Simple zeros of modular <i>L</i>
-functions

Milinovich, Micah B.; Ng, Nathan

doi:10.1112/plms/pdu041

Cited by 8 publications

(6 citation statements)

References 56 publications

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“…Proof. This follows from the direct analogue of Lemma 8.1 in [18]; we provide the details here for the sake of completeness. By Cauchy's integral formula, we have…”

Section: Introductionmentioning

confidence: 96%

“…Proof This follows from the direct analogue of [, Lemma 8.1]; we provide the details here for the sake of completeness. By Cauchy's integral formula, we have

\sum_{0 < Im (ρ) \leq T} {| ζ^{(ν)} false(ρ false) |}^{2 k} = {((), \frac{ν!}{2 π})}^{2 k} \sum_{0 < Im (ρ) \leq T} {(||, \int_{C}, \frac{ζ (ρ + α)}{α^{ν + 1}}, d, α)}^{2 k},

where C is the positively oriented circle of radius

{false(\log T false)}^{- 1}

centered at the origin.…”

Section: Introductionmentioning

confidence: 99%

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An Upper Bound for Discrete Moments of the Derivative of the Riemann Zeta‐function

Kirila

2020

Mathematika

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Assuming the Riemann hypothesis, we establish an upper bound for the 2kth discrete moment of the derivative of the Riemann zeta-function at nontrivial zeros, where k is a positive real number. Our upper bound agrees with conjectures of Gonek and Hejhal and of Hughes, Keating and O'Connell. This sharpens a result of Milinovich. Our proof builds upon a method of Adam Harper concerning continuous moments of the zeta-function on the critical line. London under an exclusive licence. 2 . Hughes et al. inserted the product over primes here was inserted in an ad hoc manner, namely from a heuristic estimate for the case k = −1/2. Bui et al. [1] used a hybrid Euler-Hadamard product model for ζ (ρ) to suggest precisely where this product over primes comes from, essentially merging ideas from number theory and random matrix theory in the same way as Gonek et al. [11] did for moments of ζ ( 1 2 + it ). These conjectures remain open, but work has been done toward the implied upper and lower bounds conditionally on RH. Milinovich and Ng [17] obtained the expected lower boundfor any natural number k. In the other direction, Milinovich [16] showed thatfor any ε > 0. The purpose of this paper is to remove the ε in the exponent here and prove the following. THEOREM 1.1. Assume RH. Let k > 0. ThenTogether with (1.2), this shows thatCompleting the square then leads us to conclude that

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“…Proof. This follows from the direct analogue of Lemma 8.1 in [18]; we provide the details here for the sake of completeness. By Cauchy's integral formula, we have…”

Section: Introductionmentioning

confidence: 96%

“…Proof This follows from the direct analogue of [, Lemma 8.1]; we provide the details here for the sake of completeness. By Cauchy's integral formula, we have

\sum_{0 < Im (ρ) \leq T} {| ζ^{(ν)} false(ρ false) |}^{2 k} = {((), \frac{ν!}{2 π})}^{2 k} \sum_{0 < Im (ρ) \leq T} {(||, \int_{C}, \frac{ζ (ρ + α)}{α^{ν + 1}}, d, α)}^{2 k},

where C is the positively oriented circle of radius

{false(\log T false)}^{- 1}

centered at the origin.…”

Section: Introductionmentioning

confidence: 99%

An Upper Bound for Discrete Moments of the Derivative of the Riemann Zeta‐function

Kirila

2020

Mathematika

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“…For example, the Riemann-von Mangoldt explicit formula, which is an asymptotic expansion of the prime-counting function, involves a sum over the non-trivial zeros of the Riemann zeta function [5]. The Riemann hypothesis, which states that all non-trivial zeros lie on the critical line ℜpsq " 1{2, has many implications including for the accurary of the error estimate of the primenumber theorem and also for a number of conjectures such as the Lindelöf hypothesis [2], the study of modular L-functions [6], and the inverse spectral problem for fractal strings [4].…”

Section: Introductionmentioning

confidence: 99%

A lower bound for the modulus of the Dirichlet eta function on partition $\mathcal{P}$ from 2-D principal component analysis

Heymann¹

2020

Preprint

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The aim of the present manuscript is to derive an expression for the lower bound of the modulus of the Dirichlet eta function on vertical lines ℜpsq " α. An approach based on a two-dimensional principal component analysis matching the dimensions of the complex plane, which is built on a parametric ellipsoidal shape, has been undertaken to achieve this result. This lower bound, which is expressed as @s P C s.t. ℜpsq ě 1 2 , |ηpsq| ě 1 ´?2 2 α where η is the Dirichlet eta function, has implications for the Riemann hypothesis as |ηpsq| ą 0 for any s P C such that ℜpsq P s 1 2 , 1r.

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“…The result can be generalised to modular forms on congruence subgroups of SL 2 (Z). Besides, the method can be used to improve the result of Milinovich and Turnage-Butterbaugh [17] and the work of Milinovich and Ng [16], but the Generalised Ramanujan Conjecture is needed for the case GL(m).…”

Section: Introductionmentioning

confidence: 99%

Higher integral moments of automorphic L-functions in short intervals

Xiao

2017

Journal of Number Theory

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Simple zeros of modular L -functions

Abstract: Abstract. Let f be a primitive holomorphic cusp form of weight k, level q, and character χ, and let L(s, f ) be its associated L-function. Assuming the generalized Riemann hypothesis for L(s, f ), we prove that the number of simple zerosfor any ε > 0 and T sufficiently large.

Cited by 8 publications

References 56 publications

An Upper Bound for Discrete Moments of the Derivative of the Riemann Zeta‐function

An Upper Bound for Discrete Moments of the Derivative of the Riemann Zeta‐function

A lower bound for the modulus of the Dirichlet eta function on partition $\mathcal{P}$ from 2-D principal component analysis

Higher integral moments of automorphic L-functions in short intervals

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