Applications of Landau’s Formula

Applications of Landau’s Formula

Niko Laaksonen

UCL

February 16th 2015 Application to Dirichlet L-functions Automorphic forms Hyperbolic Landau-type formula

Applications of Landau’s Formula

Introduction

Landau and his work Automorphic forms Hyperbolic Landau-type formula

Applications of Landau’s Formula

Introduction

Landau and his work Application to Dirichlet L-functions Hyperbolic Landau-type formula

Applications of Landau’s Formula

Introduction

Landau and his work Application to Dirichlet L-functions Automorphic forms Applications of Landau’s Formula

Introduction

Landau and his work Application to Dirichlet L-functions Automorphic forms Hyperbolic Landau-type formula “Number theory is useful, since one can graduate with it.”

Born in Berlin to a Jewish family PNT & Prime Ideal Theorem Dismissal from Göttingen

Applications of Landau’s Formula

Edmund Landau (1877–1938) Born in Berlin to a Jewish family PNT & Prime Ideal Theorem Dismissal from Göttingen

Applications of Landau’s Formula

Edmund Landau (1877–1938)

“Number theory is useful, since one can graduate with it.” PNT & Prime Ideal Theorem Dismissal from Göttingen

Applications of Landau’s Formula

Edmund Landau (1877–1938)

“Number theory is useful, since one can graduate with it.”

Born in Berlin to a Jewish family Dismissal from Göttingen

Applications of Landau’s Formula

Edmund Landau (1877–1938)

“Number theory is useful, since one can graduate with it.”

Born in Berlin to a Jewish family PNT & Prime Ideal Theorem Applications of Landau’s Formula

Edmund Landau (1877–1938)

“Number theory is useful, since one can graduate with it.”

Born in Berlin to a Jewish family PNT & Prime Ideal Theorem Dismissal from Göttingen Here ρ = β + iγ are the nontrivial zeros of ζ. ( log p, if x = pn, The von Mangoldt function Λ(x) = 0, otherwise.

Applications of Landau’s Formula

Landau’s Formula

From Über die Nullstellen der Zetafunktion (1911): Theorem (Landau’s Formula) For a fixed x > 1,

X T xρ = − Λ(x) + O(log T ). 2π 0<γ≤T ( log p, if x = pn, The von Mangoldt function Λ(x) = 0, otherwise.

Applications of Landau’s Formula

Landau’s Formula

From Über die Nullstellen der Zetafunktion (1911): Theorem (Landau’s Formula) For a fixed x > 1,

X T xρ = − Λ(x) + O(log T ). 2π 0<γ≤T

Here ρ = β + iγ are the nontrivial zeros of ζ. Applications of Landau’s Formula

Landau’s Formula

From Über die Nullstellen der Zetafunktion (1911): Theorem (Landau’s Formula) For a fixed x > 1,

X T xρ = − Λ(x) + O(log T ). 2π 0<γ≤T

Here ρ = β + iγ are the nontrivial zeros of ζ. ( log p, if x = pn, The von Mangoldt function Λ(x) = 0, otherwise. + O(x log 2xT log log 3x)

  x    1  +O log x min T, + O log 2T min T, , hxi log x where hxi denotes the distance from x to the nearest prime power other than x itself.

This is uniform in x!

Applications of Landau’s Formula

Gonek’s Version

Theorem (Gonek–Landau Formula (1993)) Let x, T > 1. Then

X T xρ = − Λ(x) 2π 0<γ≤T This is uniform in x!

Applications of Landau’s Formula

Gonek’s Version

Theorem (Gonek–Landau Formula (1993)) Let x, T > 1. Then

X T xρ = − Λ(x) + O(x log 2xT log log 3x) 2π 0<γ≤T

  x    1  +O log x min T, + O log 2T min T, , hxi log x where hxi denotes the distance from x to the nearest prime power other than x itself. Applications of Landau’s Formula

Gonek’s Version

Theorem (Gonek–Landau Formula (1993)) Let x, T > 1. Then

X T xρ = − Λ(x) + O(x log 2xT log log 3x) 2π 0<γ≤T

  x    1  +O log x min T, + O log 2T min T, , hxi log x where hxi denotes the distance from x to the nearest prime power other than x itself.

This is uniform in x! Applications of Landau’s Formula

Results of Gonek

Corollary 1 (Gonek) 1 Under the RH, for large T and α real with |α| ≤ 2π log T , we have

X 1 2 |ζ( 2 + i(γ + 2πα/ log T )| = 0<γ≤T ! sin πα2 T   1 − log2 T + O T log7/4 T . πα 2π Applications of Landau’s Formula

Results of Gonek II

Corollary 2 (Gonek) Assume the RH and fix 0 < α < 1. Then as T → ∞

2 X sin( α (γ − γ0) log T )  1 α T 2 ∼ + log T. α (γ − γ0) log T α 3 2π 0<γ,γ0≤T 2 Grand Simplicity Hypothesis (Wintner, Hooley, Montgomery)  1 γ : L( 2 + iγ, χ) = 0, χ primitive is linearly independent over Q.

R. Murty and K. Murty (1994): If F,G ∈ S then

|ZF (T )∆ZG(T )| = o(T ) =⇒ F = G,

where ZF (T ) is the set of zeros of F (s) in Re(s) ≥ 1/2, | Im(s) ≤ T , ∆ is the symmetric difference.

Applications of Landau’s Formula

Value Distribution of L(s, χ)

Fujii (1976): A positive proportion of zeros of L(s, ψ)L(s, χ) are distinct. R. Murty and K. Murty (1994): If F,G ∈ S then

|ZF (T )∆ZG(T )| = o(T ) =⇒ F = G,

where ZF (T ) is the set of zeros of F (s) in Re(s) ≥ 1/2, | Im(s) ≤ T , ∆ is the symmetric difference.

Applications of Landau’s Formula

Value Distribution of L(s, χ)

Fujii (1976): A positive proportion of zeros of L(s, ψ)L(s, χ) are distinct.

Grand Simplicity Hypothesis (Wintner, Hooley, Montgomery)  1 γ : L( 2 + iγ, χ) = 0, χ primitive is linearly independent over Q. Applications of Landau’s Formula

Value Distribution of L(s, χ)

Fujii (1976): A positive proportion of zeros of L(s, ψ)L(s, χ) are distinct.

Grand Simplicity Hypothesis (Wintner, Hooley, Montgomery)  1 γ : L( 2 + iγ, χ) = 0, χ primitive is linearly independent over Q.

R. Murty and K. Murty (1994): If F,G ∈ S then

|ZF (T )∆ZG(T )| = o(T ) =⇒ F = G,

where ZF (T ) is the set of zeros of F (s) in Re(s) ≥ 1/2, | Im(s) ≤ T , ∆ is the symmetric difference. Let χ, ψ be real, primitive and non-principal 1 characters with distinct prime moduli q and `. Fix σ ∈ ( 2 , 1). Then, for a positive proportion of the nontrivial zeros of ζ with γ > 0, the values of L(σ + iγ, χ) and L(σ + iγ, ψ) are linearly independent over R.

Theorem 2 (L–Petridis 2012) Two Dirichlet L-functions with distinct primitive non-principal characters attain different values at cT non-trivial zeros of ζ up to height T , for some positive constant c.

Applications of Landau’s Formula

Our Theorems

Theorem 1 (L–Petridis 2011) Assume the RH. , primitive and non-principal 1 characters with distinct prime moduli q and `. Fix σ ∈ ( 2 , 1). Then, for a positive proportion of the nontrivial zeros of ζ with γ > 0, the values of L(σ + iγ, χ) and L(σ + iγ, ψ) are linearly independent over R.

Theorem 2 (L–Petridis 2012) Two Dirichlet L-functions with distinct primitive non-principal characters attain different values at cT non-trivial zeros of ζ up to height T , for some positive constant c.

Applications of Landau’s Formula

Our Theorems

Theorem 1 (L–Petridis 2011) Assume the RH. Let χ, ψ be real and non-principal 1 characters with distinct prime moduli q and `. Fix σ ∈ ( 2 , 1). Then, for a positive proportion of the nontrivial zeros of ζ with γ > 0, the values of L(σ + iγ, χ) and L(σ + iγ, ψ) are linearly independent over R.

Theorem 2 (L–Petridis 2012) Two Dirichlet L-functions with distinct primitive non-principal characters attain different values at cT non-trivial zeros of ζ up to height T , for some positive constant c.

Applications of Landau’s Formula

Our Theorems

Theorem 1 (L–Petridis 2011) Assume the RH. Let χ, ψ be real, primitive 1 Fix σ ∈ ( 2 , 1). Then, for a positive proportion of the nontrivial zeros of ζ with γ > 0, the values of L(σ + iγ, χ) and L(σ + iγ, ψ) are linearly independent over R.

Theorem 2 (L–Petridis 2012) Two Dirichlet L-functions with distinct primitive non-principal characters attain different values at cT non-trivial zeros of ζ up to height T , for some positive constant c.

Applications of Landau’s Formula

Our Theorems

Theorem 1 (L–Petridis 2011) Assume the RH. Let χ, ψ be real, primitive and non-principal characters with distinct prime moduli q and `. Then, for a positive proportion of the nontrivial zeros of ζ with γ > 0, the values of L(σ + iγ, χ) and L(σ + iγ, ψ) are linearly independent over R.

Theorem 2 (L–Petridis 2012) Two Dirichlet L-functions with distinct primitive non-principal characters attain different values at cT non-trivial zeros of ζ up to height T , for some positive constant c.

Applications of Landau’s Formula

Our Theorems

Theorem 1 (L–Petridis 2011) Assume the RH. Let χ, ψ be real, primitive and non-principal 1 characters with distinct prime moduli q and `. Fix σ ∈ ( 2 , 1). Theorem 2 (L–Petridis 2012) Two Dirichlet L-functions with distinct primitive non-principal characters attain different values at cT non-trivial zeros of ζ up to height T , for some positive constant c.

Applications of Landau’s Formula

Our Theorems

Theorem 1 (L–Petridis 2011) Assume the RH. Let χ, ψ be real, primitive and non-principal 1 characters with distinct prime moduli q and `. Fix σ ∈ ( 2 , 1). Then, for a positive proportion of the nontrivial zeros of ζ with γ > 0, the values of L(σ + iγ, χ) and L(σ + iγ, ψ) are linearly independent over R. Theorem 2 (L–Petridis 2012) Two Dirichlet L-functions with distinct primitive non-principal characters attain different values at cT non-trivial zeros of ζ up to height T , for some positive constant c.

Applications of Landau’s Formula

Our Theorems Theorem 1 (L–Petridis 2011) Assume the RH. Let χ, ψ be real, primitive and non-principal 1 characters with distinct prime moduli q and `. Fix σ ∈ ( 2 , 1). Then, for a positive proportion of the nontrivial zeros of ζ with γ > 0, the values of L(σ + iγ, χ) and L(σ + iγ, ψ) are linearly independent over R.

ρ

1

1 Re s = 2 Theorem 2 (L–Petridis 2012) Two Dirichlet L-functions with distinct primitive non-principal characters attain different values at cT non-trivial zeros of ζ up to height T , for some positive constant c.

Applications of Landau’s Formula

Our Theorems Theorem 1 (L–Petridis 2011) Assume the RH. Let χ, ψ be real, primitive and non-principal 1 characters with distinct prime moduli q and `. Fix σ ∈ ( 2 , 1). Then, for a positive proportion of the nontrivial zeros of ζ with γ > 0, the values of L(σ + iγ, χ) and L(σ + iγ, ψ) are linearly independent over R.

σ + iγ

1

1 Re s = 2 Applications of Landau’s Formula

Our Theorems

Theorem 1 (L–Petridis 2011) Assume the RH. Let χ, ψ be real, primitive and non-principal 1 characters with distinct prime moduli q and `. Fix σ ∈ ( 2 , 1). Then, for a positive proportion of the nontrivial zeros of ζ with γ > 0, the values of L(σ + iγ, χ) and L(σ + iγ, ψ) are linearly independent over R.

Theorem 2 (L–Petridis 2012) Two Dirichlet L-functions with distinct primitive non-principal characters attain different values at cT non-trivial zeros of ζ up to height T , for some positive constant c. To obtain complex characters need Im L(2σ, χψ) 6= 0

The conditions for the characters in Theorem 1 are very restrictive

It is difficult to extend Theorem 1 to the critical line In Theorem 2 we fail to obtain positive proportion (would prove simple zeros for L!) Not dependent on RH

Applications of Landau’s Formula

Remarks

In particular, Theorem 1 implies that the arguments of the L-functions are distinct To obtain complex characters need Im L(2σ, χψ) 6= 0 It is difficult to extend Theorem 1 to the critical line In Theorem 2 we fail to obtain positive proportion (would prove simple zeros for L!) Not dependent on RH

Applications of Landau’s Formula

Remarks

In particular, Theorem 1 implies that the arguments of the L-functions are distinct The conditions for the characters in Theorem 1 are very restrictive It is difficult to extend Theorem 1 to the critical line In Theorem 2 we fail to obtain positive proportion (would prove simple zeros for L!) Not dependent on RH

Applications of Landau’s Formula

Remarks

In particular, Theorem 1 implies that the arguments of the L-functions are distinct The conditions for the characters in Theorem 1 are very restrictive To obtain complex characters need Im L(2σ, χψ) 6= 0 In Theorem 2 we fail to obtain positive proportion (would prove simple zeros for L!) Not dependent on RH

Applications of Landau’s Formula

Remarks

In particular, Theorem 1 implies that the arguments of the L-functions are distinct The conditions for the characters in Theorem 1 are very restrictive To obtain complex characters need Im L(2σ, χψ) 6= 0 It is difficult to extend Theorem 1 to the critical line Not dependent on RH

Applications of Landau’s Formula

Remarks

In particular, Theorem 1 implies that the arguments of the L-functions are distinct The conditions for the characters in Theorem 1 are very restrictive To obtain complex characters need Im L(2σ, χψ) 6= 0 It is difficult to extend Theorem 1 to the critical line In Theorem 2 we fail to obtain positive proportion (would prove simple zeros for L!) Applications of Landau’s Formula

Remarks

In particular, Theorem 1 implies that the arguments of the L-functions are distinct The conditions for the characters in Theorem 1 are very restrictive To obtain complex characters need Im L(2σ, χψ) 6= 0 It is difficult to extend Theorem 1 to the critical line In Theorem 2 we fail to obtain positive proportion (would prove simple zeros for L!) Not dependent on RH Fix a prime P , define A(γ) = B(s, P )(L(s, χ)L(s, ψ) − L(s, χ)L(s, ψ)), where B(s, P ) is a specific Dirichlet polynomial. Suppose we know that for a constant c 6= 0, X X A(γ) ∼ cN(T ) and |A(γ)|2  N(T ), 0<γ≤T 0<γ≤T then X C-S | P A(γ)|2 |c|2N(T )2 1 ≥  = |c|2N(T ). P |A(γ)|2 N(T ) 0<γ≤T A(γ)6=0

Applications of Landau’s Formula

Proof of Theorem 1 Notice z, w ∈ C are L.I. over R iff |zw − zw| > 0. Suppose we know that for a constant c 6= 0, X X A(γ) ∼ cN(T ) and |A(γ)|2  N(T ), 0<γ≤T 0<γ≤T then X C-S | P A(γ)|2 |c|2N(T )2 1 ≥  = |c|2N(T ). P |A(γ)|2 N(T ) 0<γ≤T A(γ)6=0

Applications of Landau’s Formula

Proof of Theorem 1 Notice z, w ∈ C are L.I. over R iff |zw − zw| > 0. Fix a prime P , define A(γ) = B(s, P )(L(s, χ)L(s, ψ) − L(s, χ)L(s, ψ)), where B(s, P ) is a specific Dirichlet polynomial. then X C-S | P A(γ)|2 |c|2N(T )2 1 ≥  = |c|2N(T ). P |A(γ)|2 N(T ) 0<γ≤T A(γ)6=0

Applications of Landau’s Formula

Proof of Theorem 1 Notice z, w ∈ C are L.I. over R iff |zw − zw| > 0. Fix a prime P , define A(γ) = B(s, P )(L(s, χ)L(s, ψ) − L(s, χ)L(s, ψ)), where B(s, P ) is a specific Dirichlet polynomial. Suppose we know that for a constant c 6= 0, X X A(γ) ∼ cN(T ) and |A(γ)|2  N(T ), 0<γ≤T 0<γ≤T Applications of Landau’s Formula

Proof of Theorem 1 Notice z, w ∈ C are L.I. over R iff |zw − zw| > 0. Fix a prime P , define A(γ) = B(s, P )(L(s, χ)L(s, ψ) − L(s, χ)L(s, ψ)), where B(s, P ) is a specific Dirichlet polynomial. Suppose we know that for a constant c 6= 0, X X A(γ) ∼ cN(T ) and |A(γ)|2  N(T ), 0<γ≤T 0<γ≤T then X C-S | P A(γ)|2 |c|2N(T )2 1 ≥  = |c|2N(T ). P |A(γ)|2 N(T ) 0<γ≤T A(γ)6=0 Series definition not valid! So use approximate functional equation Then we can apply (uniform) Landau’s formula We need the B(s, P ) factor to have the mean value not being trivially 0 It also takes some effort to prove that the c we get is nonzero! The 4th moment is harder to estimate but we only need an upper bound

Applications of Landau’s Formula

Proof of Theorem 1 (cont.)

Ideally we would like to just apply Landau’s formula to L(s, χ)L(s, ψ) at the appropriate points Then we can apply (uniform) Landau’s formula We need the B(s, P ) factor to have the mean value not being trivially 0 It also takes some effort to prove that the c we get is nonzero! The 4th moment is harder to estimate but we only need an upper bound

Applications of Landau’s Formula

Proof of Theorem 1 (cont.)

Ideally we would like to just apply Landau’s formula to L(s, χ)L(s, ψ) at the appropriate points Series definition not valid! So use approximate functional equation We need the B(s, P ) factor to have the mean value not being trivially 0 It also takes some effort to prove that the c we get is nonzero! The 4th moment is harder to estimate but we only need an upper bound

Applications of Landau’s Formula

Proof of Theorem 1 (cont.)

Ideally we would like to just apply Landau’s formula to L(s, χ)L(s, ψ) at the appropriate points Series definition not valid! So use approximate functional equation Then we can apply (uniform) Landau’s formula It also takes some effort to prove that the c we get is nonzero! The 4th moment is harder to estimate but we only need an upper bound

Applications of Landau’s Formula

Proof of Theorem 1 (cont.)

Ideally we would like to just apply Landau’s formula to L(s, χ)L(s, ψ) at the appropriate points Series definition not valid! So use approximate functional equation Then we can apply (uniform) Landau’s formula We need the B(s, P ) factor to have the mean value not being trivially 0 The 4th moment is harder to estimate but we only need an upper bound

Applications of Landau’s Formula

Proof of Theorem 1 (cont.)

Ideally we would like to just apply Landau’s formula to L(s, χ)L(s, ψ) at the appropriate points Series definition not valid! So use approximate functional equation Then we can apply (uniform) Landau’s formula We need the B(s, P ) factor to have the mean value not being trivially 0 It also takes some effort to prove that the c we get is nonzero! Applications of Landau’s Formula

Proof of Theorem 1 (cont.)

Ideally we would like to just apply Landau’s formula to L(s, χ)L(s, ψ) at the appropriate points Series definition not valid! So use approximate functional equation Then we can apply (uniform) Landau’s formula We need the B(s, P ) factor to have the mean value not being trivially 0 It also takes some effort to prove that the c we get is nonzero! The 4th moment is harder to estimate but we only need an upper bound The B(s, p) factor is now simpler which makes proving c 6= 0 easier (so we get complex characters) A more natural approach, but the analysis is more difficult (Garunkstis–Kalpokas–Steuding, 2010)

Applications of Landau’s Formula

Proof of Theorem 2

Similar to Theorem 1, but we start from 1 Z ζ0 (s)B(s, p)L(s, χ) ds. 2πi C ζ

The role of Landau’s Formula is replaced by the Modified Gonek Lemma A more natural approach, but the analysis is more difficult (Garunkstis–Kalpokas–Steuding, 2010)

Applications of Landau’s Formula

Proof of Theorem 2

Similar to Theorem 1, but we start from 1 Z ζ0 (s)B(s, p)L(s, χ) ds. 2πi C ζ

The role of Landau’s Formula is replaced by the Modified Gonek Lemma The B(s, p) factor is now simpler which makes proving c 6= 0 easier (so we get complex characters) Applications of Landau’s Formula

Proof of Theorem 2

Similar to Theorem 1, but we start from 1 Z ζ0 (s)B(s, p)L(s, χ) ds. 2πi C ζ

The role of Landau’s Formula is replaced by the Modified Gonek Lemma The B(s, p) factor is now simpler which makes proving c 6= 0 easier (so we get complex characters) A more natural approach, but the analysis is more difficult (Garunkstis–Kalpokas–Steuding, 2010) f(γz) = f(z), for all γ ∈ Γ (∆ + λ)f = 0, where ∆ is the Laplacian on H

Γ acts on H as Möbius transformations az + b a b γz = , γ = ∈ Γ. cz + d c d Then Γ\H is a Riemann surface

−0.5 0.5

f : H −→ C is called an automorphic form if

Applications of Landau’s Formula

Upper Half-Plane

Let H = {z ∈ C : Im z > 0} and Γ = SL2(Z) f(γz) = f(z), for all γ ∈ Γ (∆ + λ)f = 0, where ∆ is the Laplacian on H

Then Γ\H is a Riemann surface

−0.5 0.5

f : H −→ C is called an automorphic form if

Applications of Landau’s Formula

Upper Half-Plane

Let H = {z ∈ C : Im z > 0} and Γ = SL2(Z) Γ acts on H as Möbius transformations az + b a b γz = , γ = ∈ Γ. cz + d c d f(γz) = f(z), for all γ ∈ Γ (∆ + λ)f = 0, where ∆ is the Laplacian on H

f : H −→ C is called an automorphic form if

Applications of Landau’s Formula

Upper Half-Plane

Let H = {z ∈ C : Im z > 0} and Γ = SL2(Z) Γ acts on H as Möbius transformations az + b a b γz = , γ = ∈ Γ. cz + d c d Then Γ\H is a Riemann surface

−0.5 0.5 f(γz) = f(z), for all γ ∈ Γ (∆ + λ)f = 0, where ∆ is the Laplacian on H

Applications of Landau’s Formula

Upper Half-Plane

Let H = {z ∈ C : Im z > 0} and Γ = SL2(Z) Γ acts on H as Möbius transformations az + b a b γz = , γ = ∈ Γ. cz + d c d Then Γ\H is a Riemann surface

−0.5 0.5

f : H −→ C is called an automorphic form if (∆ + λ)f = 0, where ∆ is the Laplacian on H

Applications of Landau’s Formula

Upper Half-Plane

Let H = {z ∈ C : Im z > 0} and Γ = SL2(Z) Γ acts on H as Möbius transformations az + b a b γz = , γ = ∈ Γ. cz + d c d Then Γ\H is a Riemann surface

−0.5 0.5

f : H −→ C is called an automorphic form if f(γz) = f(z), for all γ ∈ Γ Applications of Landau’s Formula

Upper Half-Plane

Let H = {z ∈ C : Im z > 0} and Γ = SL2(Z) Γ acts on H as Möbius transformations az + b a b γz = , γ = ∈ Γ. cz + d c d Then Γ\H is a Riemann surface

−0.5 0.5

f : H −→ C is called an automorphic form if f(γz) = f(z), for all γ ∈ Γ (∆ + λ)f = 0, where ∆ is the Laplacian on H No! isospectral 6≡ isometric

1 λ = s(1 − s) with s = 2 + it or 0 ≤ s ≤ 1 Since Γ\H is non-compact ∆ has both a discrete and a continuous spectrum

Maass cusp forms φn form an orthonormal system of eigenfunctions of ∆ Eisenstein series: X E(z, s) = (Im(γz))s

γ∈Γ∞\Γ

Applications of Landau’s Formula

Spectrum of ∆ Can you hear the shape of a drum? 1 λ = s(1 − s) with s = 2 + it or 0 ≤ s ≤ 1 Since Γ\H is non-compact ∆ has both a discrete and a continuous spectrum

Maass cusp forms φn form an orthonormal system of eigenfunctions of ∆ Eisenstein series: X E(z, s) = (Im(γz))s

γ∈Γ∞\Γ

Applications of Landau’s Formula

Spectrum of ∆ Can you hear the shape of a drum? No! isospectral 6≡ isometric Since Γ\H is non-compact ∆ has both a discrete and a continuous spectrum

Maass cusp forms φn form an orthonormal system of eigenfunctions of ∆ Eisenstein series: X E(z, s) = (Im(γz))s

γ∈Γ∞\Γ

Applications of Landau’s Formula

Spectrum of ∆ Can you hear the shape of a drum? No! isospectral 6≡ isometric

1 λ = s(1 − s) with s = 2 + it or 0 ≤ s ≤ 1 Maass cusp forms φn form an orthonormal system of eigenfunctions of ∆ Eisenstein series: X E(z, s) = (Im(γz))s

γ∈Γ∞\Γ

Applications of Landau’s Formula

Spectrum of ∆ Can you hear the shape of a drum? No! isospectral 6≡ isometric

1 λ = s(1 − s) with s = 2 + it or 0 ≤ s ≤ 1 Since Γ\H is non-compact ∆ has both a discrete and a continuous spectrum Eisenstein series: X E(z, s) = (Im(γz))s

γ∈Γ∞\Γ

Applications of Landau’s Formula

Spectrum of ∆ Can you hear the shape of a drum? No! isospectral 6≡ isometric

1 λ = s(1 − s) with s = 2 + it or 0 ≤ s ≤ 1 Since Γ\H is non-compact ∆ has both a discrete and a continuous spectrum

Maass cusp forms φn form an orthonormal system of eigenfunctions of ∆ Applications of Landau’s Formula

Spectrum of ∆ Can you hear the shape of a drum? No! isospectral 6≡ isometric

1 λ = s(1 − s) with s = 2 + it or 0 ≤ s ≤ 1 Since Γ\H is non-compact ∆ has both a discrete and a continuous spectrum

Maass cusp forms φn form an orthonormal system of eigenfunctions of ∆ Eisenstein series: X E(z, s) = (Im(γz))s

γ∈Γ∞\Γ The isometries on H are Möbius transformations On H the geodesics are lines and semi-circles perpendicular to the real axis Hyperbolic motions correspond to closed geodesics on Γ\H!

Theorem (Prime Geodesic Theorem)  x Let πΓ(x) = # {γ} : N(γ) ≤ x , then πΓ(x) ∼ log x .

Applications of Landau’s Formula

Prime Geodesics

Classify γ 6= I as elliptic, parabolic, hyperbolic On H the geodesics are lines and semi-circles perpendicular to the real axis Hyperbolic motions correspond to closed geodesics on Γ\H!

Theorem (Prime Geodesic Theorem)  x Let πΓ(x) = # {γ} : N(γ) ≤ x , then πΓ(x) ∼ log x .

Applications of Landau’s Formula

Prime Geodesics

Classify γ 6= I as elliptic, parabolic, hyperbolic The isometries on H are Möbius transformations Hyperbolic motions correspond to closed geodesics on Γ\H!

Theorem (Prime Geodesic Theorem)  x Let πΓ(x) = # {γ} : N(γ) ≤ x , then πΓ(x) ∼ log x .

Applications of Landau’s Formula

Prime Geodesics

Classify γ 6= I as elliptic, parabolic, hyperbolic The isometries on H are Möbius transformations On H the geodesics are lines and semi-circles perpendicular to the real axis Theorem (Prime Geodesic Theorem)  x Let πΓ(x) = # {γ} : N(γ) ≤ x , then πΓ(x) ∼ log x .

Applications of Landau’s Formula

Prime Geodesics

Classify γ 6= I as elliptic, parabolic, hyperbolic The isometries on H are Möbius transformations On H the geodesics are lines and semi-circles perpendicular to the real axis Hyperbolic motions correspond to closed geodesics on Γ\H! Applications of Landau’s Formula

Prime Geodesics

Classify γ 6= I as elliptic, parabolic, hyperbolic The isometries on H are Möbius transformations On H the geodesics are lines and semi-circles perpendicular to the real axis Hyperbolic motions correspond to closed geodesics on Γ\H!

Theorem (Prime Geodesic Theorem)  x Let πΓ(x) = # {γ} : N(γ) ≤ x , then πΓ(x) ∼ log x . S(T,X) = O(T 2)

S(T,X) = O(X11/48+T )= ⇒ O(x35/48+) in PGT

S(T,X) = O(X1/8T 5/4(log T )2)= ⇒ O(x7/10+)

For X > 1, define X S(T,X) = Xitj

|tj |≤T

Essential in the proof of Prime Geodesic Theorem Trivial estimate (Weyl’s Law)

Iwaniec (1984)

Luo–Sarnak (1995)

Applications of Landau’s Formula

Exponential Sum

1 2 Let λj = 4 + tj be the eigenvalues of ∆ S(T,X) = O(T 2)

S(T,X) = O(X11/48+T )= ⇒ O(x35/48+) in PGT

S(T,X) = O(X1/8T 5/4(log T )2)= ⇒ O(x7/10+)

Essential in the proof of Prime Geodesic Theorem Trivial estimate (Weyl’s Law)

Iwaniec (1984)

Luo–Sarnak (1995)

Applications of Landau’s Formula

Exponential Sum

1 2 Let λj = 4 + tj be the eigenvalues of ∆ For X > 1, define X S(T,X) = Xitj

|tj |≤T S(T,X) = O(T 2)

S(T,X) = O(X11/48+T )= ⇒ O(x35/48+) in PGT

S(T,X) = O(X1/8T 5/4(log T )2)= ⇒ O(x7/10+)

Trivial estimate (Weyl’s Law)

Iwaniec (1984)

Luo–Sarnak (1995)

Applications of Landau’s Formula

Exponential Sum

1 2 Let λj = 4 + tj be the eigenvalues of ∆ For X > 1, define X S(T,X) = Xitj

|tj |≤T

Essential in the proof of Prime Geodesic Theorem S(T,X) = O(X11/48+T )= ⇒ O(x35/48+) in PGT

S(T,X) = O(X1/8T 5/4(log T )2)= ⇒ O(x7/10+)

S(T,X) = O(T 2) Iwaniec (1984)

Luo–Sarnak (1995)

Applications of Landau’s Formula

Exponential Sum

1 2 Let λj = 4 + tj be the eigenvalues of ∆ For X > 1, define X S(T,X) = Xitj

|tj |≤T

Essential in the proof of Prime Geodesic Theorem Trivial estimate (Weyl’s Law) S(T,X) = O(X11/48+T )= ⇒ O(x35/48+) in PGT

S(T,X) = O(X1/8T 5/4(log T )2)= ⇒ O(x7/10+)

Iwaniec (1984)

Luo–Sarnak (1995)

Applications of Landau’s Formula

Exponential Sum

1 2 Let λj = 4 + tj be the eigenvalues of ∆ For X > 1, define X S(T,X) = Xitj

|tj |≤T

Essential in the proof of Prime Geodesic Theorem Trivial estimate (Weyl’s Law) S(T,X) = O(T 2) S(T,X) = O(X1/8T 5/4(log T )2)= ⇒ O(x7/10+)

S(T,X) = O(X11/48+T )= ⇒ O(x35/48+) in PGT Luo–Sarnak (1995)

Applications of Landau’s Formula

Exponential Sum

1 2 Let λj = 4 + tj be the eigenvalues of ∆ For X > 1, define X S(T,X) = Xitj

|tj |≤T

Essential in the proof of Prime Geodesic Theorem Trivial estimate (Weyl’s Law) S(T,X) = O(T 2) Iwaniec (1984) S(T,X) = O(X1/8T 5/4(log T )2)= ⇒ O(x7/10+)

Luo–Sarnak (1995)

Applications of Landau’s Formula

Exponential Sum

1 2 Let λj = 4 + tj be the eigenvalues of ∆ For X > 1, define X S(T,X) = Xitj

|tj |≤T

Essential in the proof of Prime Geodesic Theorem Trivial estimate (Weyl’s Law) S(T,X) = O(T 2) Iwaniec (1984) S(T,X) = O(X11/48+T )= ⇒ O(x35/48+) in PGT S(T,X) = O(X1/8T 5/4(log T )2)= ⇒ O(x7/10+)

Applications of Landau’s Formula

Exponential Sum

1 2 Let λj = 4 + tj be the eigenvalues of ∆ For X > 1, define X S(T,X) = Xitj

|tj |≤T

Essential in the proof of Prime Geodesic Theorem Trivial estimate (Weyl’s Law) S(T,X) = O(T 2) Iwaniec (1984) S(T,X) = O(X11/48+T )= ⇒ O(x35/48+) in PGT Luo–Sarnak (1995) Applications of Landau’s Formula

Exponential Sum

1 2 Let λj = 4 + tj be the eigenvalues of ∆ For X > 1, define X S(T,X) = Xitj

|tj |≤T

Essential in the proof of Prime Geodesic Theorem Trivial estimate (Weyl’s Law) S(T,X) = O(T 2) Iwaniec (1984) S(T,X) = O(X11/48+T )= ⇒ O(x35/48+) in PGT Luo–Sarnak (1995) S(T,X) = O(X1/8T 5/4(log T )2)= ⇒ O(x7/10+) Difficult since no analogue of Gauss’ geometric method! Define (for smooth, compactly supported f) Z Nf (X) = f(z)N(z, z, X) dµ(z) Γ\H Apply pre-trace formula to a mollified version of

χ[0,(X−2)/4](u)

S(T,X) appears when estimating the contribution of the discrete spectrum

Applications of Landau’s Formula

Hyperbolic Lattice Counting

−1 X Let N(z, w, X) = #{γ ∈ Γ: γz ∈ Bw(cosh 2 )} Define (for smooth, compactly supported f) Z Nf (X) = f(z)N(z, z, X) dµ(z) Γ\H Apply pre-trace formula to a mollified version of

χ[0,(X−2)/4](u)

S(T,X) appears when estimating the contribution of the discrete spectrum

Applications of Landau’s Formula

Hyperbolic Lattice Counting

−1 X Let N(z, w, X) = #{γ ∈ Γ: γz ∈ Bw(cosh 2 )} Difficult since no analogue of Gauss’ geometric method! Apply pre-trace formula to a mollified version of

χ[0,(X−2)/4](u)

S(T,X) appears when estimating the contribution of the discrete spectrum

Applications of Landau’s Formula

Hyperbolic Lattice Counting

−1 X Let N(z, w, X) = #{γ ∈ Γ: γz ∈ Bw(cosh 2 )} Difficult since no analogue of Gauss’ geometric method! Define (for smooth, compactly supported f) Z Nf (X) = f(z)N(z, z, X) dµ(z) Γ\H S(T,X) appears when estimating the contribution of the discrete spectrum

Applications of Landau’s Formula

Hyperbolic Lattice Counting

−1 X Let N(z, w, X) = #{γ ∈ Γ: γz ∈ Bw(cosh 2 )} Difficult since no analogue of Gauss’ geometric method! Define (for smooth, compactly supported f) Z Nf (X) = f(z)N(z, z, X) dµ(z) Γ\H Apply pre-trace formula to a mollified version of

χ[0,(X−2)/4](u) Applications of Landau’s Formula

Hyperbolic Lattice Counting

−1 X Let N(z, w, X) = #{γ ∈ Γ: γz ∈ Bw(cosh 2 )} Difficult since no analogue of Gauss’ geometric method! Define (for smooth, compactly supported f) Z Nf (X) = f(z)N(z, z, X) dµ(z) Γ\H Apply pre-trace formula to a mollified version of

χ[0,(X−2)/4](u)

S(T,X) appears when estimating the contribution of the discrete spectrum Also for hyperbolic lattice counting (on average) O(x1/2+)

Applications of Landau’s Formula

The Conjecture

Conjecture (Petridis–Risager 2014) For X > 1, 1+ε ε S(T,X) ε T X .

This implies the best possible error term in PGT O(x3/4+) Applications of Landau’s Formula

The Conjecture

Conjecture (Petridis–Risager 2014) For X > 1, 1+ε ε S(T,X) ε T X .

This implies the best possible error term in PGT O(x3/4+) Also for hyperbolic lattice counting (on average) O(x1/2+) Theorem 3 (L 2014) For a fixed X > 1,

|F | sin(T log X) T S(T,X) = T + (X1/2 − X−1/2)−1Λ (X) π log X π Γ 2T  T  + X−1/2Λ(X1/2) + O π log T

Applications of Landau’s Formula

Landau-type Formula ( log(N(p)), if X = N(p)`, ` ∈ N, Let ΛΓ(X) = 0, otherwise. Applications of Landau’s Formula

Landau-type Formula ( log(N(p)), if X = N(p)`, ` ∈ N, Let ΛΓ(X) = 0, otherwise.

Theorem 3 (L 2014) For a fixed X > 1,

|F | sin(T log X) T S(T,X) = T + (X1/2 − X−1/2)−1Λ (X) π log X π Γ 2T  T  + X−1/2Λ(X1/2) + O π log T |F | Z ∞ = h(r)r tanh(πr) dr 4π −∞ ∞ X X −1 + N(p)`/2 − N(p)−`/2
g(` log p) log p p `=1 X X π` Z ∞ cosh π(1 − 2`/m)r + (2m sin )−1 h(r) dr m cosh πr R 0<`

Applications of Landau’s Formula

Proof By Selberg Trace Formula:

X 1 Z ∞ −ϕ0 1  h(tj) + h(r) + ir dr 4π −∞ ϕ 2 tj >0 ∞ X X −1 + N(p)`/2 − N(p)−`/2
g(` log p) log p p `=1 X X π` Z ∞ cosh π(1 − 2`/m)r + (2m sin )−1 h(r) dr m cosh πr R 0<`

Applications of Landau’s Formula

Proof By Selberg Trace Formula:

X 1 Z ∞ −ϕ0 1  h(tj) + h(r) + ir dr 4π −∞ ϕ 2 tj >0 |F | Z ∞ = h(r)r tanh(πr) dr 4π −∞ X X π` Z ∞ cosh π(1 − 2`/m)r + (2m sin )−1 h(r) dr m cosh πr R 0<`

Applications of Landau’s Formula

Proof By Selberg Trace Formula:

X 1 Z ∞ −ϕ0 1  h(tj) + h(r) + ir dr 4π −∞ ϕ 2 tj >0 |F | Z ∞ = h(r)r tanh(πr) dr 4π −∞ ∞ X X −1 + N(p)`/2 − N(p)−`/2
g(` log p) log p p `=1 + trash...

Applications of Landau’s Formula

Proof By Selberg Trace Formula:

X 1 Z ∞ −ϕ0 1  h(tj) + h(r) + ir dr 4π −∞ ϕ 2 tj >0 |F | Z ∞ = h(r)r tanh(πr) dr 4π −∞ ∞ X X −1 + N(p)`/2 − N(p)−`/2
g(` log p) log p p `=1 X X π` Z ∞ cosh π(1 − 2`/m)r + (2m sin )−1 h(r) dr m cosh πr R 0<`

Proof By Selberg Trace Formula:

X 1 Z ∞ −ϕ0 1  h(tj) + h(r) + ir dr 4π −∞ ϕ 2 tj >0 |F | Z ∞ = h(r)r tanh(πr) dr 4π −∞ ∞ X X −1 + N(p)`/2 − N(p)−`/2
g(` log p) log p p `=1 X X π` Z ∞ cosh π(1 − 2`/m)r + (2m sin )−1 h(r) dr m cosh πr R 0<`

Applications of Landau’s Formula

Proof (cont.)

ir −ir Set h(r) = (χ[−T,T ] ∗ ψ)(r)(X + X ) Trickiest term to estimate is the contribution of continuous spectrum However, after a suitable change of contour we can use an argument similar to Landau’s original one

Applications of Landau’s Formula

Proof (cont.)

ir −ir Set h(r) = (χ[−T,T ] ∗ ψ)(r)(X + X ) P Balance between smoothing h(tj) and other error terms However, after a suitable change of contour we can use an argument similar to Landau’s original one

Applications of Landau’s Formula

Proof (cont.)

ir −ir Set h(r) = (χ[−T,T ] ∗ ψ)(r)(X + X ) P Balance between smoothing h(tj) and other error terms Trickiest term to estimate is the contribution of continuous spectrum Applications of Landau’s Formula

Proof (cont.)

ir −ir Set h(r) = (χ[−T,T ] ∗ ψ)(r)(X + X ) P Balance between smoothing h(tj) and other error terms Trickiest term to estimate is the contribution of continuous spectrum However, after a suitable change of contour we can use an argument similar to Landau’s original one We would like an actual Landau formula! Generalisation to other spaces?

Applications of Landau’s Formula

Remarks (Theorem 3) Generalisation to other spaces?

Applications of Landau’s Formula

Remarks (Theorem 3)

We would like an actual Landau formula! Applications of Landau’s Formula

Remarks (Theorem 3)

We would like an actual Landau formula! Generalisation to other spaces? Applications of Landau’s Formula

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