Let = + i denote a nontrivial zero of ( s ) , and consider the - - PowerPoint PPT Presentation

INVESTIGATING THE VERTICAL

DISTRIBUTION OF ZEROS OF L-FUNCTIONS

Caroline Turnage-Butterbaugh Duke University

Computational Aspects of L-functions ICERM November 11, 2015

Let ρ = β + iγ denote a nontrivial zero of ζ(s), and consider the sequence of ordinates of zeros in the critical strip: 0 < γ1 ≤ γ2 ≤ . . . ≤ γn ≤ γn+1 ≤ . . . .

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Let ρ = β + iγ denote a nontrivial zero of ζ(s), and consider the sequence of ordinates of zeros in the critical strip: 0 < γ1 ≤ γ2 ≤ . . . ≤ γn ≤ γn+1 ≤ . . . . Since N(T) :=

• ρ

0<γ<T

1 ∼ T 2π log T,

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Let ρ = β + iγ denote a nontrivial zero of ζ(s), and consider the sequence of ordinates of zeros in the critical strip: 0 < γ1 ≤ γ2 ≤ . . . ≤ γn ≤ γn+1 ≤ . . . . Since N(T) :=

• ρ

0<γ<T

1 ∼ T 2π log T, the average size of γn+1 − γn is 2π log(γn).

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Let µ := lim inf

n→∞

γn+1 − γn (2π/ log γn) and λ := lim sup

n→∞

γn+1 − γn (2π/ log γn).

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Let µ := lim inf

n→∞

γn+1 − γn (2π/ log γn) and λ := lim sup

n→∞

γn+1 − γn (2π/ log γn). By definition, we have µ ≤ 1 ≤ λ,

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Let µ := lim inf

n→∞

γn+1 − γn (2π/ log γn) and λ := lim sup

n→∞

γn+1 − γn (2π/ log γn). By definition, we have µ ≤ 1 ≤ λ, and we expect µ = 0 and λ = ∞.

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Let µ := lim inf

n→∞

γn+1 − γn (2π/ log γn) and λ := lim sup

n→∞

γn+1 − γn (2π/ log γn). By definition, we have µ ≤ 1 ≤ λ, and we expect µ = 0 and λ = ∞.

CONJECTURE

Gaps between consecutive zeros of ζ(s) that are arbitrarily small/large, relative to the average gap size, appear infinitely often.

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WHAT IS KNOWN? (ASSUMING RH)

lower bound for λ upper bound for µ Gonek/Mueller 1.9 Montgomery & Odlyzko 1.9799 0.5179 Conrey, Ghosh & Gonek 2.3378 0.5172 R.R. Hall 2.6306∗ Bui, Milinovich & Ng 2.6950 0.5155 Feng & Wu 2.7327 0.5154 Bredberg 2.766∗ Bui 3.18 Preobrazhenskii 0.515396 a∗Results are unconditional, but RH must be assumed to give lower

bound on λ. a∗∗preprint

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WHAT IS KNOWN? (ASSUMING RH)

lower bound for λ upper bound for µ Gonek/Mueller 1.9 Montgomery & Odlyzko 1.9799 0.5179 Conrey, Ghosh & Gonek 2.3378 0.5172 R.R. Hall 2.6306∗ Bui, Milinovich & Ng 2.6950 0.5155 Feng & Wu 2.7327 0.5154 Bredberg 2.766∗ Bui 3.18 Preobrazhenskii 0.515396 a∗Results are unconditional, but RH must be assumed to give lower

bound on λ. a∗∗preprint

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WHAT IS KNOWN? (ASSUMING RH)

lower bound for λ upper bound for µ Gonek/Mueller 1.9 Montgomery & Odlyzko 1.9799 0.5179 Conrey, Ghosh & Gonek 2.3378 0.5172 R.R. Hall 2.6306∗ Bui, Milinovich & Ng 2.6950 0.5155 Feng & Wu 2.7327 0.5154 Bredberg 2.766∗ Bui 3.18 Preobrazhenskii 0.515396 a∗Results are unconditional, but RH must be assumed to give lower

bound on λ. a∗∗preprint

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WHAT IS KNOWN? (ASSUMING RH)

lower bound for λ upper bound for µ Gonek/Mueller 1.9 Montgomery & Odlyzko 1.9799 0.5179 Conrey, Ghosh & Gonek 2.3378 0.5172 Hall 2.6306∗ Bui, Milinovich & Ng 2.6950 0.5155 Feng & Wu 2.7327 0.5154 Bredberg 2.766∗ Bui 3.18 Preobrazhenskii 0.515396 a∗Results are unconditional; must assume RH for lower bound on λ.

a∗∗preprint

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WHAT IS KNOWN? (ASSUMING RH)

lower bound for λ upper bound for µ Gonek/Mueller 1.9 Montgomery & Odlyzko 1.9799 0.5179 Conrey, Ghosh & Gonek 2.3378 0.5172 Hall 2.6306∗ Bui, Milinovich & Ng 2.6950 0.5155 Feng & Wu 2.7327 0.5154 Bredberg 2.766∗ Bui 3.18 Preobrazhenskii 0.515396 a∗Results are unconditional; must assume RH for lower bound on λ.

a∗∗preprint

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WHAT IS KNOWN? (ASSUMING RH)

lower bound for λ upper bound for µ Gonek/Mueller 1.9 Montgomery & Odlyzko 1.9799 0.5179 Conrey, Ghosh & Gonek 2.3378 0.5172 Hall 2.6306∗ Bui, Milinovich & Ng 2.6950 0.5155 Feng & Wu 2.7327 0.5154 Bredberg 2.766∗ Bui 3.18 Preobrazhenskii 0.515396 a∗Results are unconditional; must assume RH for lower bound on λ.

a∗∗preprint

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WHAT IS KNOWN? (ASSUMING RH)

lower bound for λ upper bound for µ Gonek/Mueller 1.9 Montgomery & Odlyzko 1.9799 0.5179 Conrey, Ghosh & Gonek 2.3378 0.5172 Hall 2.6306∗ Bui, Milinovich & Ng 2.6950 0.5155 Feng & Wu 2.7327 0.5154 Bredberg 2.766∗ Bui 3.18 Preobrazhenskii 0.515396 a∗Results are unconditional; must assume RH for lower bound on λ.

a∗∗preprint

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WHAT IS KNOWN? (ASSUMING RH)

lower bound for λ upper bound for µ Gonek/Mueller 1.9 Montgomery & Odlyzko 1.9799 0.5179 Conrey, Ghosh & Gonek 2.3378 0.5172 Hall 2.6306∗ Bui, Milinovich & Ng 2.6950 0.5155 Feng & Wu 2.7327 0.5154 Bredberg 2.766∗ Bui∗,∗∗ 3.18 Preobrazhenskii∗∗ 0.515396 a∗Results are unconditional; must assume RH for lower bound on λ.

a∗∗preprint

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WHAT IS KNOWN? (ASSUMING RH)

lower bound for λ upper bound for µ Gonek/Mueller 1.9 Montgomery & Odlyzko 1.9799 0.5179 Conrey, Ghosh & Gonek 2.3378 0.5172 Hall 2.6306∗ Bui, Milinovich & Ng 2.6950 0.5155 Feng & Wu 2.7327 0.5154 Bredberg 2.766∗ Bui∗,∗∗ 3.18 Preobrazhenskii∗∗ 0.515396 a∗Results are unconditional; must assume RH for lower bound on λ.

a∗∗preprint

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GAPS BETWEEN ZEROS OF OTHER L-FUNCTIONS

• For large gaps, we consider the following degree 2

L-functions:

• ζK(s) – the Dedekind zeta-function of a quadratic number

field K with discriminant d

• L(s, f) – an automorphic L-function on GL(2) over Q, where

f is either a primitive holomorphic cusp form or a primitive Maass cusp form

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GAPS BETWEEN ZEROS OF OTHER L-FUNCTIONS

• For large gaps, we consider the following degree 2

L-functions:

• ζK(s) – the Dedekind zeta-function of a quadratic number

field K with discriminant d

• L(s, f) – an automorphic L-function on GL(2) over Q, where

f is either a primitive holomorphic cusp form or a primitive Maass cusp form

• For small gaps, we consider primitive L-functions from the

Selberg Class.

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GAPS BETWEEN ZEROS OF OTHER L-FUNCTIONS

• For large gaps, we consider the following degree 2

L-functions:

• ζK(s) – the Dedekind zeta-function of a quadratic number

field K with discriminant d

• L(s, f) – an automorphic L-function on GL(2) over Q, where

f is either a primitive holomorphic cusp form or a primitive Maass cusp form

• For small gaps, we consider primitive L-functions from the

Selberg Class. As in the case of ζ(s), we expect that

CONJECTURE

Gaps between consecutive zeros that are arbitrarily large, relative to the average gap size, appear infinitely often for both ζK(s) and L(s, f).

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LARGE GAPS BETWEEN ZEROS OF ζK(s), L(s, f)

Theorem (T., 2014)

Assuming GRH for ζK(s), we have λK ≥ 2.449.

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LARGE GAPS BETWEEN ZEROS OF ζK(s), L(s, f)

Theorem (T., 2014)

Assuming GRH for ζK(s), we have λK ≥ 2.449.

Theorem (Bui, Heap, T., 2014 (preprint))

Assuming GRH for ζK(s), we have λK ≥ 2.866.

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LARGE GAPS BETWEEN ZEROS OF ζK(s), L(s, f)

Theorem (T., 2014)

Assuming GRH for ζK(s), we have λK ≥ 2.449.

Theorem (Bui, Heap, T., 2014 (preprint))

Assuming GRH for ζK(s), we have λK ≥ 2.866.

Theorem (Barrett, McDonald, Miller, Ryan, T., Winsor, 2015)

Assuming GRH for L(s, f), we have λf ≥ 1.732. These results can be stated unconditionally if we restrict our attention to zeros on the critical line.

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LARGE GAPS - HALL’S METHOD (MODIFIED BY BREDBERG)

Wirtinger’s Inequality

Let g : [a, b] → C be continuously differentiable and suppose that g(a) = g(b) = 0. Then b

a

|g(t)|2dt ≤ b − a π 2 b

a

|g′(t)|2dt.

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LARGE GAPS - HALL’S METHOD (MODIFIED BY BREDBERG)

Wirtinger’s Inequality

Let g : [a, b] → C be continuously differentiable and suppose that g(a) = g(b) = 0. Then b

a

|g(t)|2dt ≤ b − a π 2 b

a

|g′(t)|2dt. By understanding the mean-values of g(t) and g′(t), we can

• btain a lower bound on gaps between zeros of g(t).

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IDEA OF ARGUMENT FOR ζK(s)

Let g(t) := exp (iνLt) ζK( 1

2 + it)M( 1 2 + it),

where ν is a real constant that will be chosen later, L ∼ log( √ dT) and M(s) is an amplifier of the form

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IDEA OF ARGUMENT FOR ζK(s)

Let g(t) := exp (iνLt) ζK( 1

2 + it)M( 1 2 + it),

where ν is a real constant that will be chosen later, L ∼ log( √ dT) and M(s) is an amplifier of the form M(s) =

• h1h2≤y

dr(h1)dr(h2)χd(h2)P[h1h2] (h1h2)s where y = Tθ, 0 < θ < 1/4, and dr(h) denotes the coefficients of ζ(s)r.

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IDEA OF ARGUMENT FOR ζK(s)

Let g(t) := exp (iνLt) ζK( 1

2 + it)M( 1 2 + it),

where ν is a real constant that will be chosen later, L ∼ log( √ dT) and M(s) is an amplifier of the form M(s) =

• h1h2≤y

dr(h1)dr(h2)χd(h2)P[h1h2] (h1h2)s where y = Tθ, 0 < θ < 1/4, and dr(h) denotes the coefficients of ζ(s)r. Here P[h] = P log y/h log y

• for 1 ≤ h ≤ y and P(x) is a polynomial.

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IDEA OF ARGUMENT FOR ζK(s)

λK := lim sup

n→∞

γK(n + 1) − γK(n) (π/ log

• |d|γK(n))

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IDEA OF ARGUMENT FOR ζK(s)

λK := lim sup

n→∞

γK(n + 1) − γK(n) (π/ log

• |d|γK(n))

Assume (towards contradiction) that λK ≤ κ.

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IDEA OF ARGUMENT FOR ζK(s)

λK := lim sup

n→∞

γK(n + 1) − γK(n) (π/ log

• |d|γK(n))

Assume (towards contradiction) that λK ≤ κ. Let t1 ≤ t2 ≤ . . . ≤ tN denote the zeros of g(t) in the interval [T, 2T].

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IDEA OF ARGUMENT FOR ζK(s)

λK := lim sup

n→∞

γK(n + 1) − γK(n) (π/ log

• |d|γK(n))

Assume (towards contradiction) that λK ≤ κ. Let t1 ≤ t2 ≤ . . . ≤ tN denote the zeros of g(t) in the interval [T, 2T]. By our assumption, tn+1 − tn ≤ (1 + o(1))κπ Ld .

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By Wirtinger’s Inequality, tn+1

tn

|g(t)|2dt ≤ (1 + o(1)) κ2 L2 tn+1

tn

|g′(t)|2dt.

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By Wirtinger’s Inequality, tn+1

tn

|g(t)|2dt ≤ (1 + o(1)) κ2 L2 tn+1

tn

|g′(t)|2dt. Summing for zeros between height T and 2T, we have 2T

T

|g(t)|2dt ≤ (1 + o(1)) κ2 L2 2T

T

|g′(t)|2dt.

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By Wirtinger’s Inequality, tn+1

tn

|g(t)|2dt ≤ (1 + o(1)) κ2 L2 tn+1

tn

|g′(t)|2dt. Summing for zeros between height T and 2T, we have 2T

T

|g(t)|2dt ≤ (1 + o(1)) κ2 L2 2T

T

|g′(t)|2dt. Therefore, if L2 κ2 2T

T |g(t)|2dt

2T

T |g′(t)|2dt

> 1, we may conclude that λK > κ.

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MEAN-VALUES FOR THE CASE ζK(1/2+it)

Using a special case of a result of Bettin, Bui, Li, and Radziwiłł (which computes the twisted moment of the product of four Dirichlet L-functions) we have

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MEAN-VALUES FOR THE CASE ζK(1/2+it)

Using a special case of a result of Bettin, Bui, Li, and Radziwiłł (which computes the twisted moment of the product of four Dirichlet L-functions) we have

Theorem (Bui, Heap, T., 2014 (preprint))

We have 2T

T

|g(t)|2 dt ∼ Cr(0)L2

d

(2r2 − 1)!((r − 1)!)4 T + O(TL2r2+4r+1

d

) and 2T

T

|g′(t)|2 dt ∼ Cr(1)L4

d

(2r2 − 1)!((r − 1)!)4 T + O(TL2r2+4r+3

d

) as T → ∞, where Cr(0), Cr(1) are constants depending on the coefficients of ζK(s)r and are given explicitly.

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λK := lim sup

n→∞

γK(n + 1) − γK(n) (π/ log

• |d|γK(n))

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λK := lim sup

n→∞

γK(n + 1) − γK(n) (π/ log

• |d|γK(n))

The choice of θ = 1/4, ν = 1.2773, r = 1, and P(x) = 1 − 10.8998x + 28.9444x2 − 22.1343x3 + 0.6148x4 allows us to conclude that

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λK := lim sup

n→∞

γK(n + 1) − γK(n) (π/ log

• |d|γK(n))

The choice of θ = 1/4, ν = 1.2773, r = 1, and P(x) = 1 − 10.8998x + 28.9444x2 − 22.1343x3 + 0.6148x4 allows us to conclude that

Theorem (Bui, Heap, T., 2014) (preprint)

Assuming GRH, we have λK ≥ 2.866. That is, there are infinitely many pairs of consecutive zeros of ζK(s) that are more than 2.866 times the average spacing apart.

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REMARK - LENGTH OF THE AMPLIFIER

M(s) =

• h1h2≤Tθ

dr(h1)dr(h2)χd(h2)P[h1h2] (h1h2)s , 0 < θ < 1/4

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REMARK - LENGTH OF THE AMPLIFIER

M(s) =

• h1h2≤Tθ

dr(h1)dr(h2)χd(h2)P[h1h2] (h1h2)s , 0 < θ < 1/4

Lower bounds for λK in terms of θ in the simple case r = 1, P(x) = 1.

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REMARK - LENGTH OF THE AMPLIFIER

M(s) =

• h1h2≤Tθ

dr(h1)dr(h2)χd(h2)P[h1h2] (h1h2)s , 0 < θ < 1/4

Lower bounds for λK in terms of θ in the simple case r = 1, P(x) = 1.

• The function is not strictly

increasing as θ → 1, (which is widely believed to be the largest value in the range of validity for twisted moment results).

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REMARK - LENGTH OF THE AMPLIFIER

M(s) =

• h1h2≤Tθ

dr(h1)dr(h2)χd(h2)P[h1h2] (h1h2)s , 0 < θ < 1/4

Lower bounds for λK in terms of θ in the simple case r = 1, P(x) = 1.

• The function is not strictly

increasing as θ → 1, (which is widely believed to be the largest value in the range of validity for twisted moment results).

• This phenomenon has also

been observed by Bredberg and Bui when studying large gaps between zeros

• f ζ(s).

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REMARK - HIGHER MOMENTS?

It is not clear that emulating higher moments should necessarily lead to larger gaps in our framework.

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REMARK - HIGHER MOMENTS?

It is not clear that emulating higher moments should necessarily lead to larger gaps in our framework. The basic point is that the coefficients in the denominator of the ratio 2T

T |g(t)|dt

2T

T |g′(t)|dt

can often be larger than that of the numerator when one considers higher moments.

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Theorem (Barett, McDonald, Miller, Ryan, T., Winsor, 2015)

Let L ∈ S be primitive of degree mL. Assume GRH and Hypothesis

• A. Then there is a computable nontrivial upper bound on µL

depending on mL. In particular, mL upper bound for µL 1 0.606894 2 0.822897 3 0.905604 4 0.942914 5 0.962190 . . . . . . where the nontrivial upper bounds for µL approach 1 as mL increases.

The case mL = 1 has previously been shown by Carneiro, Chandee, Littmann, and Milinovich in 2014.

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METHODS OVERVIEW

• We utilize an argument of Goldston, Gonek, ¨

Ozl¨ uk, and Snyder with a new modification of Carneiro, Chandee, Littmann, and Milinovich.

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METHODS OVERVIEW

• We utilize an argument of Goldston, Gonek, ¨

Ozl¨ uk, and Snyder with a new modification of Carneiro, Chandee, Littmann, and Milinovich.

• Murty and Perelli proved a general pair correlation result

for all primitive L-functions in the Selberg Class for restricted support inversely proportional to the degree of the function, assuming the Generalized Riemann Hypothesis and their Hypothesis A.

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METHODS OVERVIEW

• We utilize an argument of Goldston, Gonek, ¨

Ozl¨ uk, and Snyder with a new modification of Carneiro, Chandee, Littmann, and Milinovich.

• Murty and Perelli proved a general pair correlation result

for all primitive L-functions in the Selberg Class for restricted support inversely proportional to the degree of the function, assuming the Generalized Riemann Hypothesis and their Hypothesis A.

• Hypothesis A is a mild assumption concerning the

correlation of the coefficients of L-functions at primes and prime powers.

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