On the splitting conjecture in the hybrid model for the Riemann zeta function

Heap, Winston

doi:10.48550/arxiv.2102.02092

Cited by 2 publications

(2 citation statements)

References 34 publications

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“…On [GHK07, p. 511], it is suggested that their splitting conjecture holds for a much wider range of X and T with X = o(T ). Recently, Heap [Hea21] has justified this suggestion, proving on RH that the splitting conjecture for ζ(s) holds to order for a much wider range of X, and establishing the splitting conjecture for k = 1 and k = 2 for wider ranges both with and without RH.…”

Section: Introductionmentioning

confidence: 95%

Moments of the Hurwitz zeta function on the critical line

Sahay¹

2021

Preprint

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We study the moments M k (T ; α) = 2T T |ζ(s, α)| 2k dt of the Hurwitz zeta function ζ(s, α) on the critical line, s = 1/2+it.We conjecture, in analogy with the Riemann zeta function, that M k (T ; α) ∼ c k (α)T (log T ) k 2 . In the case of α ∈ Q, we use heuristics from analytic number theory and random matrix theory to compute c k (α). In the process, we investigate moments of products of Dirichlet L-functions on the critical line. We provide several pieces of evidence for our conjectures, in particular by proving some of them for the cases k = 1, 2 and α ∈ Q.

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Section: Introductionmentioning

confidence: 95%

Moments of the Hurwitz zeta function on the critical line

Sahay¹

2021

Preprint

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“…On [17, p. 511], it is suggested that this splitting conjecture holds for a much wider range of X and T with X = o(T). Recently, Heap [24] has justified this suggestion. He proved on RH that the splitting conjecture for ζ (s) holds for every k > 0 and a much wider range of X provided one requires only an order of magnitude result, instead of an asymptotic.…”

Section: Anurag Sahaymentioning

confidence: 99%

Moments of the Hurwitz zeta function on the critical line

Sahay¹

2022

Math. Proc. Camb. Phil. Soc.

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We study the moments $M_k(T;\,\alpha) = \int_T^{2T} |\zeta(s,\alpha)|^{2k}\,dt$ of the Hurwitz zeta function $\zeta(s,\alpha)$ on the critical line, $s = 1/2 + it$ with a rational shift $\alpha \in \mathbb{Q}$ . We conjecture, in analogy with the Riemann zeta function, that $M_k(T;\,\alpha) \sim c_k(\alpha) T (\!\log T)^{k^2}$ . Using heuristics from analytic number theory and random matrix theory, we conjecturally compute $c_k(\alpha)$ . In the process, we investigate moments of products of Dirichlet L-functions on the critical line. We prove some of our conjectures for the cases $k = 1,2$ .

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On the splitting conjecture in the hybrid model for the Riemann zeta function

Cited by 2 publications

References 34 publications

Moments of the Hurwitz zeta function on the critical line

Moments of the Hurwitz zeta function on the critical line

Moments of the Hurwitz zeta function on the critical line

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