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MATHEMATICS INAUGURAL ARTICLE d,n Theorem 1. If d ≥ 1, then Jγ (X ) is hyperbolic for all sufficiently large n. An effective proof of Theorem 1 for small d gives the following theorem.

d,n Theorem 2. If 1 ≤ d ≤ 8, then Jγ (X ) is hyperbolic for every n ≥ 0. Theorem 1 follows from a general phenomenon that Jensen polynomials for a wide class of sequences α can be modeled by the Hd (X ), which we define (in a somewhat nonstandard normalization) as the orthogonal polynomials for the 2 measure µ(X ) = e−X /4 or more explicitly by the

∞ d 2 3 X t −t2+Xt 2 t 3 t H (X ) = e = 1 + X t + (X − 2) + (X − 6X ) + ··· . [3] d d! 2! 3! d=0

More precisely, we will prove the following general theorem describing the limiting behavior of Jensen polynomials of sequences with appropriate growth.

Theorem 3. Let {α(n)}, {A(n)}, and {δ(n)} be three sequences of positive real numbers with δ(n) tending to zero and satisfying

 α(n + j )    log = A(n)j − δ(n)2j 2 + o δ(n)d as n → ∞, [4] α(n)

for some d ≥ 1 and all 0 ≤ j ≤ d. Then, we have

 −d   δ(n) d,n δ(n) X − 1 lim Jα = Hd (X ), [5] n→∞ α(n) exp(A(n))

uniformly for X in any compact of R. Since the Hermite polynomials have distinct roots, and since this property of a polynomial with real coefficients is invariant under small deformation, we immediately deduce the following corollary.

d,n Corollary 4. The Jensen polynomials Jα (X ) for a sequence α: N → R satisfying the conditions in Theorem 3 are hyperbolic for all but finitely many values n. Theorem 1 is a special case of this corollary. Namely, we shall use Theorem 9 to prove that the Taylor coefficients {γ(n)} satisfy the required growth conditions in Theorem 3 for every d ≥ 2. Theorem 3 in the case of the is the derivative aspect Gaussian unitary ensemble (GUE) model prediction for the zeros of Jensen polynomials. To make this precise, recall that Dyson (8), Montgomery (9), and Odlyzko (10) conjecture that the nontrivial zeros of the Riemann zeta function are distributed like the eigenvalues of random Hermitian matrices. These eigenvalues satisfy Wigner’s Semicircular Law, as do the roots of the Hermite polynomials Hd (X ), when suitably d,0 1  normalized, as d → +∞ (see chapter 3 of ref. 11). The roots of Jγ (X ), as d → +∞, approximate the zeros of Λ 2 + z (see ref. 1 or lemma 2.2 of ref. 12), and so GUE predicts that these roots also obey the Semicircular Law. Since the derivatives of 1  d,n Λ 2 + z are also predicted to satisfy GUE, it is natural to consider the limiting behavior of Jγ (X ) as n → +∞. The work here proves that these derivative aspect limits are the Hermite polynomials Hd (X ), which, as mentioned above, satisfy GUE in degree aspect. Returning to the general case of sequences with suitable growth conditions, Theorem 3 has applications in where the hyperbolicity of polynomials determines the log-concavity of enumerative statistics. For example, see the classic theorem by Heilmann and Leib (13), along with works by Chudnovsky and Seymour (14), Haglund (15), Haglund et al. (16), Stanley (17), and Wagner (18), to name a few. Theorem 3 represents a criterion for establishing the hyperbolicity of polynomials in enumerative combinatorics. The theorem reduces the problem to determining whether suitable asymptotics hold. Here, we were motivated by a conjecture of Chen, d,n Jia, and Wang concerning the Jensen polynomials Jp (X ), where p(n) is the partition function. Nicolas (19) and Desalvo and Pak 2,n 3,n (20) proved that Jp (X ) is hyperbolic for n ≥ 25, and, more recently, Chen et al. proved (21) that Jp (X ) is hyperbolic for n ≥ 94, inspiring them to state as a conjecture the following result.

d,n Theorem 5 (Chen–Jia–Wang Conjecture). For every integer d ≥ 1, there exists an integer N (d) such that Jp (X ) is hyperbolic for n ≥ N (d). Table 1 gives the conjectured minimal value for N (d) for d = 2j with 1 ≤ j ≤ 5. More precisely, for each d ≤ 32 it gives the smallest d,n integer such that Jp (X ) is hyperbolic for N(d) ≤ n ≤ 50,000.

Remark 6: Larson and Wagner (22) have made the proof of Theorem 5 effective by a brute-force implementation of Hermite’s criterion (see theorem C of ref. 3). They showed that the values in the table are correct for d = 4 and d = 5 and that 2 N (d) ≤ (3d)24d (50d)3d in general. The true values are presumably much smaller and are probably of only polynomial growth, the numbers N (d) in the table being approximately of size 10 d 2 log d.

Table 1. Conjectured minimal values of N(d) d 1 2 4 8 16 32 N(d) 1 25 206 1,269 6,917 35,627

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MATHEMATICS INAUGURAL ARTICLE 2, n 3, n Table 3. The polynomials bJp and bJp

2, n 3, n n bJp (X) bJp (X) 100 ≈ 0.9993X2 + 0.0731X − 1.9568 ≈ 0.9981X3 + 0.2072X2 − 5.9270X + 1.1420 200 ≈ 0.9997X2 + 0.0459X − 1.9902 ≈ 0.9993X3 + 0.1284X2 − 5.9262X − 1.4818 300 ≈ 0.9998X2 + 0.0346X − 1.9935 ≈ 0.9996X3 + 0.0965X2 − 5.9497X − 1.3790 400 ≈ 0.9999X2 + 0.0282X − 1.9951 ≈ 0.9998X3 + 0.0786X2 − 5.9621X − 1.2747 ...... 108 ≈ 0.9999X2 + 0.0000X − 1.9999 ≈ 0.9999X3 + 0.0000X2 − 5.9999X − 0.0529

because the inner sum, which is the (d − k)th difference of the polynomial j 7→ j i evaluated at j = 0, vanishes for i < d − k and equals A(n) −t2 (d − k)! for i = d − k. Theorem 8 follows, and Theorem 3 is just the special case E(n) = e and F (t) = e .  3. Proof of Theorem 7 Assume that f is a of (possibly fractional) weight k on SL2(Z) (possibly with multiplier system) and with a pole of (possibly fractional) order m > 0 at infinity and write its Fourier expansion at infinity as in [7]. It is standard, either by the circle method of Hardy–Ramanujan–Rademacher or by using Poincar´e series (for example, see ref. 23), that the Fourier coefficients of f have the asymptotic form √ k−1 √  C 2π mn  af (n) = Af n 2 Ik−1(4π mn) + O n e , [9]

as n → ∞ for some nonzero constants Af [an explicit multiple of af (−m)] and C , where Iκ(x) denotes the usual I -. In view of the expansion of Bessel functions at infinity, this implies that af (n) has an to all orders in 1/n of the form √ 4π mn 2k−3  c1 c2  a (n) ∼ e n 4 exp c + + + ··· , f 0 n n2

for some constants c0, c1, ... depending on f [and in fact only on m and k if we normalize the leading coefficient af (−m) of f to be equal to 1]. This gives an asymptotic expansion

  ∞ ! i ∞ i−1 i ! i af (n + j ) √ X 1/2 j 2k − 3 X (−1) j X −k j log ∼ 4π m 1 + i + ck i+k , [10] af (n) i i− 2 4 i n i n i=1 n i=1 i,k≥1

p valid to all orders in n, and it follows that the sequence {af (n)} satisfies the hypotheses of Theorem 3 with A(n) = 2π m/n + O(1/n) and δ(n) = (π/2)1/2m1/4n−3/4 + O(n−5/4). Theorem 7 then follows from the corollary to Theorem 3.

(n) 1 4. Asymptotics for Λ ( 2 ) ¶ (n) 1  Previous work of Coffey (6) and Pustyl’nikov (7) offer asymptotics for the derivatives Λ 2 . Here, we follow a slightly different approach and obtain effective asymptotics, a result which is of independent interest. To describe our asymptotic expansion, we first give a formula for these derivatives in terms of an auxiliary function, whose asymptotic expansion we shall then determine. Following Riemann (cf. chapter 8 of ref. 25), we have

Z ∞ Z ∞ s −1 1  s 1−s  dt Λ(s) = t 2 θ0(t) dt = + t 2 + t 2 θ0(t) , 0 s(s − 1) 1 t

P∞ −πk2t 1 −1/2 −1/2 where θ0(t) = k=1 e = 2 (t − 1) + t θ0(1/t). It follows that

F (n) Λ(n) 1 = − 2n+2 n! + , [11] 2 2n−1

for n > 0 (both are of course zero for n odd), where F (n) is defined for any real n ≥ 0 by

Z ∞ n −3/4 F (n) = (log t) t θ0(t) dt. [12] 1

In particular, if n is a positive integer, then the Taylor coefficients γ(n) defined in [1] satisfy

! ! 2n n! 2n n! 32 F (2n − 2) − F (2n) γ(n) = · 8 Λ(2n−2) 1 − Λ(2n) 1  = · 2 . [13] (2n)! 2 2 2 (2n)! 22n−1

¶It is interesting to note that Hadamard obtained rough estimates for these derivatives in 1893. His formulas are correctly reprinted on p. 125 of ref. 24.

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− − − − − [12] γ 3, n L t A  2 1+ (1 n +1 n 2n −1 − 2.0000 2.0016 2.0029 2.0061 2.0199 n 2 4 e n = − L/4−n − = π λ)a n the and ) ! sgvnt l resin orders all to given is 2  t λ) − F − 4 3 b 20 − π 3 hoe 9 Theorem L −3/4 )) (2n  a f  2 4 3 , f /L+34 (t − + 4 ε so , ( + ((1 e 4 ε (log = ) − L/4−n + e f log( + 4 3 a −πλ A )= 2)  f 11ε 91ε i (t 180 24 log λ (i  (a )a ) /L+34 imply [13] and 2 n + 1 2 ≈ ≈ ≈ ≈ ≈ = γ ≥ λ sue t nqemxmmat maximum unique its assumes a t )) )/f + (n = + ) 0.9931X 0.9911X 0.9872X 0.9769X 0.9999X f . e r oyoil fdegree of polynomials are 3) r l ierepesosin expressions linear all are n (t −C in ε L(π b ) t n 2 17ε (a 3 1 −3/4 )  24  h iertr aihsb h hieof choice the by vanishes term linear The λ. λ + around + hr h oiattr feach of term dominant the where ), + 1 n e 2 3 + 1 /2 L yteaypoi expansion asymptotic the by ε n 3 3 3 3 3 b + e 4 4 2 + + + + + + 2 O −π  17ε b + 1 n 0.4136X 0.0009X 0.4705X 0.5625X 0.7570X + 4 3 24  1 t n t )  · · · fo o n econsider we on, now (from n A 4 = + n ahcoefficient each and A b J : = 2−ε i γ 3, 1 + 3  a 16 are n 3 λ . . . F (X ε 2 2 2 2 2 b sgvnby given is 3 2  , 5 − − − − − (n L + ) + 5.9501X 5.9999X 5.9374X 5.9153X 5.8690X . sn t enn qainabove. equation defining its using ), A (n ε 4 6 6 λ ∞), →  4 + n n − − − − − bi + · · · t 0.6623 0.0014 0.7580 0.9159 1.2661 /3c ihcefiinsin coefficients with = NSLts Articles Latest PNAS 32 5  b a , k in . where , eog to belongs n ihcoefficients with A n i a sfie and fixed as sgoverned is = Q(L), e L | a sthe is Q[ε]. the , f8 of 5 [14] the

MATHEMATICS INAUGURAL ARTICLE Plugging in t = (1 + λ)a immediately gives the asymptotic expansion

Z ∞ Z ∞ −C λ2/2 3 4  f (t) dt = a f (a) e 1 + A3λ + A4λ + ··· dλ 1 −1+1/a r 2π  3A 15A (2i − 1)!!A  = a f (a) 1 + 4 + 6 + ··· + 2i + ··· . C C 2 C 3 C i

(Here, only the part of the with C λ2 < B log n, where B is any function of n going to infinity as n does, contributes.) This equality and the expression in Theorem 9 are interpreted as asymptotic expansions. Although these series themselves may not converge for a fixed n, we may truncate the resulting approximation at O(n−A) for some A > 0, and as n → +∞, this approximation becomes true to the specified precision. Substituting into this expansion the formulas for C and Ai in terms of n, we obtain the statement of the theorem with F (n) replaced by the integral over f (t), with only A2i (i ≤ 3k) contributing to bk . But then the −3πt −K same asymptotic formula holds also for F (n), since the ratio f (t)/θ0(t) = 1 + e + ··· is equal to 1 + O(n ) for any K > 0 for t near a.  5. Proof of Theorems 1 and 2 A. Proof of Theorem 1. For each d ≥ 1, we use Theorem 3 with sequences {A(n)} and {δ(n)} for which

  d γ(n + j ) 2 2 X i  d  log = A(n)j − j δ(n) + g (n)j + o δ(n) , [15] γ(n) i i=3

i  for all 0 ≤ j ≤ d, where gi (n) = o δ(n) . Stirling’s formula, [13], and [14] give

n−2 n+ 1 1 n r e n 2 (1 + )Lb 2π  L n 3  b (n)   1  γ(n) = 12n · exp − b + 1 + 1 b 1 + O , [16] n+ 1 1 2−ε 2nb−3n b 2 (1 + ) K 4 L 4 nb n b 12nb

−1 −2 where nb : = 2n − 2, L : = L(nb), and K : = K (nb) : = L(nb) + L(nb) nb − 3/4. The L(nb) are values of a nonvanishing for <(n) > 1, and so for |j | < n − 1, we have the Taylor expansion

L(n + 2j ) X j m L(j ; n) : = b = 1 + ` (n) . L(n) m m! b m≥1

n If J = λ(n − 1) with −1 < λ < 1, then the asymptotic L(n) ≈ log( log n ) implies

L (n(λ + 1)) lim L(J , n) = lim b = 1. n→+∞ n→+∞ L(nb)   2 −8(nb−3/4L)(1+L/2) 1 In particular, we have `1(n) = K ·L2 and `2(n) = K 3·L5 and `m (n) = o (n−1)m . By a similar argument applied to

b (n+2j ) K (n + 2j ) j m 1 + 1 b j m b X nb+2j X K(j ; n) : = = 1 + km (n) and B(j ; n) : = b (n) = 1 + βm (n) , K (nb) m! 1 + 1 b m! m≥1 nb m≥1

    1 2(L+1) 2nb(L+2) 1 we find that βm (n) = o (n−1)m+1 , k1(n) = K ·L2 − K 2L4 , and km (n) = o (n−1)m for m ≥ 2.

6, n Table 5. The polynomials bJγ

6, n n bJγ (X) 100 ≈ 0.912X6 + 3.086X5 − 24.114X4 − 55.652X3 + 133.109X2 + 151.696X − 85.419 200 ≈ 0.950X6 + 2.374X5 − 26.625X4 − 42.824X3 + 153.246X2 + 115.849X − 100.510 300 ≈ 0.965X6 + 2.011X5 − 27.608X4 − 36.282X3 + 161.084X2 + 97.843X − 106.295 400 ≈ 0.973X6 + 1.780X5 − 28.139X4 − 32.111X3 + 165.303X2 + 86.428X − 109.388 ...... 1010 ≈ 0.999X6 + 0.000X5 − 29.999X4 − 0.008X3 + 179.999X2 + 0.020X − 119.999

6 of 8 | www.pnas.org/cgi/doi/10.1073/pnas.1902572116 Griffin et al. Downloaded by guest on September 25, 2021 Downloaded by guest on September 25, 2021 xml 11: Example where the let we convenience, For Examples 6. that out turns It satisfy coefficients degree even the while satisfy M M n for formulas the Using Therefore, rfne al. et Griffin L Steffen Larson, Hannah ACKNOWLEDGMENTS. Jensen partition 3 and 2 degree X the that indeed 3 Table in observe we data, these With by modeled result. are the polynomials illustrate better and [ quickly more of approximations one-term the shown be can it [16], of form effective an Using 1 which for then 2. Theorem of Proof the of Sketch B. 0 the for bounds The choose will we that idea oko aaadctda otoei eto .MG n ..wr upre yNFGat M-529 n M-610.KO a upre by supported was K.O. DMS-1601306. and DMS-1502390 Grants NSF by supported were K.O. and M.G. 4. Section in Fund. footnote Candler a Griggs as Asa cited Hadamard of work that k ttrsotthat out turns It ≤ δ < ≤ ε ε 6 sn hsciein eas hs vectors chose also we criterion, this Using ial,w ocuei al ihdt o h ere6rnraie esnpolynomials 3. Jensen and renormalized 2 6 degrees degree of the case the for in data (18) with using 5 function Table zeta in Riemann conclude the we for Finally, 1 Theorem illustrates 4 Table Let d d − j 10 < o hc h eurdieulte odfor hold inequalities required the which for (n hoe 1 Theorem ≤ 30X R 6 m 10 ) en elnumbers real define 8, (j and → 1 = 6 J b ; . 4 n g γ d 0. 180X + m 4 eilsrt h aeof case the illustrate We ,n /24 ) δ Therefore, (n ≤ 6 sntreal. not is (6) eteapoiainfor approximation the be (X mle that implies M d = ) and ) ≤ ε shproi if hyperbolic is 4 g 2 etakteMxPac nttt o ahmtc nBn o t upr n optlt;WlimY .Ce,Rc Kreminski, Rick Chen, C. Y. William hospitality; and support its for Bonn in Mathematics for Institute Planck Max the thank We O 1 8 m 104 = : k bih ee ank n a anrfrdsusosrltdt hswr;adJcusG Jacques and work; this to related discussions for Wagner Ian and Sarnak, Peter obrich, − ¨ − sn emt’ rtro setermCo e.3). ref. of C theorem (see criterion Hermite’s using (n ` = 120. (n 1 16.05 A(n ) (n hoe 3 Theorem ,w a choose may we [10], Using −1/2. − n h smttc bv ml the imply above asymptotics the and ), g g J b δ < H lim 1) ) 1 2 α (n d (n (n δ ∼ ` 2 C 10 ,n 1−m 2 (n (X = : ) n (n (n log = ) = ) g (X 6 →+∞ 1 ) ie tteedo eto losatisfy also 3 Section of end the at given 10] . = ) 0 Let (n 2 ), ,  ) − n leri aiuain give manipulations algebraic and , j − ≤ β < ple,adiscrlaygives corollary its and applies, r and ) J eoeteplnmaswihcneg to converge which polynomials the denote ) b 1+ (1 d X n b γ s 1 by A(n d and J b n b 1 k  γ ,n 4 4 = 4,n 0 2 γ (4` + (β d (n k − (X β < nL ,n − 4 1 o( + log(1 λ) k ) n d b δ < ε (n (X 2 + L ,n using (n and 2 2 d = : ) 3 4,n 1, 2 and 1  ) ( = : j − k 2 = ) ) · (n )/γ bv,w define we above, + γ K ∼ that h < δ −12 n γ (n + ) δ k (n n H b ε H (n p ε (n (n ≥ 28 ) γ 4 ` d )e 3 d ε < (n ) 1 0 (0 = : (d −g (X n M b + (X ) ) and lim = λ) β < (n A( δ .I elet we If [18]. in as be < −d ` ) .W hnexpand then We [16]. from obtained (n ), ε j 2 d 2 n = ) ) d : = ) C ) (n (n (k ) ε 2 L 4,n h ro olw rmtefc htw on utbecocsfrwhich for choices suitable found we that fact the from follows proof The . and ) · j . d n (n J 1,7. 0.813, 1.384, 041, )) ) 1 + L oti n,if end, this To ). · (d A(n X →+∞ γ A(n d < e o all for , L P ,n 3 δ j − 2L ( 2+16 + −12 ) 1 + − − n o(1)  = ) k d < log = : ) 1), ) =0 2 6X log δ L 2 − exp(A(n j hoe 1. Theorem 14.25 (n 2 k − √ . . . h + ,adas htfrsfcetylarge sufficiently for that also and [15], in term error 145.70δ omk s fteeieulte,frpstv integers positive for inequalities, these of use make To . + 1 = . k R 24n ) n b 2π X X ` (J , ` 1 ε .20 1 −1 k o all for (n  d (n ; ehv ofimdtehproiiyo the of hyperbolicity the confirmed have We . − 4 (0)) n )) nL C 4 ) − ) δ )−J (n J 1 n b · 2 n (n (n n L  2 2 L 24n 3/2 = 1,0.804) 313, ) fpstv ubr n signs and numbers positive of n ,12 ),  2L − = H , − 24 2 + h w-emapoiain converge approximations two-term the 3, Theorem β < λ(n j log ≥ −1 + d 1 X ) k k (X =0 d . 7 1 1 L 4,n + L  and (n 2 − and 2 − β ) − nL 4 O · < k n 1) ) b d .W o illustrate now We [5]. in 16.01 K 2 2 ,n 1 +  δ 0, 1  for X (n + ≤ For n − O log k 2−ε = ) 1 j n J b , −1  δ L J ≤ (L (n R . n 4 n γ 6,n q  lnsfrbign hi teto oold to attention their bringing for elinas (j ´ < λ < ial,w eemndnumbers determined we Finally, 8. · 2−ε ≥ ) 1 2) + K . β < (X ; (24n 100 n 2  =: )= ) 0 12π . 4,n , −1) hnacluainreveals calculation a then 1, hc ovreto converge which h d erecoefficients degree odd the < P 3/2 NSLts Articles Latest PNAS 12. s m d − ≥1 , s hoe 7 Theorem (24n d g −1 m 288 (n −1) , . . . )j J 2 b , m γ . d n s ,n Although ihthe with , H 0 ih[6], with ehave we (X {± ∈ 6 | (X n ) f8 of 7 [17] [18] [19] and = ) for 1}

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