a ) + s n (2 n θ ) . On varying a , the first term essentiallyvaries along the whole set of values of s n , whereas the second term remains constant. Soone nearly reduces to study s n ( u ) + s n ( v ) for independent variables u, v , and one can alsoiterate (and compare with a compatible probabilistic distribution). This possibly leads to abetter understanding of the image of f ( re ( θ )) for fixed r approaching , and in turn to anunderstanding of the distribution of the values attained by the function. Appendix by Francesco Veneziano
Finding a zero of S ( w ) . Here we will prove computationally that the holomorphic function S ( w ) defined in (5) and expressed by the series S ( w ) = ∞ (cid:88) n =0 (cid:18) e ( w n ) + e (cid:18) w − n +1 (cid:19) − e (cid:18) − n +1 (cid:19)(cid:19) has a zero in the upper-half plane. This fact, together with Proposition 1.1, provides analternative proof of the main assertion that the Fredholm series has infinitely many zeroes,independent from the study of the quantities c m carried out in Section 3.2.Thanks to uniform convergence on compacts, we can write S ( k ) ( w ) = ∞ (cid:88) n =0 (cid:32) (2 n +1 πi ) k e ( w n ) + (cid:18) πi n (cid:19) k e (cid:18) w − n +1 (cid:19)(cid:33) . Writing t = (cid:61) w we have(18) (cid:12)(cid:12)(cid:12) S ( k ) ( w ) (cid:12)(cid:12)(cid:12) ≤ (2 π ) k ∞ (cid:88) n =0 kn e − n +1 πt + π k ∞ (cid:88) n =0 − kn e − πt n . UMBERTO ZANNIER
Set now a = log k πt , and let n be an integer, n ≥ (cid:100) a (cid:101) + 1 . By standard calculus we find(19) kn e − n +1 πt = e kn log 2 − n +1 πt ≤ e − k e − (log 2)22 kn . Splitting the first sum of (18) at the summand (cid:100) a (cid:101) we get (cid:12)(cid:12)(cid:12) S ( k ) ( w ) (cid:12)(cid:12)(cid:12) ≤ (2 π ) k (cid:100) a (cid:101) (cid:88) n =0 kn e − n +1 πt + (2 π ) k ∞ (cid:88) n = (cid:100) a (cid:101) +1 kn e − n +1 πt + π k ∞ (cid:88) n =0 − kn e − πt n . Estimating the first and third terms with geometrical sums, putting these terms together, andusing (19) to estimate the middle term we get, for k ≥ , (cid:12)(cid:12)(cid:12) S ( k ) ( w ) (cid:12)(cid:12)(cid:12) ≤ (2 π ) k k − k ( (cid:100) a (cid:101) +1) + 3 k +1 ≤
43 ( π a +2 ) k + 3 k +1 . From the definition of a and the elementary inequality k ! ≤ k k e k − we eventually obtain(20) (cid:12)(cid:12)(cid:12) S ( k ) ( w ) (cid:12)(cid:12)(cid:12) ≤ (cid:18) et (cid:19) k k ! + 3 k +1 . Consider now a point w on the upper-half plane, with imaginary part t . For a point w atdistance ρ from w using (20) we have that | S ( w ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ (cid:88) k =0 S ( k ) ( w ) k ! ( w − w ) k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ | S (cid:48) ( w ) | ρ − | S ( w ) | − ∞ (cid:88) k =2 (cid:12)(cid:12) S ( k ) ( w ) (cid:12)(cid:12) k ! ρ k ≥ | S (cid:48) ( w ) | ρ − | S ( w ) | − ∞ (cid:88) k =2 (cid:18) eρt (cid:19) k − ∞ (cid:88) k =2 (3 ρ ) k k != | S (cid:48) ( w ) | ρ − | S ( w ) | − e ρ t ( t − eρ ) − e ρ + 9 ρ + 3 . Now we can finally set w = − . . i and ρ = 0 . and check numericallythat | S ( w ) | > | S ( w ) | on the circle of centre w and radius ρ , thus proving that within thatdisk lies a zero of S ( w ) . Some zeroes of the Fredholm series.
As shown, the Fredholm series has infinitely manyzeroes. We list here the approximate values of a few of them, other than the zero at z = 0 andthe only other real zero at z = − . . . . − . ± . i, − . ± . i, − . ± . i, − . ± . i, − . ± . i, − . ± . i, . ± . i, . ± . i, . ± . i, . ± . i, . ± . i, . ± . i. Of course the zeroes tend to accumulate towards the unit circle. Figure 1 shows the zeroesof the 13th partial sum which lie in the interior of the unit circle (there are 1126 of them); inorder to visualize them better, we have rescaled their distances from the origin according tothe hyperbolic metric in the Poincaré disk model. In other words, each zero ρe iθ in the unitcircle has been mapped in the picture to the point log (cid:16) ρ − ρ (cid:17) e iθ .Francesco VenezianoUniversità di GenovaVia Dodecaneso 35, 16146 Genova, ITALYe-mail: [email protected] NOTE ON THE ZEROES OF THE FREDHOLM SERIES 9 - - - - - Figure 1.References [1] - G. H. Hardy, Ramanujan 1940, Cambridge Univ. Press.[2] - W. Hayman, Meromorphic Functions, Oxford Uni. Press, 1964.[3] - G. H. Hardy, E. M. Wright, An Introduction to the Theory of Numbers, IV Ed., 1975, OxfordUniv. Press.[4] - A. J. Kempner, On transcendental Numbers, Trans. Amer. Math. Soc. (1916), 476–482.[5] - K. Mahler, Arithmetische Eigenschaften der Losungen einer Klasse von Funktionalgleichungen,Math. Ann. 101 (1929), 342–366.[6] - K. Mahler, Fifty Years as a Mathematician, Journal of Number Theory (1982), 121–155.[7] - K. Mahler, On a Special Function, Journal of Number Theory (1980), 20–26.[8] - D. Masser, Auxiliary Polynomials in Number Theory, 2016, Cambridge University Press,[9] - D. Masser, Transcendence and Linear Independence on Commutative Group Varieties, in Dio-phantine Approximation, F. Amoroso and U. Zannier Eds., Springer LNM, 1819, 2000.[10] - R. Remmert, Classical Topics in Complex Function Theory, 1991, Springer Verlag GTM.[11] - W. Rudin, Analisi Reale e Complessa, Translation from the English version ( Real and ComplexAnalysis
McGrew Hill) by Maria Luisa and Edoardo Vesentini, 1974, Boringhieri.[12] - J. Shallit, Real Numbers with Bounded Partial Quotients, a Survey, L’Ens. Math.38