new a symplectic test of the l-functions ratios conjecture · 2008. 5. 13. · introduction ratios...
TRANSCRIPT
Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
A Symplectic Test of the L-FunctionsRatios Conjecture
Steven J MillerBrown University
[email protected]@aya.yale.edu
http://www.math.brown.edu/∼sjmiller
Cornell University, June 5th, 2008
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
L-functions
Riemann zeta function:
ζ(s) =
∞∑
n=1
1ns =
∏
p
(1 − 1
ps
)−1
.
Dirichlet L-functions:
L(s, χd) =
∞∑
n=1
χd(n)
ns =∏
p
(1 − χd(p)
ps
)−1
.
Elliptic curve L-functions: build up with data related tonumber of solutions modulo p.
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
Properties of zeros of L-functions
infinitude of primes, primes in arithmetic progression.
Chebyshev’s bias: π3,4(x) ≥ π1,4(x) ‘most’ of the time.
Birch and Swinnerton-Dyer conjecture.
Goldfeld, Gross-Zagier: bound for h(D) fromL-functions with many central point zeros.
Even better estimates for h(D) if a positivepercentage of zeros of ζ(s) are at most 1/2 − ε of theaverage spacing to the next zero.
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
Distribution of zeros
ζ(s) 6= 0 for Re(s) = 1: π(x), πa,q(x).
GRH: error terms.
GSH: Chebyshev’s bias.
Analytic rank, adjacent spacings: h(D).
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
Measures of Spacings: n-Level Correlations2 NICHOLAS M. KATZ AND PETER SARNAK
Figure 1. Nearest neighbor spacings among 70 million zeroes be-yond the 1020-th zero of zeta, verses µ1(GUE).
RH (needless to say, in the numerical experiments reported on below all zeroesfound were on the line Re(s) = 1
2 ) and order the ordinates γ:
. . . . . . γ−1 ≤ 0 ≤ γ1 ≤ γ2 . . . .(4)
Then γj = −γ−j, j = 1, 2, . . . , and in fact γ1 is rather large, being equal to14.1347 . . . . It is known (apparently already to Riemann) that
#{j : 0 ≤ γj ≤ T } ∼ T log T
2π, as T → ∞.(5)
In particular, the mean spacing between the γ′js tends to zero as j → ∞. In order
to examine the (statistical) law of the local spacings between these numbers were-normalize (or “unfold” as it is sometimes called) as follows:Set
γj =γj log γj
2πfor j ≥ 1.(6)
The consecutive spacings δj are defined to be
δj = γj+1 − γj , j = 1, 2, . . . .(7)
More generally, the k − th consecutive spacings are
δ(k)j = γj+k − γj , j = 1, 2, . . . .(8)
What laws (i.e. distributions), if any, do these numbers obey?During the years 1980-present, Odlyzko [OD] has made an extensive and pro-
found numerical study of the zeroes and in particular their local spacings. Hefinds that they obey the laws for the (scaled) spacings between the eigenvalues of
n-level correlation: {αj} an increasing sequence ofnumbers, B ⊂ Rn−1 a compact box:
limN→∞
#
{(αj1 − αj2, . . . , αjn−1 − αjn
)∈ B, ji 6= jk ≤ N
}
N
Odlyzko: Normalized spacings of ζ(s) starting at 1020
Montgomery, Hejhal: Pair, triple correlations of ζ(s)Rudnick-Sarnak: n-level correlations for allautomorphic cupsidal L-fnsKatz-Sarnak: n-level correlations for the classicalcompact groupsinsensitive to any finite set of zeros
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
Measures of Spacings: n-Level Density and Families
Let gi be even Schwartz functions whose FourierTransform is compactly supported, L(s, f ) an L-functionwith zeros 1
2 + iγf and conductor Qf :
Dn,f (g) =∑
j1,...,jnji 6=±jk
g1
(γf ,j1
log Qf
2π
)· · ·gn
(γf ,jn
log Qf
2π
)
Properties of n-level density:� Individual zeros contribute in limit� Most of contribution is from low zeros� Average over similar L-functions (family)To any geometric family, Katz-Sarnak predict then-level density depends only on a symmetry group (aclassical compact group) attached to the family.
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
n-Level Density
n-level density: F = ∪FN a family of L-functions orderedby conductors, gk an even Schwartz function:
Dn,F(g) = limN→∞
1|FN|
∑
f∈FN
∑
j1,...,jnji 6=±jk
g1
(log Qf
2πγj1;f
)· · ·gn
(log Qf
2πγjn;f
)
As N → ∞, 1-level density converges to∫
g(x)W1,G(F)(x)dx =
∫g(u)W1,G(F)(u)du.
Conjecture (Katz-Sarnak)(In the limit) Distribution of zeros near central point agreeswith distribution of eigenvalues near 1 of a classicalcompact group.
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
1-Level Densities
Let G be one of the classical compact groups: Unitary,Symplectic, Orthogonal (or SO(even), SO(odd)).If supp(g) ⊂ (−1, 1), 1-level density of G is
g(0) − cGg(0)
2,
where
cG =
0 G is Unitary1 G is Symplectic
−1 G is Orthogonal.
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
Some Results
Orthogonal:� Iwaniec-Luo-Sarnak: 1-level density for H±
k (N), Nsquare-free.� Miller, Young: families of elliptic curves.� Güloglu: 1-level for {Symr f : f ∈ Hk(1)}, r odd.Symplectic:� Rubinstein: n-level densities for L(s, χd).� Güloglu: 1-level for {Symr f : f ∈ Hk(1)}, r even.Unitary:� Hughes-Rudnick, Miller: families of primitiveDirichlet characters.
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
Identifying the Symmetry Groups
Often an analysis of the monodromy group in thefunction field case suggests the answer.All simple families studied to date are built from GL1or GL2 L-functions.Tools: Explicit Formula, Orthogonality of Characters /Petersson Formula.How to identify symmetry group in general? Onepossibility is by the signs of the functional equation:Folklore Conjecture: If all signs are even and nocorresponding family with odd signs, Symplecticsymmetry; otherwise SO(even). (False!)
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
Explicit Formula
π: cuspidal automorphic representation on GLn.Qπ > 0: analytic conductor of L(s, π) =
∑λπ(n)/ns.
By GRH the non-trivial zeros are 12 + iγπ,j.
Satake parameters {απ,i(p)}ni=1;
λπ(pν) =∑n
i=1 απ,i(p)ν.
L(s, π) =∑
nλπ(n)
ns =∏
p
∏ni=1 (1 − απ,i(p)p−s)
−1.
∑
j
g(
γπ,jlog Qπ
2π
)= g(0)−2
∑
p,ν
g(
ν log plog Qπ
)λπ(pν) log ppν/2 log Qπ
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
Some Results: Rankin-Selberg Convolution of Families
Symmetry constant: cL = 0 (resp, 1 or -1) if family L hasunitary (resp, symplectic or orthogonal) symmetry.
Rankin-Selberg convolution: Satake parameters forπ1,p × π2,p are
{απ1×π2(k)}nmk=1 = {απ1(i) · απ2(j)} 1≤i≤n
1≤j≤m.
Theorem (Dueñez-Miller)If F and G are nice families of L-functions, thencF×G = cF · cG.
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
Ratios Conjecture
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
History
Farmer (1993): Considered∫ T
0
ζ(s + α)ζ(1 − s + β)
ζ(s + γ)ζ(1 − s + δ)dt ,
conjectured (for appropriate values)
T(α + δ)(β + γ)
(α + β)(γ + δ)− T 1−α−β (δ − β)(γ − α)
(α + β)(γ + δ).
Conrey-Farmer-Zirnbauer (2007): conjectureformulas for averages of products of L-functions overfamilies:
RF =∑
f∈F
ωfL(
12 + α, f
)
L(1
2 + γ, f) .
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
Uses of the Ratios Conjecture
Applications:� n-level correlations and densities;� mollifiers;� moments;� vanishing at the central point;
Advantages:� RMT models often add arithmetic ad hoc;� predicts lower order terms, often to square-rootlevel.
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
Inputs for 1-level density
Approximate Functional Equation:
L(s, f ) =∑
m≤x
am
ms + εXL(s)∑
n≤y
an
n1−s ;
� ε sign of the functional equation,� XL(s) ratio of Γ-factors from functional equation.
Explicit Formula: g Schwartz test function,
∑
f∈F
ωf
∑
γ
g(
γlog Nf
2π
)=
12πi
∫
(c)
−∫
(1−c)
R′F(· · · )g (· · · )
� R′F(r) = ∂
∂αRF (α, γ)
∣∣∣α=γ=r
.
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
Procedure
Use approximate functional equation to expandnumerator.Expand denominator by generalized Mobius function:cusp form
1L(s, f )
=∑
h
µf (h)
hs ,
where µf (h) is the multiplicative function equaling 1for h = 1, −λf (p) if n = p, χ0(p) if h = p2 and 0otherwise.Execute the sum over F , keeping only main(diagonal) terms.Extend the m and n sums to infinity (complete theproducts).Differentiate with respect to the parameters.
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
Main Results
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
Symplectic Families
Fundamental discriminants: d square-free and 1modulo 4, or d/4 square-free and 2 or 3 modulo 4.Associated character χd :� χd(−1) = 1 say d even;� χd(−1) = −1 say d odd.� even (resp., odd) if d > 0 (resp., d < 0).
Will study following families:� even fundamental discriminants at most X ;� {8d : 0 < d ≤ X , d an odd, positive square-freefundamental discriminant}.
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
Prediction from Ratios Conjecture
1
X∗
∑
d≤X
∑
γd
g(
γdlog X
2π
)=
1
X∗ log X
∫∞
−∞g(τ)
∑
d≤X
[log
d
π+
1
2
Γ′
Γ
(1
4±
iπτ
log X
)]dτ
+2
X∗ log X
∑
d≤X
∫∞
−∞g(τ)
[ζ′
ζ
(1 +
4πiτ
log X
)+ A′
D
(2πiτ
log X;
2πiτ
log X
)
− e−2πiτ log(d/π)/ log XΓ(
14 − πiτ
log X
)
Γ(
14 + πiτ
log X
) ζ
(1 −
4πiτ
log X
)AD
(−
2πiτ
log X;
2πiτ
log X
)]dτ + O(X− 1
2 +ε),
with
AD(−r , r) =∏
p
(1 − 1
(p + 1)p1−2r −1
p + 1
)·(
1 − 1p
)−1
A′D(r ; r) =
∑
p
log p(p + 1)(p1+2r − 1)
.
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
Prediction from Ratios Conjecture
Main term is
1X ∗
∑
d≤X
∑
γd
g(
γdlog X
2π
)=
∫ ∞
−∞
g(x)
(1 − sin(2πx)
2πx
)dx
+ O(
1log X
),
which is the 1-level density for the scaling limit ofUSp(2N). If supp(g) ⊂ (−1, 1), then the integral of g(x)against − sin(2πx)/2πx is −g(0)/2.
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
Prediction from Ratios Conjecture
Assuming RH for ζ(s), for supp(g) ⊂ (−σ, σ) ⊂ (−1, 1):
−2
X∗ log X
∑
d≤X
∫∞
−∞g(τ) e
−2πiτ log(d/π)log X
Γ(
14 − πiτ
log X
)
Γ(
14 + πiτ
log X
) ζ
(1 −
4πiτ
log X
)AD
(−
2πiτ
log X;
2πiτ
log X
)dτ
= −g(0)
2+ O(X− 3
4 (1−σ)+ε);
the error term may be absorbed into the O(X−1/2+ε) errorif σ < 1/3.
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
Main Results
Theorem (M– ’07)
Let supp(g) ⊂ (−σ, σ), assume RH for ζ(s). 1-LevelDensity agrees with prediction from Ratios Conjecture
up to O(X−(1−σ)/2+ε) for the family of quadraticDirichlet characters with even fundamentaldiscriminants at most X;up to O(X−1/2 + X−(1− 3
2 σ)+ε + X− 34 (1−σ)+ε) for our
sub-family. If σ < 1/3 then agrees up to O(X−1/2+ε).
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
Numerics (J. Stopple): 1,003,083 negative fundamentaldiscriminants −d ∈ [1012, 1012 + 3.3 · 106]
Histogram of normalized zeros (γ ≤ 1, about 4 million).� Red: main term. � Blue: includes O(1/ log X ) terms.
� Green: all lower order terms.24
Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
Sketch of Proofs
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
Ratios Calculation
Hardest piece to analyze is
R(g; X ) = − 2X ∗ log X
∑
d≤X
∫ ∞
−∞
g(τ)e−2πiτ log(d/π)log X
Γ(
14 − πiτ
log X
)
Γ(
14 + πiτ
log X
)
· ζ
(1 − 4πiτ
log X
)AD
(− 2πiτ
log X;
2πiτlog X
)dτ,
AD(−r , r) =∏
p
(1 − 1
(p + 1)p1−2r −1
p + 1
)·(
1 − 1p
)−1
.
Proof: shift contours, keep track of poles of ratios of Γ andzeta functions.
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
Ratios Calculation: Weaker result for supp(g) ⊂ (−1, 1).
d -sum is X ∗e−2πi(1− log πlog X )τ
(1 − 2πiτ
log X
)−1+ O(X 1/2);
decay of g restricts τ -sum to |τ | ≤ log X , Taylorexpand everything but g: small error term and
∫
|τ |≤log Xg(τ)
N∑
n=−1
an
logn X(2πiτ)ne−2πi(1− log π
log X )τdτ
=N∑
n=−1
an
logn X
∫
|τ |≤log X(2πiτ)ng(τ)e−2πi(1− log π
log X )τdτ ;
from decay of g can extend the τ -integral to R
(essential that N is fixed and finite!), for n ≥ 0 get theFourier transform of g(n) (the nth derivative of g) at1 − π
log X , vanishes if supp(g) ⊂ (−1, 1).27
Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
Number Theory Sums
Seven = − 2X ∗
∑
d≤X
∞∑
`=1
∑
p
χd(p)2 log pp` log X
g(
2log p`
log X
)
Sodd = − 2X ∗
∑
d≤X
∞∑
`=0
∑
p
χd(p) log pp(2`+1)/2 log X
g(
log p2`+1
log X
).
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
Number Theory Sums
LemmaLet supp(g) ⊂ (−σ, σ) ⊂ (−1, 1). Then
Seven = −g(0)
2+
2log X
∫ ∞
−∞
g(τ)ζ ′
ζ
(1 +
4πiτlog X
)dτ
+2
log X
∫ ∞
−∞
g(τ)A′D
(2πiτlog X
;2πiτlog X
)+ O(X− 1
2+ε)
Sodd = O(X− 1−σ2 log6 X ).
If instead we consider the family of characters χ8d for odd,positive square-free d ∈ (0, X ) (d a fundamentaldiscriminant), then
Sodd = O(X−1/2+ε + X−(1− 32 σ)+ε).
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
Analysis of Seven
χd(p)2 = 1 except when p|d . Replace χd(p)2 with 1, andsubtract off the contribution from when p|d :
Seven = −2∞∑
`=1
∑
p
log pp` log X
g(
2log p`
log X
)
+2
X ∗
∑
d≤X
∞∑
`=1
∑
p|d
log pp` log X
g(
2log p`
log X
)
= Seven;1 + Seven;2.
Lemma (Perron’s Formula)
Seven;1 = −g(0)
2+
2log X
∫ ∞
−∞
g(τ)ζ ′
ζ
(1 +
4πiτlog X
)dτ.
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
Analysis of Seven: Seven;2
This piece gives us∫
g(τ)A′D(− · · · , · · · ).
Main ideas:� Restrict to p ≤ X 1/2.� For p < X 1/2:
∑d≤X ,p|d 1 = X∗
p+1 + O(X 1/2).� Use Fourier Transform to expand g.
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
Analysis of Sodd
Sodd = − 2X ∗
∞∑
`=0
∑
p
log pp(2`+1)/2 log X
g(
log p2`+1
log X
)∑
d≤X
χd(p).
Jutila’s bound
∑
1<n≤Nn non−square
∣∣∣∣∣∣∑
0<d≤Xd fund. disc.
χd(n)
∣∣∣∣∣∣
2
� NX log10 N.
Proof: Cauchy-Schwarz and Jutila: p2`+1 non-square:
∞∑
`=0
∑
p(2`+1)/2≤Xσ
∣∣∣∣∣∑
d≤X
χd(p)
∣∣∣∣∣
2
1/2
� X1+σ
2 log5 X .
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
Analysis of Sodd: Extending Support
More technical, replace Jutila’s bound by applyingPoisson Summation to character sums.
LemmaLet supp(g) ⊂ (−σ, σ) ⊂ (−1, 1). For family{8d : 0 < d ≤ X , d an odd, positive square-freefundamental discriminant}, Sodd = O(X− 1
2+ε + X−(1− 32 σ)+ε).
In particular, if σ < 1/3 then Sodd = O(X−1/2+ε).
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
Conclusions
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
Conclusions
Ratios Conjecture gives detailed predictions (up toX 1/2+ε).
Number Theory agrees with predictions for suitablyrestricted test functions.
Numerics quite good.
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
Appendix
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
RH and the Prime Number Theorem
From ζ(s) =∑
n−s =∏
p (1 − p−s)−1, logarithmic derivative is
−ζ ′(s)
ζ(s)=∑ Λ(n)
ns ,
where Λ(n) = log p if n = pk and is 0 otherwise.
Take Mellin transform, integrate and shift contour. Find∑
n≤x
Λ(n) = x −∑
ρ
xρ
ρ,
where ρ = 1/2 + iγ runs over non-trivial zeros of ζ(s).
Partial summation gives Prime Number Theorem (to first order, thereare x/ log x primes at most x) if Reρ < 1.
The smaller max Re(ρ) is, the better the error term in the PrimeNumber Theorem. The Riemann Hypothesis (RH) says Re(ρ) = 1/2.
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
Primes in Arithmetic Progression
To study number primes p ≡ a mod q, use
L(s, χ) =∑ χ(n)
ns =∏
p
(1 − χ(p)
ps
)−1
.
Key sum: 1φ(q)
∑χ mod q χ(n) is 1 if n ≡ 1 mod q and 0 otherwise.
Similar arguments give
∑
p≡a mod q
log pps = − 1
φ(q)
∑
χ mod q
L′(s, χ)
L(s, χ)χ(a) + Good(s).
Note: To understand {p ≡ a mod q} need to understand all L(s, χ);see benefit of studying a family.
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
GSH and Chebyshev’s Bias
π3,4(x) ≥ π1,4(x) and π2,3(x) ≥ π1,3(x) ‘most’ of the time. Useanalytic density:
Denan(S) = lim sup1
log T
∫
S∩[2,T ]
dtt
.
Have π3,4(x) ≥ π1,4(x) with analytic density .9959 (first flip at 26861);π2,3(x) ≥ π1,3(x) with analytic density .9990 (first flip ≈ 6 · 1011).
Non-residues beat residues. Key ingredient Generalized SimplicityHypothesis (GSH): the zeros of L(s, χ) are linearly independent overQ.
Structure of zeros important: GSH used to show a flow on a torus isfull (becomes equidistributed).
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
Class Number
Class number: measures failure of unique factorization (order of idealclass group).
Imaginary quadratic field Q(√
D), fundamental discriminant D < 0, Igroup of non-zero fractional ideals, P subgroup of principal ideals,H = I/P class group, h(D) = #H the class number. Dirichlet proved
L(1, χD) =2πh(D)
wD√
D,
where χD the quadratic character and wD = 2 if D < −4, 4 if D = −4and 6 if D = −3.
Theorem: h(D) = 1 ⇔ −D ∈ {3, 4, 7, 8, 11, 19, 43, 67, 163}.
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
Class Number and Distribution of Zeros I
Expect√
|D|
log log |D| � h(D) �√|D| log log |D|. Siegel proved
h(D) > c(ε)|D|1/2−ε (but ineffective).
Goldfeld, Gross-Zagier: f primitive cusp form of weight k , level N,trivial central character, suppose m = ords=1/2L(s, f )L(s, χD) ≥ 3,g = m − 1 or m − 2 so that (−1)g = ω(f )ω(fχD ) (signs of fnal eqs).Then have effective bound
h(D) � (log |D|)g−1∏
p|D
(1 +
1p
)−3(1 +
λ(p)√
pp + 1
)−1
.
Good result from using an elliptic curve that vanishes to order 3 ats = 1/2, application of many zeros at central point.
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
Class Number and Distribution of Zeros II
Assume a positive percent of zeros (or cT log T/(log |D|)A) of zeroswith γ ≤ T ) of ζ(s) are at most 1/2 − ε of the average spacing fromthe next zero ζ(s). Then h(D) �
√|D|/(log |D|)B , all constants
computable.
See actual spacings between zeros are tied to number theory (havepositive percent are less than half the average spacing if GUEConjecture holds for adjacent spacings).
Instead of 1/2 − ε, under RH have: .68 (Montgomery), .5179(Montgomery-Odlyzko), .5171 (Conrey-Ghosh-Gonek), .5169(Conrey-Iwaniec) (Montgomery says led to pair correlation conjectureby looking at gaps between zeros of ζ(s) and h(D)).
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
References
43
Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
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E. Bogomolny, O. Bohigas, P. Leboeuf and A. G. Monastra, Onthe spacing distribution of the Riemann zeros: corrections to theasymptotic result, Journal of Physics A: Mathematical andGeneral 39 (2006), no. 34, 10743–10754.
E. B. Bogomolny and J. P. Keating, Gutzwiller’s trace formula andspectral statistics: beyond the diagonal approximation, Phys.Rev. Lett. 77 (1996), no. 8, 1472–1475.
B. Conrey and D. Farmer, Mean values of L-functions andsymmetry, Internat. Math. Res. Notices 2000, no. 17, 883–908.
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Introduction Ratios Conjecture Main Results Proofs Conclusions Appendix Refs
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