Cotangent sums, quantum modular forms, and the generalized Riemann hypothesis

Lewis, James B.; Zagier, Don

doi:10.1007/s40687-018-0159-8

Cited by 9 publications

(8 citation statements)

References 11 publications

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“…We now precisely state our main results, introduced above, extending work in [4, 11], and relating to work in [12]. In what follows, we let

0 < h < k

with

gcd (h, k) = 1

(as the case of

(h, k) = (0, 1)

was treated in [4]).…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…Our second main result in this paper leads to what may essentially be viewed as an infinite family of (true) quantum modular forms, arising from period functions of our twisted Eisenstein series E ± s (h, k; τ ) (see Theorem 2 for a precise statement). We now precisely state our main results, introduced above, extending work in [4,11], and relating to work in [12]. In what follows, we let 0 < h < k with gcd(h, k) = 1 (as the case of (h, k) = (0, 1) was treated in [4]).…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…Results by Bettin and Conrey on such period functions in [4], and later related work by Lewis and Zagier in [12] which gives equivalent criteria for the Riemann hypothesis in this context, are additionally motivated by connections to quantum modular forms. Loosely speaking, quantum modular forms are functions

f (x)

defined for

x \in Q

, and whose errors to modularity there, namely the functions

h_{γ, κ} false(x false) : = f false(x false) - χ_{γ} {(C x + D)}^{- κ} f false(γ x false)

, where

γ = false(\begin{matrix} A & B \\ C & D \end{matrix} false) \in S L_{2} false(double-struckZ false)

κ \in \frac{1}{2} Z

, and

false| χ_{γ} false| = 1

, extend to suitably analytic or continuous functions in

double-struckR

.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

See 2 more Smart Citations

Twisted Eisenstein series, cotangent‐zeta sums, and quantum modular forms

Folsom

2020

Trans. London Math. Soc.

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“…We now precisely state our main results, introduced above, extending work in [4, 11], and relating to work in [12]. In what follows, we let

0 < h < k

with

gcd (h, k) = 1

(as the case of

(h, k) = (0, 1)

was treated in [4]).…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

f (x)

defined for

x \in Q

, and whose errors to modularity there, namely the functions

h_{γ, κ} false(x false) : = f false(x false) - χ_{γ} {(C x + D)}^{- κ} f false(γ x false)

, where

γ = false(\begin{matrix} A & B \\ C & D \end{matrix} false) \in S L_{2} false(double-struckZ false)

κ \in \frac{1}{2} Z

, and

false| χ_{γ} false| = 1

, extend to suitably analytic or continuous functions in

double-struckR

.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Twisted Eisenstein series, cotangent‐zeta sums, and quantum modular forms

Folsom

2020

Trans. London Math. Soc.

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“…In order to prove Theorem 1.6 (1), we adapt the proof of Proposition 10 in [12] which pertained to a similar function. We first note that we can extend G k (z) to C \ R by defining…”

Section: Proofs Of the Theoremsmentioning

confidence: 99%

Eichler integrals of Eisenstein series as $q$-brackets of weighted $t$-hook functions on partitions

Bringmann¹,

Ono²,

Wagner³

2020

Preprint

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We consider the t-hook functions on partitions f a,t : P → C defined bywhere H t (λ) is the multiset of partition hook numbers that are multiples of t. The Bloch-Okounkov q-brackets f a,t q include Eichler integrals of the classical Eisenstein series. For even a ≥ 2, we show that these q-brackets are natural pieces of weight 2 − a sesquiharmonic and harmonic Maass forms, while for odd a ≤ −1, we show that they are holomorphic quantum modular forms. We use these results to obtain new formulas of Chowla-Selberg type, and asymptotic expansions involving values of the Riemann zeta-function and Bernoulli numbers. We make use of work of Berndt, Han and Ji, and Zagier.f q := λ∈P f (λ)q |λ| λ∈P q |λ| ∈ C[[q]],

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“…Suitable modifications can be made to restrict the domain of r γ to appropriate subsets of Q and allow both multiplier systems and transformations on subgroups of SL 2 (Z). Since their inception, there has been substantial interest in studying these modular objects which emerge in diverse contexts: Maass forms [9], supersymmetric quantum field theory [12], topological invariants for plumbed 3-manifolds [5], [10], [11], combinatorics [13], [20], unified Witten-Reshetikhin-Turaev invariants [19] and L-functions [24], [26]. For more examples, see Chapter 21 in [4].…”

Section: Introductionmentioning

confidence: 99%

Quantum modularity of partial theta series with periodic coefficients

Goswami¹,

Osburn²

2020

Preprint

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We explicitly prove the quantum modularity of partial theta series with even or odd periodic coefficients. As an application, we show that the Kontsevich-Zagier series Ft(q) which matches (at a root of unity) the colored Jones polynomial for the family of torus knots T (3, 2 t ), t ≥ 2, is a weight 3/2 quantum modular form. This generalizes Zagier's result on the quantum modularity for the "strange" series F (q).

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Cotangent sums, quantum modular forms, and the generalized Riemann hypothesis

Cited by 9 publications

References 11 publications

Twisted Eisenstein series, cotangent‐zeta sums, and quantum modular forms

Twisted Eisenstein series, cotangent‐zeta sums, and quantum modular forms

Eichler integrals of Eisenstein series as $q$-brackets of weighted $t$-hook functions on partitions

Quantum modularity of partial theta series with periodic coefficients

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