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Polynomial Versions of Integer Partitions andTheir Zeros
Robert Boyer
Rutgers University – Experimental Math Seminar
February 9, 2012
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Joint Work
William M. Y. Goh
Daniel T. Parry
1 Partition Polynomials: Asymptotics and Zeros,Contemporary Mathematics, Volume 457 (2008), 99-112(with Bill Goh)
2 On the Zeros of Plane Partition Polynomials, (with DanielParry), Electronic Journal of Combinatorics, Volume 18 (2)(2012), # P30 (with Daniel Parry) 26 pages
3 Phase Calculations for Planar Partition Polynomials,accepted by Rocky Mountain Journal of Mathematics (withDaniel Parry)
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Main Examples
Partition Polynomials:n∑
k=1
pk(n)xk
Plane Partition Polynomials:n∑
k=1
ppk (n)xk
Odd Partition Polynomials:n∑
k=1
poddk (n)xk
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Polynomial Versions of Different Partition Numbers
Example 1 Partition Polynomialp(n): partitions of npk (n): partition of n with exactly k parts
Fn(x) =∑n
k=1 pk (n)xk
F4(x) = x4 + x3 + 2x2 + xsince4 = 4, = 3 + 1, = 2 + 2, = 2 + 1 + 1, = 1 + 1 + 1 + 1
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Plane Partitions
Example 2 Plane Partition Polynomial
• Plane partition of n is an array πi ,j of positive integers withsum n whose rows and columns are decreasing.• Its trace is the sum of its diagonal entries.
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Plane Partitions
Example 2 Plane Partition Polynomial
• Plane partition of n is an array πi ,j of positive integers withsum n whose rows and columns are decreasing.• Its trace is the sum of its diagonal entries.
PL(n): plane partitions of n
PLk (n): plane partitions of n with trace k
Qn(x) =∑n
k=1 PLk (n)xk
Q4(x) = x4 + 2x3 + 6x2 + 4x
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Plane Partitions of 4 and Their Traces
PL1(4) = 4, PL2(4) = 6, PL3(4) = 2, PL4(4) = 1• Trace = 4,4
• Trace = 3
3 1 ,31
,
• Trace = 2
2 2 ,22
, 2 1 1 ,2 11
,211
,1 11 1
,
• Trace = 1
1 1 1 1 ,1 1 11
,1 111
,
1111
Conclude4 3 2
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Last Polynomial Example
Example 3 Odd Partition Polynomial
podd(n): partitions of n all of whose parts are oddpodd ,k (n): partitions of n with exactly k parts all of which areoddFodd ,n(x) =
∑nk=1 podd ,k(n)xk
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Last Polynomial Example
Example 3 Odd Partition Polynomial
podd(n): partitions of n all of whose parts are oddpodd ,k (n): partitions of n with exactly k parts all of which areoddFodd ,n(x) =
∑nk=1 podd ,k(n)xk
Other natural examples: use partitions whose parts lie in anarithmetic progression, partitions whose parts are congruent toseveral residue classes modulo a, etc.
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Zeros of the basic examples of degree 1000
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Partition Polynomial of degree 1000
10.50
-0.5 0-1
-0.5
0.5
-1
1
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Plane Partition Polynomial of degree 1000
K1.0 K0.5 0 0.5 1.0
K1.0
K0.5
0.5
1.0
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Odd Partition Polynomial of degree 1000
K1.0 K0.5 0 0.5 1.0
K1.0
K0.5
0.5
1.0
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Zeros Accumulate on Unit Circle
Theorem
Zeros accumulate to the unit circle as n → ∞
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Zeros Accumulate on Unit Circle
Theorem
Zeros accumulate to the unit circle as n → ∞
• Proof uses general techniques.
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Theorem of Erdös and Turán
Theorem
Let q(x) be the polynomial∑n
k=0 akxk of degree n withnon-zero constant term a0 6= 0. For 0 ≤ θ1 < θ2 ≤ 2π,
|# {z : arg z ∈ [θ1, θ2], q(z) = 0} −θ2 − θ1
2πn
∣
∣
∣
∣
< 16
√
n ln( |a0| + |a1| + · · · + |an|√
a0an
)
.
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Asymptotics for the Partition Numbers
p(n) ∼ 1
4n√
3exp
(
π
√
2n3
)
PL(n) ∼ ζ(3)7/36eζ′(−1)
211/36√
πn25/36exp
(
3 3√
ζ(3)(n
2
)2/3)
.
podd(n) ∼ 2−5/4 3−1/4 n−3/4 exp
(
π
√
n3
)
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Arguments of Zeros Are Uniformly Distributed
• qn(x) =
n∑
k=1
pk (n)xk−1
• deg(qn) = n − 1, aj = pj+1(n), a0 = an−1 = 1
•n−1∑
k=0
ak =
n∑
k=1
pk (n) = p(n) ∼ 1
4n√
3exp
(
π
√
2n3
)
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Arguments of Zeros Are Uniformly Distributed,continued
• Average number of zeros of qn(x) in the sectorθ1 < arg x < θ2
=1
n − 1# {z : arg z ∈ [θ1, θ2], qn(z) = 0}
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Arguments of Zeros Are Uniformly Distributed,continued
• Average number of zeros of qn(x) in the sectorθ1 < arg x < θ2
=1
n − 1# {z : arg z ∈ [θ1, θ2], qn(z) = 0}
By Theorem of Erdös and Turán:
1n − 1
∣
∣
∣
∣
# {z : arg z ∈ [θ1, θ2], qn(z) = 0} − θ2 − θ1
2π(n − 1)
∣
∣
∣
∣
is bounded by
16
√
ln( |a0| + |a1| + · · · + |an−1|√
a0an−1
)
= 16
√
ln[p(n)]√
n − 1
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Arguments of Zeros Are Uniformly Distributed,continued
16
√
ln[p(n)]√
n − 1∼ 16
1√n − 1
√
ln[
1
4n√
3eπ
√2n/3
]
= 161√
n − 1
√
− ln(4n√
3) + π√
2n/3
Conclude: the arguments of the zeros of the partitionpolynomials are uniformly distributed.
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Average Number of Zeros
10.50
-0.5 0-1
-0.5
0.5
-1
1
Elementary arguments show:average number of zeros both inside and outside the unit circlego to 0 as n → ∞.
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Average Number of Zeros
10.50
-0.5 0-1
-0.5
0.5
-1
1
Elementary arguments show:average number of zeros both inside and outside the unit circlego to 0 as n → ∞.
Conclusion: zeros are uniformly distributed around the unitcircle.
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Average Number of Zeros
10.50
-0.5 0-1
-0.5
0.5
-1
1
Elementary arguments show:average number of zeros both inside and outside the unit circlego to 0 as n → ∞.
Conclusion: zeros are uniformly distributed around the unitcircle.
Problem: Ignore the contribution of zeros inside the unit disk.
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Zero Attractor
Definition
Informally, the Zero Attractor for {Fn(x)} is the set of limits ofthe zeros of Fn(x).
Formally, the Zero Attractor for {Fn(x)} is the limit of the setsZ (Fn) in the space of compact subsets of the complex plane inthe Hausdorff metric.
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Structure of Asymptotics
Assumption: asymptotics of Fn(x) inside unit disk D have thefollowing form:D is the union of disjoint open regions R1, R2, . . . (up to theirboundaries) and
Fn(x) ∼ a1,n(x)en1/2L1(x), x ∈ R1,
Fn(x) ∼ a2,n(x)en1/2L2(x), x ∈ R2, · · ·
where aj ,n(x) 6= 0 on Rj , both aj ,n(x), Lj(x) are analytic on Rj ,
and aj ,n(x) = o(en1/2Lj (x)).
RegionsR1 = {x : ℜL1(x) > ℜL2(x),ℜL3(x), · · · }R2 = {x : ℜL2(x) > ℜL1(x),ℜL3(x), · · · }, · · ·
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Informal Principle to Find Zero Attractor
• For adjacent regions, say R1 and R2, their common boundaryis a subset of the level set ℜL1(x) = ℜL2(x).
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Informal Principle to Find Zero Attractor
• For adjacent regions, say R1 and R2, their common boundaryis a subset of the level set ℜL1(x) = ℜL2(x).
• The zeros of Fn(x) cannot converge to a point interior to anyof the regions R(j); that is, the zeros accumulate along theircommon boundary.
•Conclusion: The zero attractor =⋃
j≥1
boundary(Rj)
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Summary
• Need polynomial analogues of the partition numberasymptotics to find their zero attractor.
• Zeros converge to the boundaries of the regions where thesense of the asymptotics change.
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Plots of Zeros
Asymptotics and Their Regions
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Example 1Partition Polynomials
Example 2Plane Partition Polynomials
Example 3Odd Partition Polynomials
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Partition Polynomial of degree 1000
10.50
-0.5 0-1
-0.5
0.5
-1
1
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Partition Polynomial Zero Attractor
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Partition Polynomial Zero Attractor in Upper HalfPlane
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Partition Polynomial Zero Attractor Closeup
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Partition Polynomial Zero Attractor with Degree70,000 Zeros
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Partition Polynomial Zeros of Degree 25,000, 30,000,40,000, 50,000, 60,000, 70,000
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Dilogarithm
Dilogarithm: Li2(x) =
∞∑
m=1
xm
m2
Functions for Partition Polynomial Asymptotics
Lk(x) =1k
√
Li2(xk )
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Asymptotics of Partition Polynomials
Theorem
Fn(x) ∼√
1 − x√
L1(x)1
n3/4exp
(
2√
nL1(x))
, x ∈ R1
Fn(x) ∼ a2(x)√
L2(x)(−1)n
n3/4exp
(
2√
nL2(x))
x ∈ R2
Fn(x) ∼ a3,n(x)√
L3(x)1
n3/4exp
(
2√
nL3(x))
, x ∈ R3
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Zero Attractor of Partition Polynomials
Three Regions:
R1 = {x : ℜ[L1(x)] > max(ℜ[L2(x)],ℜ[L3(x)])},R2 = {x : ℜ[L2x)] > max(ℜ[L1(x)],ℜ[L3(x)])},R3 = {x : ℜ[L3(x)] > max(ℜ[L1(x)],ℜ[L2(x)])}.
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Zero Attractor of Partition Polynomials
Three Regions:
R1 = {x : ℜ[L1(x)] > max(ℜ[L2(x)],ℜ[L3(x)])},R2 = {x : ℜ[L2x)] > max(ℜ[L1(x)],ℜ[L3(x)])},R3 = {x : ℜ[L3(x)] > max(ℜ[L1(x)],ℜ[L2(x)])}.
Zero attractor consists of portions of the three level sets
ℜ[L1(x)] = ℜ[L2(x)], ℜ√
Li2(x) =12ℜ√
Li2(x2)
ℜ[L1(x)] = ℜ[L3(x)], ℜ√
Li2(x) =13ℜ√
Li2(x3)
ℜ[L2(x) = ℜ[L3(x)],12ℜ√
Li2(x2) =13ℜ√
Li2(x3)
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Special Points on Zero Attractor
Special Points:
• Three curves intersect at (−0.692206, 0.6913717)
• Solution to ℜL1(eit) = 13ℜL3(eit) is 2.06672966
(where “green curve” intersects unit circle)
• Solution to 12ℜL2(eit) = 1
3ℜL3(eit) is 2.36170417(where “blue curve” intersects unit circle)
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Plane Partition Polynomial of degree 500
K1.0 K0.5 0 0.5 1.0
K1.0
K0.5
0.5
1.0
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Plane Partition Polynomial of degree 2200
K1.0 K0.5 0 0.5 1.0
K1.0
K0.5
0.5
1.0
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Zero Attractor for Plane Partition Polynomials
K1.0 K0.5 0 0.5 1.0
K1.0
K0.5
0.5
1.0
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
K1.0 K0.5 0 0.5 1.0
K1.0
K0.5
0.5
1.0
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Closeup of Zero Attractor for Plane PartitionPolynomials with degree 800 zeros
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
• Special points on zero attractor
on unit circle ei2.989863546
on negative axis −0.8250030529
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Trilogarithm
Trilogarithm: Li3(x) =∞∑
m=1
xm
m3
Functions for Plane Partition Polynomial Asymptotics
Lk(x) =1k
3√
Li3(xk ), k = 1, 2, . . .
Level Set for Zero Attractor
ℜL1(x) = ℜL2(x)
Interval in Zero Attractor [−0.8250030529, 0]
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Asymptotics of Plane Partition Polynomials
Theorem
(a) Let x ∈ R1 \ [x∗, 0],
Qn(x) ∼ 12√
1 − x
√
L1(x)
6πn4/3exp
(
32 n2/3L1(x)
)
.
(b) Let x ∈ R2,
Qn(x) ∼ (−1)n 24√
1 − x2 8
√
1 − x1 + x
√
L2(x)
6πn4/3exp
(
32n2/3 L2(x)
)
.
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Odd Partition Polynomial of degree 1000
K1.0 K0.5 0 0.5 1.0
K1.0
K0.5
0.5
1.0
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Odd Partition Polynomial of degree 900
K1.0 K0.5 0 0.5 1.0
K1.0
K0.5
0.5
1.0
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Odd Partition Polynomial Zero Attractor
K1.0 K0.5 0 0.5 1.0
0.2
0.4
0.6
0.8
1.0
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Odd Partition Zero Attractor Closeup
K0.10 K0.05 0.00 0.05 0.10
0.92
0.94
0.96
0.98
1.00
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Odd Partition Zero Attractor with Composite Zeros ofDegree 2n, n = 12, · · · , 15
K0.10 K0.05 0.00 0.05 0.10
0.8
0.9
1.0
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Asymptotics of Odd Partition Polynomials
Functions for the Asymptotics
L2j+1(x) =1
2j + 1
√
Li2(x2j+1), j = 0, 1, 2, . . .
L2j(x) =1j
√
Li2(−x j), j = 1, 2, . . .
Level Sets for Zero Attractor
ℜL1(x) = ℜL4(x), ℜL2(x) = ℜL4(x)
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Generating Functions G(x , q)
Partition Polynomials
∞∏
ℓ=1
11 − xqℓ
=
∞∑
n=0
Fn(x) qn
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Generating Functions G(x , q)
Partition Polynomials
∞∏
ℓ=1
11 − xqℓ
=
∞∑
n=0
Fn(x) qn
Plane Partition Polynomials
∞∏
ℓ=1
1(1 − xqℓ)ℓ
=
∞∑
n=0
Qn(x) qn
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Generating Functions G(x , q)
Partition Polynomials
∞∏
ℓ=1
11 − xqℓ
=
∞∑
n=0
Fn(x) qn
Plane Partition Polynomials
∞∏
ℓ=1
1(1 − xqℓ)ℓ
=
∞∑
n=0
Qn(x) qn
Odd Partition Polynomials
∞∏
ℓ=1
11 − xq2ℓ−1 =
∞∑
n=0
Fodd ,n(x) qn
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Circle Method
Cauchy Integral Formula: for any radius 0 < r < 1
Fn(x) =1
2πi
∫
|u|=r
G(x , u)
un+1 du
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Circle Method
Cauchy Integral Formula: for any radius 0 < r < 1
Fn(x) =1
2πi
∫
|u|=r
G(x , u)
un+1 du
Subdivide the circle |u| = r into subarcs relative to rationalpoints distributed around the circle using Farey fractions.
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Circle Method
Cauchy Integral Formula: for any radius 0 < r < 1
Fn(x) =1
2πi
∫
|u|=r
G(x , u)
un+1 du
Subdivide the circle |u| = r into subarcs relative to rationalpoints distributed around the circle using Farey fractions.
Farey fractions of order N
FN = {h/k : (h, k) = 1, 1 ≤ k ≤ N}
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Circle Method
Cauchy Integral Formula: for any radius 0 < r < 1
Fn(x) =1
2πi
∫
|u|=r
G(x , u)
un+1 du
Subdivide the circle |u| = r into subarcs relative to rationalpoints distributed around the circle using Farey fractions.
Farey fractions of order N
FN = {h/k : (h, k) = 1, 1 ≤ k ≤ N}
Special intervalsLet h1/k1 < h/k < h2/k2 be three consecutive Farey fractionsof order N:
(
h + h1
k + k1− h
k,h + h2
k + k2− h
k
)
with corresponding circular arc ξ(N)h,k : re2πiθ
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Circle Method - Continued
Circle Radius, Farey Fraction OrderBoth the radius r and the order N are functions of n and x
r = r(x , n) N = N(x , n).
Order of Farey fractions: N = N(x , n) = δn1/2
Radius of integration circle: r(x , n) = exp(
−ℜLm(x)
2πn1/2
)
,
(x ∈ R(m), m = 1, 2, 3)
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Circle Method - Continued
Circle Radius, Farey Fraction OrderBoth the radius r and the order N are functions of n and x
r = r(x , n) N = N(x , n).
Order of Farey fractions: N = N(x , n) = δn1/2
Radius of integration circle: r(x , n) = exp(
−ℜLm(x)
2πn1/2
)
,
(x ∈ R(m), m = 1, 2, 3)
Question: Which Farey arcs contribute as n → ∞? (“majorarcs")
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
G(x , u) in neighborhood of e2πih/k
By using series expansion of the logarithm and the identity
e−τ =1
2πi
∫ σ+i∞
σ−i∞Γ(s)τ−s ds, σ > 1,ℜτ > 0
ln[G(x , e−w+2πih/k)] =1
2πi
∫ σ+i∞
σ−i∞Qh,k(x , s) τ−s ds, σ > 1,ℜw > 0
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
G(x , u) in neighborhood of e2πih/k
By using series expansion of the logarithm and the identity
e−τ =1
2πi
∫ σ+i∞
σ−i∞Γ(s)τ−s ds, σ > 1,ℜτ > 0
ln[G(x , e−w+2πih/k)] =1
2πi
∫ σ+i∞
σ−i∞Qh,k(x , s) τ−s ds, σ > 1,ℜw > 0
where
Qh,k(x , s) =
∞∑
m,ℓ=1
xℓ
ℓe2πimℓh/k (mℓ)−s
=
∞∑
ℓ=1
xℓ
ℓs+1
∞∑
m=1
e2πimℓh/k
ms , σ > 1
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
• Given1
2πi
∫ σ+i∞
σ−i∞Qh,k(x , s) τ−s ds, σ > 1,,
we want to shift the vertical line to the left half plane.
• The residue of Qh,k (x , s) at s = 1 will give the dominantcontribution to the integral.
• Need useful form of Qh,k(x , s) to get the residues.
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Special Functions
Polylogarithm: Lis(x) =∞∑
n=1
xn
ns
Zeta Function: ζ(s) =∞∑
n=1
1ns
Hurwitz Zeta Function: ζ(s, a) =∞∑
n=0
1(n + a)s
Lerch Phi Function: Φ(x , s, a) =
∞∑
n=0
xn
(n + a)s
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Expansion of Qh,k(x , s) in terms of special functions
Qh,k (x , s) =ζ(s)
ks+1 Lis+1(xk ) +
k−1∑
r=1
x r
ks+1 Φ(xk , s + 1, r/k) Lis(e2πirh/k )
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Expansion of Qh,k(x , s) in terms of special functions
Qh,k (x , s) =ζ(s)
ks+1 Lis+1(xk ) +
k−1∑
r=1
x r
ks+1 Φ(xk , s + 1, r/k) Lis(e2πirh/k )
ζ(s) has a simple pole at s = 1 with residue 1. This gives theonly contribution:
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Expansion of Qh,k(x , s) in terms of special functions
Qh,k (x , s) =ζ(s)
ks+1 Lis+1(xk ) +
k−1∑
r=1
x r
ks+1 Φ(xk , s + 1, r/k) Lis(e2πirh/k )
ζ(s) has a simple pole at s = 1 with residue 1. This gives theonly contribution:
Theorem
The residue of Qh,k(x , s) at s = 1 is
1k2 Li2(x
k )
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Regions R1, R2, R3
Theorem
ℜLk(x) < max[ℜL1(x),ℜL2(x),ℜL3(x)], k ≥ 4, x 6= 0.
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Regions R1, R2, R3
Theorem
ℜLk(x) < max[ℜL1(x),ℜL2(x),ℜL3(x)], k ≥ 4, x 6= 0.
Hence the regions Rm, with m = 1, 2, 3 have the stronger form
Rm ={
x : ℜLm(x) > ℜLj(x), j 6= m}
.
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Estimation of Integrals from Circle Method
• Assume x ∈ R(m), m = 1, 2, or 3
The main contribution to the integral∫
ξ(n)h,k
G(x , u)
un+1 du
is gotten from the integral Ih,k ,n(x) where
Ih,k ,n(x) =1
2πn1/2
∫ 2πn1/2/[k(k+k ′′)]
2πn1/2/[k(k+k ′)]exp
[
n1/2 Φ(x , z)]
dz, x ∈ Rm
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Estimation of Integrals from Circle Method
• Assume x ∈ R(m), m = 1, 2, or 3
The main contribution to the integral∫
ξ(n)h,k
G(x , u)
un+1 du
is gotten from the integral Ih,k ,n(x) where
Ih,k ,n(x) =1
2πn1/2
∫ 2πn1/2/[k(k+k ′′)]
2πn1/2/[k(k+k ′)]exp
[
n1/2 Φ(x , z)]
dz, x ∈ Rm
Φ(x , z) =Lm(x)2
ℜLk(x) − iz+ (ℜLm(x) − iz), x ∈ Rm
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Saddle Point MethodAssume that ℜB(x) has a unique maximum on (a, b) andℜB(x0) < 0.
∫ b
aet B(x) dx ∼ etB(x0)
√
2π
−tB′′(x0)
∂
∂zΦ(x , z) =
i Lm(x)2
(ℜLm(x) − iz)2 − i = 0 =⇒ z0 = −ℑLm(x)
∂2
∂2zΦ(x , z) =
i Lm(x)2
(ℜLm(x) − iz)3 (−2)(−i) = −2Lm(x)2
(ℜLm(x) − iz)3
ℜ ∂2
∂2zΦ(x , z0) = −2ℜ 1
Lm(x)< 0
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Conclusion of Estimation Using Saddle Point Method
Ih,k ,n(x) ∼ 12πn1/2
√π√
Lm(x)1
n1/4en1/2Lm(x)
=1
2√
π
√
Lm(x)1
n3/4en1/2Lm(x)
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Conclusion of Estimation Using Saddle Point Method
Ih,k ,n(x) ∼ 12πn1/2
√π√
Lm(x)1
n1/4en1/2Lm(x)
=1
2√
π
√
Lm(x)1
n3/4en1/2Lm(x)
• A more detailed analysis is needed to get the full asymptoticexpansion of the polynomials.
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
I applied the above method to the followingexamples but I do not have proofs.
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Polynomials for partitions whose parts arecongruent to either 1 or 2 modulo 3
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Degree 400
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Zeros from several polynomials
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Attractor with zeros in second quadrant
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Zero Attractor
L1(x) =√
2Li2(x)
L2(x) =12
√
2Li2(x2)
L3(x) =13
[
2Li2(x3) + xΦ(x3, 2, 1/3) · e2πi/3
+ x2Φ(x3, 2, 2/3) · (e2πi/3)2
+ xΦ(x3, 2, 1/3) · (e2πi/3)2 + x2Φ(x3, 2, 2/3)(e4πi/3)2]1/2
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Zero Attractor
Zero attractor• level set ℜL1 = ℜL3 which is near the imaginary axis andconnects 0 with e1.59829i (solution to ℜL1(eit) = ℜL3(eit).
• level set ℜL2 = ℜL3 which is near −1 and connects thepoints −0.897454 on the negative axis with e2.92246i (which aresolutions to ℜL2(t) = ℜL3(t) for t ∈ (−1, 0) andℜL2(eit) = ℜL3(eit)).
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Polynomials for partitions whose parts arecongruent to either 0 or 2 modulo 3
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Parts congruent to either 0 or 2 modulo 3
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Parts congruent to either 0 or 2 modulo 3 – Closeup
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Zero Attractor
L1(x) =√
2Li2(x), L2(x) =12
√
2Li2(x2)
L3(x) =13
[
Li2(x3) + xΦ(x3, 2, 1/3) + x2Φ(x3, 2, 2/3)
+ xΦ(x3, 2, 1/3) · (e2πi/3)2 + x2Φ(x3, 2, 2/3)(e4πi/3)2]1/2
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Zero Attractor
Zero attractor• [−0.55008, 0]
• level set ℜL1 = ℜL3 connects 0 to e1.60212i (solution toℜL1(eit) = ℜL3(eit));
• level set ℜL2 = ℜL3 connects −0.55008 to e2.84288i (solutionto ℜL2(eit) = ℜL3(eit))
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Polynomials for partitions whose parts arecongruent to either 1 or 3 modulo 4
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Parts congruent to either 1 or 3 modulo 4 - Degree1500
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Parts congruent to either 1 or 3 modulo 4 - Degrees3000, 4500, 6000 – Closeup to imaginary axis
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Zero Attractor Closeup with zeros of degree 1500,3000, 45000, 6000
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
For 1 ≤ b < a, let
Kb,a;h,k(x) =1
ak2 Li2(xk ) +
k−1∑
r=1, k |ra
x r e2πirbh/k
ak2 Φ(xk , 2, r/k)
Lh,k(x) =[
K1,4;h,k(x) + K3,4;h,k(x)]1/2
L1(x) = L0,1(x), L2(x) = L1,2(x), L4(x) = L1,4(x).
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Zero Attractor
• imaginary axis interval [0, 0.97474] i (solution toℜL1(it) = ℜL4(it) for 0 < t < 1) [ℜL1(x) > max{ℜLk (x) : k ≥ 2}on disk of radius ≃ 0.97474]
• level set ℜL2 = ℜL4 (in second quadrant) that connects0.97474 i to the point on the unit circle e1.60867i (solution toℜL2(eit) = ℜL4(eit);
• level set ℜL1 = ℜL4 (in second quadrant) that connects0.97474 i to the point on the unit circle e1.53291i (solution toℜL1(eit) = ℜL4(eit).
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Parts congruent to either 1 or 4 modulo 5 – Degree6000
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Zero Attractor and Zeros for Parts congruent to either1 or 4 modulo 5
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Closeup of Zero Attractor with zeros of degree 3000,4000, 5000, 6000
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros
Functions for Asymptotics and Zero Attractor
L1(x) =[
K1,5;0,1(x) + K4,5;0,1(x)]1/2
L5(x) =[
K1,5;3,5(x) + K4,5;3,5(x)]1/2
• e1.581091115i is the solution to ℜ[L1(eit) = ℜ[L5(eit)]
• Level set ℜL1(x) = ℜL5(x)
Robert Boyer Polynomial Versions of Integer Partitions and Their Zeros