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 jαj ≤ 2π log T , we have X 1 2 jζ( 2 + i(γ + 2πα= log T )j = 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 ! 1 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 2 S then jZF (T )∆ZG(T )j = o(T ) =) F = G; where ZF (T ) is the set of zeros of F (s) in Re(s) ≥ 1=2, j 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; )L(s; χ) are distinct. R. Murty and K. Murty (1994): If F; G 2 S then jZF (T )∆ZG(T )j = o(T ) =) F = G; where ZF (T ) is the set of zeros of F (s) in Re(s) ≥ 1=2, j 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; )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; )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 2 S then jZF (T )∆ZG(T )j = o(T ) =) F = G; where ZF (T ) is the set of zeros of F (s) in Re(s) ≥ 1=2, j Im(s) ≤ T , ∆ is the symmetric difference. Let χ, be real, primitive and non-principal 1 characters with distinct prime moduli q and `. Fix σ 2 ( 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; ) 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 ( 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; ) 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, be real and non-principal 1 characters with distinct prime moduli q and `. Fix σ 2 ( 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; ) 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, be real, primitive 1 Fix σ 2 ( 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; ) 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, 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; ) 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, be real, primitive and non-principal 1 characters with distinct prime moduli q and `. Fix σ 2 ( 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, be real, primitive and non-principal 1 characters with distinct prime moduli q and `. Fix σ 2 ( 2 ; 1).