On a variance of Hecke eigenvalues in arithmetic progressions

Lau, Yuk-Kam; Zhao, Liang

doi:10.1016/j.jnt.2011.12.011

Cited by 20 publications

(19 citation statements)

References 13 publications

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“…For recent work on asymptotics of sums of d 3 over arithmetic progressions, see [15] and the literature cited therein. The variance Var(S d 2 ;X;Q ) of S d 2 has been studied by Motohashi [32], Blomer [3], Lau and Zhao [28], the result being [28] (we assume Q prime for simplicity):…”

Section: The Divisor Function In Arithmetic Progressionsmentioning

confidence: 99%

Sums of divisor functions in $$\mathbb {F}_q[t]$$ F q [ t ] and matrix integrals

Keating¹,

Rodgers

Roditty-Gershon

et al. 2017

Math. Z.

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We study the mean square of sums of the kth divisor function d k (n) over short intervals and arithmetic progressions for the rational function field over a finite field of q elements. In the limit as q → ∞ we establish a relationship with a matrix integral over the unitary group. Evaluating this integral enables us to compute the mean square of the sums of d k (n) in terms of a lattice point count. This lattice point count can in turn be calculated in terms of a certain piecewise polynomial function, which we analyse. Our results suggest general conjectures for the corresponding classical problems over the integers, which agree with the few cases where the answer is known.

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Section: The Divisor Function In Arithmetic Progressionsmentioning

confidence: 99%

Sums of divisor functions in $$\mathbb {F}_q[t]$$ F q [ t ] and matrix integrals

Keating¹,

Rodgers

Roditty-Gershon

et al. 2017

Math. Z.

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show abstract

“…Banks, Heath-Brown and Shparlinski [1] have considered the average over a and proved that for any ε > 0 there exists some δ > 0 such that for a sufficiently large X 1≤a≤q gcd(a,q)=1 |E(X; q, a)| ≤ X 1−δ holds uniformly for q < X 1−ε . For other examples, see [2,4,6,7,15].…”

mentioning

confidence: 99%

Divisor problem in arithmetic progressions modulo a prime power

Liu

Shparlinski

Zhang

2018

Advances in Mathematics

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“…In this way Lester [23] has evaluated the variance for c ∈ (k − 1, k). It is likely that a similar argument could be used to verify Conjecture 1 in this restricted range for all k (indeed, this is close to the strategy of [22] in the case k = 2).…”

Section: Introductionmentioning

confidence: 56%

The variance of divisor sums in arithmetic progressions

Rodgers

Soundararajan

2017

Forum Mathematicum

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We study the variance of sums of the $k$-fold divisor function $d_k(n)$ over sparse arithmetic progressions, with averaging over both residue classes and moduli. In a restricted range, we confirm an averaged version of a recent conjecture about the asymptotics of this variance. This result is closely related to moments of Dirichlet $L$-functions and our proof relies on the asymptotic large sieve.Comment: 30 pages. Incorporates referee suggestion

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On a variance of Hecke eigenvalues in arithmetic progressions

Cited by 20 publications

References 13 publications

Sums of divisor functions in $$\mathbb {F}_q[t]$$ F q [ t ] and matrix integrals

Sums of divisor functions in $$\mathbb {F}_q[t]$$ F q [ t ] and matrix integrals

Divisor problem in arithmetic progressions modulo a prime power

The variance of divisor sums in arithmetic progressions

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