Upper bounds for the moments of derivatives of Dirichlet L-functions

Sono, Keiju

doi:10.2478/s11533-013-0382-x

Cited by 2 publications

(4 citation statements)

References 15 publications

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“…3 together with Lemma 2.6 give directly the following corollary. For a Dirichlet character χ modulo q such that χ 2 = χ 0 , the inequalitylog |L(σ + it, χ)| Re p x χ( p) p σ +λ/log x+it log(x/ p) log x + λ) 2 log q + log + T log x + O(log log log q)holds uniformly for |t| T < log A q and 1/2 σ 1/2 + λ/log x.The next result was proved by Sono[20, Lemma 4.3] and is a q-analogue of [21,Lemma 3]. Suppose that x 2 and k is an integer such that x k < q.…”

mentioning

confidence: 72%

“…5]) that the moments at the central point satisfy the asymptotic formulas

\begin{matrix} true \\ right M_{2 k} (q) = \sum_{χ \in X_{q}^{*}} {| L (1 / 2, χ) |}^{2 k} \sim C_{k} q {log}^{k^{2}} q, C_{k} > 0 . \end{matrix}

Even though the asymptotic formulas are not known for

k ⩾ 3

, lower bounds of the expected order of magnitude

\begin{matrix} true \\ right \sum_{χ \in X_{q}^{*}} {| L (1 / 2, χ) |}^{2 k} ≫ q {log}^{k^{2}} q \end{matrix}

have been given by Rudnick and Soundararajan [18] for

q

prime. Assuming the generalized Riemann hypothesis (GRH), Soundararajan mentioned in [21] that we could derive the upper bound

M_{2 k} (q) ≪ q {log}^{k^{2} + ε} q

(indeed, it follows from the work of Sono [20]). We can generalize, in some way, these moments using shifts and consider

\begin{matrix} true \\ right \sum_{χ \in X_{q}^{*}} L (() \frac{1}{2} + i t_{1}, χ) \dots L (() \frac{1}{2} + i t_{2 k}, χ), \end{matrix}

where

(t_{1}, \dots, t_{2 k})

is a sequence of real numbers.…”

Section: Introductionmentioning

confidence: 99%

“…Even though the asymptotic formulas are not known for k 3, lower bounds of the expected order of magnitude χ∈X * q |L(1/2, χ)| 2k q log k 2 q have been given by Rudnick and Soundararajan [18] for q prime. Assuming the generalized Riemann hypothesis (GRH), Soundararajan mentioned in [21] that we could derive the upper bound M 2k (q) q log k 2 + q (indeed, it follows from the work of Sono [20]). We can generalize, in some way, these moments using shifts and consider χ∈X * q L 1 2 + it 1 , χ · · · L 1 2 + it 2k , χ , (1.2) where (t 1 , .…”

mentioning

confidence: 99%

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Shifted Moments of ‐functions and Moments of Theta Functions

Munsch¹

2016

Mathematika

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mentioning

confidence: 72%

“…5]) that the moments at the central point satisfy the asymptotic formulas

\begin{matrix} true \\ right M_{2 k} (q) = \sum_{χ \in X_{q}^{*}} {| L (1 / 2, χ) |}^{2 k} \sim C_{k} q {log}^{k^{2}} q, C_{k} > 0 . \end{matrix}

Even though the asymptotic formulas are not known for

k ⩾ 3

, lower bounds of the expected order of magnitude

\begin{matrix} true \\ right \sum_{χ \in X_{q}^{*}} {| L (1 / 2, χ) |}^{2 k} ≫ q {log}^{k^{2}} q \end{matrix}

have been given by Rudnick and Soundararajan [18] for

q

prime. Assuming the generalized Riemann hypothesis (GRH), Soundararajan mentioned in [21] that we could derive the upper bound

M_{2 k} (q) ≪ q {log}^{k^{2} + ε} q

(indeed, it follows from the work of Sono [20]). We can generalize, in some way, these moments using shifts and consider

\begin{matrix} true \\ right \sum_{χ \in X_{q}^{*}} L (() \frac{1}{2} + i t_{1}, χ) \dots L (() \frac{1}{2} + i t_{2 k}, χ), \end{matrix}

where

(t_{1}, \dots, t_{2 k})

is a sequence of real numbers.…”

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Shifted Moments of ‐functions and Moments of Theta Functions

Munsch¹

2016

Mathematika

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show abstract

“…Since the detailed history can be found in many papers and textbooks (for example, see [2,4,[6][7][8][16][17][18][19]), here we introduce only some of recent results. As a lower bound, by an elegant argument using Sylvester's sequences, Radziwiłł and Soundararajan [13] obtained so called continuous lower bound 4 T (log T ) k 2 for all k > 1.…”

Section: Introductionmentioning

confidence: 99%