I NVESTIGATING 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 + i γ denote a nontrivial zero of ζ ( s ) , and consider the sequence of ordinates of zeros in the critical strip: 0 < γ 1 ≤ γ 2 ≤ . . . ≤ γ n ≤ γ n + 1 ≤ . . . . 1 / 14
Let ρ = β + i γ denote a nontrivial zero of ζ ( s ) , and consider the sequence of ordinates of zeros in the critical strip: 0 < γ 1 ≤ γ 2 ≤ . . . ≤ γ n ≤ γ n + 1 ≤ . . . . Since 1 ∼ T � N ( T ) := 2 π log T , ρ 0 <γ< T 1 / 14
Let ρ = β + i γ denote a nontrivial zero of ζ ( s ) , and consider the sequence of ordinates of zeros in the critical strip: 0 < γ 1 ≤ γ 2 ≤ . . . ≤ γ n ≤ γ n + 1 ≤ . . . . Since 1 ∼ T � N ( T ) := 2 π log T , ρ 0 <γ< T 2 π the average size of γ n + 1 − γ n is log ( γ n ) . 1 / 14
Let γ n + 1 − γ n γ n + 1 − γ n µ := lim inf and λ := lim sup ( 2 π/ log γ n ) . ( 2 π/ log γ n ) n →∞ n →∞ 2 / 14
Let γ n + 1 − γ n γ n + 1 − γ n µ := lim inf and λ := lim sup ( 2 π/ log γ n ) . ( 2 π/ log γ n ) n →∞ n →∞ By definition, we have µ ≤ 1 ≤ λ, 2 / 14
Let γ n + 1 − γ n γ n + 1 − γ n µ := lim inf and λ := lim sup ( 2 π/ log γ n ) . ( 2 π/ log γ n ) n →∞ n →∞ By definition, we have µ ≤ 1 ≤ λ, and we expect µ = 0 λ = ∞ . and 2 / 14
Let γ n + 1 − γ n γ n + 1 − γ n µ := lim inf and λ := lim sup ( 2 π/ log γ n ) . ( 2 π/ log γ n ) n →∞ n →∞ By definition, we have µ ≤ 1 ≤ λ, and we expect µ = 0 λ = ∞ . and C ONJECTURE Gaps between consecutive zeros of ζ ( s ) that are arbitrarily small/large, relative to the average gap size, appear infinitely often. 2 / 14
W HAT 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 2 . 6306 ∗ R.R. Hall Bui, Milinovich & Ng 2 . 6950 0 . 5155 Feng & Wu 2 . 7327 0 . 5154 2 . 766 ∗ Bredberg 3 . 18 Bui 0 . 515396 Preobrazhenskii a ∗ Results are unconditional, but RH must be assumed to give lower bound on λ . a ∗∗ preprint 2 / 14
W HAT 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 2 . 6306 ∗ R.R. Hall Bui, Milinovich & Ng 2 . 6950 0 . 5155 Feng & Wu 2 . 7327 0 . 5154 2 . 766 ∗ Bredberg 3 . 18 Bui 0 . 515396 Preobrazhenskii a ∗ Results are unconditional, but RH must be assumed to give lower bound on λ . a ∗∗ preprint 2 / 14
W HAT 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 2 . 6306 ∗ R.R. Hall Bui, Milinovich & Ng 2 . 6950 0 . 5155 Feng & Wu 2 . 7327 0 . 5154 2 . 766 ∗ Bredberg 3 . 18 Bui 0 . 515396 Preobrazhenskii a ∗ Results are unconditional, but RH must be assumed to give lower bound on λ . a ∗∗ preprint 2 / 14
W HAT IS KNOWN ? ( ASSUMING RH) lower bound for λ upper bound for µ Gonek/Mueller 1 . 9 1 . 9799 0 . 5179 Montgomery & Odlyzko 2 . 3378 0 . 5172 Conrey, Ghosh & Gonek 2 . 6306 ∗ Hall Bui, Milinovich & Ng 2 . 6950 0 . 5155 Feng & Wu 2 . 7327 0 . 5154 2 . 766 ∗ Bredberg Bui 3 . 18 Preobrazhenskii 0 . 515396 a ∗ Results are unconditional; must assume RH for lower bound on λ . a ∗∗ preprint 2 / 14
W HAT IS KNOWN ? ( ASSUMING RH) lower bound for λ upper bound for µ Gonek/Mueller 1 . 9 1 . 9799 0 . 5179 Montgomery & Odlyzko 2 . 3378 0 . 5172 Conrey, Ghosh & Gonek 2 . 6306 ∗ Hall Bui, Milinovich & Ng 2 . 6950 0 . 5155 Feng & Wu 2 . 7327 0 . 5154 2 . 766 ∗ Bredberg Bui 3 . 18 Preobrazhenskii 0 . 515396 a ∗ Results are unconditional; must assume RH for lower bound on λ . a ∗∗ preprint 2 / 14
W HAT IS KNOWN ? ( ASSUMING RH) lower bound for λ upper bound for µ Gonek/Mueller 1 . 9 1 . 9799 0 . 5179 Montgomery & Odlyzko 2 . 3378 0 . 5172 Conrey, Ghosh & Gonek 2 . 6306 ∗ Hall Bui, Milinovich & Ng 2 . 6950 0 . 5155 Feng & Wu 2 . 7327 0 . 5154 2 . 766 ∗ Bredberg Bui 3 . 18 Preobrazhenskii 0 . 515396 a ∗ Results are unconditional; must assume RH for lower bound on λ . a ∗∗ preprint 2 / 14
W HAT IS KNOWN ? ( ASSUMING RH) lower bound for λ upper bound for µ Gonek/Mueller 1 . 9 1 . 9799 0 . 5179 Montgomery & Odlyzko 2 . 3378 0 . 5172 Conrey, Ghosh & Gonek 2 . 6306 ∗ Hall Bui, Milinovich & Ng 2 . 6950 0 . 5155 Feng & Wu 2 . 7327 0 . 5154 2 . 766 ∗ Bredberg Bui 3 . 18 Preobrazhenskii 0 . 515396 a ∗ Results are unconditional; must assume RH for lower bound on λ . a ∗∗ preprint 2 / 14
W HAT IS KNOWN ? ( ASSUMING RH) lower bound for λ upper bound for µ Gonek/Mueller 1 . 9 1 . 9799 0 . 5179 Montgomery & Odlyzko 2 . 3378 0 . 5172 Conrey, Ghosh & Gonek 2 . 6306 ∗ Hall Bui, Milinovich & Ng 2 . 6950 0 . 5155 Feng & Wu 2 . 7327 0 . 5154 2 . 766 ∗ Bredberg Bui ∗ , ∗∗ 3 . 18 Preobrazhenskii ∗∗ 0 . 515396 a ∗ Results are unconditional; must assume RH for lower bound on λ . a ∗∗ preprint 2 / 14
W HAT IS KNOWN ? ( ASSUMING RH) lower bound for λ upper bound for µ Gonek/Mueller 1 . 9 1 . 9799 0 . 5179 Montgomery & Odlyzko 2 . 3378 0 . 5172 Conrey, Ghosh & Gonek 2 . 6306 ∗ Hall Bui, Milinovich & Ng 2 . 6950 0 . 5155 Feng & Wu 2 . 7327 0 . 5154 2 . 766 ∗ Bredberg Bui ∗ , ∗∗ 3 . 18 Preobrazhenskii ∗∗ 0 . 515396 a ∗ Results are unconditional; must assume RH for lower bound on λ . a ∗∗ preprint 2 / 14
G APS BETWEEN ZEROS OF OTHER L - FUNCTIONS • For large gaps, we consider the following degree 2 L -functions: • ζ K ( 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 3 / 14
G APS BETWEEN ZEROS OF OTHER L - FUNCTIONS • For large gaps, we consider the following degree 2 L -functions: • ζ K ( 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. 3 / 14
G APS BETWEEN ZEROS OF OTHER L - FUNCTIONS • For large gaps, we consider the following degree 2 L -functions: • ζ K ( 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 C ONJECTURE Gaps between consecutive zeros that are arbitrarily large, relative to the average gap size, appear infinitely often for both ζ K ( s ) and L ( s , f ) . 3 / 14
L ARGE GAPS BETWEEN ZEROS OF ζ K ( s ) , L ( s , f ) Theorem (T., 2014) Assuming GRH for ζ K ( s ) , we have λ K ≥ 2 . 449 . 4 / 14
L ARGE GAPS BETWEEN ZEROS OF ζ K ( s ) , L ( s , f ) Theorem (T., 2014) Assuming GRH for ζ K ( s ) , we have λ K ≥ 2 . 449 . Theorem (Bui, Heap, T., 2014 (preprint)) Assuming GRH for ζ K ( s ) , we have λ K ≥ 2 . 866 . 4 / 14
L ARGE GAPS BETWEEN ZEROS OF ζ K ( s ) , L ( s , f ) Theorem (T., 2014) Assuming GRH for ζ K ( s ) , we have λ K ≥ 2 . 449 . Theorem (Bui, Heap, T., 2014 (preprint)) Assuming GRH for ζ K ( s ) , we have λ K ≥ 2 . 866 . Theorem (Barrett, McDonald, Miller, Ryan, T., Winsor, 2015) Assuming GRH for L ( s , f ) , we have λ f ≥ 1 . 732 . These results can be stated unconditionally if we restrict our attention to zeros on the critical line. 4 / 14
L ARGE G APS - H ALL ’ S M ETHOD ( MODIFIED BY B REDBERG ) Wirtinger’s Inequality Let g : [ a , b ] → C be continuously differentiable and suppose that g ( a ) = g ( b ) = 0 . Then � b � 2 � b � b − a | g ( t ) | 2 dt ≤ | g ′ ( t ) | 2 dt . π a a 5 / 14
L ARGE G APS - H ALL ’ S M ETHOD ( MODIFIED BY B REDBERG ) Wirtinger’s Inequality Let g : [ a , b ] → C be continuously differentiable and suppose that g ( a ) = g ( b ) = 0 . Then � b � 2 � b � b − a | g ( t ) | 2 dt ≤ | g ′ ( t ) | 2 dt . π a a By understanding the mean-values of g ( t ) and g ′ ( t ) , we can obtain a lower bound on gaps between zeros of g ( t ) . 5 / 14
I DEA OF ARGUMENT FOR ζ K ( s ) Let g ( t ) := exp ( i ν L t ) ζ K ( 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 6 / 14
I DEA OF ARGUMENT FOR ζ K ( s ) Let g ( t ) := exp ( i ν L t ) ζ K ( 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 d r ( h 1 ) d r ( h 2 ) χ d ( h 2 ) P [ h 1 h 2 ] � M ( s ) = ( h 1 h 2 ) s h 1 h 2 ≤ y where y = T θ , 0 < θ < 1 / 4 , and d r ( h ) denotes the coefficients of ζ ( s ) r . 6 / 14
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