Shifted Moments of ‐functions and Moments of Theta Functions

Munsch, Marc

doi:10.1112/s0025579316000218

Cited by 13 publications

(26 citation statements)

References 20 publications

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“…The proof of Theorem 1.6 is given in Section 2.5. Under the assumption of the Generalized Riemann Hypothesis, Munsch [22] obtains the following upper bounds…”

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confidence: 99%

Upper and Lower Bounds for Higher Moments of Theta Functions

Munsch

Shparlinski

2016

Q J Math

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“…The proof of Theorem 1.6 is given in Section 2.5. Under the assumption of the Generalized Riemann Hypothesis, Munsch [22] obtains the following upper bounds…”

mentioning

confidence: 99%

Upper and Lower Bounds for Higher Moments of Theta Functions

Munsch

Shparlinski

2016

Q J Math

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“…As initiated in previous works [28–30, 32], we aim at similar results for moments of theta functions

ϑ (x; χ)

associated with Dirichlet characters, as defined by

ϑ false(x; χ false) = \sum_{n ⩾ 1} χ false(n false) {normale}^{- π n^{2} x / p} false(χ \in X_{p}^{+} false),

where

X_{p}^{+}

denotes the subgroup of order

\frac{1}{2} false(p - 1 false)

of even Dirichlet characters modulo

p

. It was conjectured by Louboutin [27] that

ϑ (1; χ) \neq 0

for every nontrivial character modulo a prime — see [11] for a case of vanishing in the composite case.…”

Section: Introduction and Statements Of Resultsmentioning

confidence: 93%

“…Lower bounds providing the expected order for the moments are obtained in [32]. Conditionally on Generalized Riemann Hypothesis, nearly optimal upper bounds are established in [30].…”

Section: Introduction and Statements Of Resultsmentioning

confidence: 99%

Small Gál sums and applications

Bretèche

Munsch²,

Tenenbaum

2020

J. London Math. Soc.

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In recent years, maximizing Gál sums regained interest due to a firm link with large values of L‐functions. In the present paper, we initiate an investigation of small sums of Gál type, with respect to the L1‐norm. We also consider the intertwined question of minimizing weighted versions of the usual multiplicative energy. We apply our estimates to: (i) a logarithmic refinement of Burgess' bound on character sums, improving previous results of Kerr, Shparlinski and Yau; (ii) an improvement on earlier lower bounds by Louboutin and the second author for the number of nonvanishing theta functions associated to Dirichlet characters; and (iii) new lower bounds for low moments of character sums.

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“…Assuming the Generalised Riemann Hypothesis for Dirichlet L-functions, Munsch [16] proved almost sharp upper bounds for the 2k-th moment of theta functions θ(1, χ) as the character χ varies over non-principal Dirichlet characters mod q, for each fixed k ∈ N. He did this by writing θ(1, χ) as a Perron integral involving the L-function L(s, χ), and then expanding the 2k-th power and bounding the averages of products 2k j=1 |L(1/2 + it j , χ)| that emerge. This is interesting here because for even characters χ, θ(1, χ) behaves roughly like n≤ √ q χ(n), which is modelled by the sum n≤ √ q f (n) of a Steinhaus random multiplicative function.…”

Section: Introductionmentioning

confidence: 99%

“…But for smaller q this kind of argument doesn't seem operable to prove sharp bounds, indeed one has already lost too much information in applying the triangle inequality to the Perron integral. Nevertheless, it might permit a relatively straightforward extension of Munsch's [16] results to non-integer k ≥ 5.…”

Section: Introductionmentioning

confidence: 99%

Moments of random multiplicative functions, II: High moments

Harper

2019

Alg. Number Th.

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We determine the order of magnitude of E| n≤x f (n)| 2q up to factors of size e O(q 2 ) , where f (n) is a Steinhaus or Rademacher random multiplicative function, for all real 1 ≤ q ≤ c log x log log x . In the Steinhaus case, we show that E| n≤x f (n)| 2q = e O(q 2 ) x q ( log x q log(2q) ) (q−1) 2 on this whole range. In the Rademacher case, we find a transition in the behaviour of the moments when q ≈ (1 + √ 5)/2, where the size starts to be dominated by "orthogonal" rather than "unitary" behaviour. We also deduce some consequences for the large deviations of n≤x f (n).The proofs use various tools, including hypercontractive inequalities, to connect E| n≤x f (n)| 2q with the q-th moment of an Euler product integral. When q is large, it is then fairly easy to analyse this integral. When q is close to 1 the analysis seems to require subtler arguments, including Doob's L p maximal inequality for martingales.where the constant C St (q) satisfies C St (q) = e −q 2 log q−q 2 log log q+O(q 2 ) for large q. For Rademacher random multiplicative f (n), when q = 1 we have that E| n≤x f (n)| 2 = n≤x, n squarefree 1 ∼ (6/π 2 )x, and for fixed integer q ≥ 2 we havewhere the constant C Rad (q) satisfies C Rad (q) = e −2q 2 log q−2q 2 log log q+O(q 2 ) for large q. As described in [9, 10], we actually have much more precise information about the constants C St (q), C Rad (q) (for example they factor into explicit "arithmetic" and "geometric" parts), but this will not be important for our purposes here.We would like to have information about E| n≤x f (n)| 2q when q ≥ 1 is not necessarily integral, and that allows q to vary as a function of x rather than being fixed.Regarding uniformity in q, Theorem 4.1 of Granville and Soundararajan [5] implies that for Steinhaus random multiplicative f (n), and uniformly for all large x and integers

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Shifted Moments of ‐functions and Moments of Theta Functions

Abstract: Abstract. Assuming the Riemann Hypothesis, Soundararajan showed in [18] that

Cited by 13 publications

References 20 publications

Upper and Lower Bounds for Higher Moments of Theta Functions

Upper and Lower Bounds for Higher Moments of Theta Functions

Small Gál sums and applications

Moments of random multiplicative functions, II: High moments

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