Moments and oscillations of exponential sums related to cusp forms

Vesalainen, Esa V.

doi:10.1017/s030500411600061x

Cited by 5 publications

(11 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The case k = 1 was considered by Hafner and Ivić [19] who essentially obtained the bound ≪ M 1/3+ϑ/3 . Similar reduction for certain ranges of k in the case of holomorphic cusp forms have recently been proved by Vesalainen [50]. The proof is analogous to the approach of [22].…”

Section: The Results: Bounds For Short Exponential Sums With Applicatsupporting

confidence: 65%

“…Proof of Corollary 5. This is proved in exactly the same way as the corresponding result, Theorem 1, in [50], except that Theorem 3 above is used to estimate the smoothing error. The only change in the computations when M 1/4 ≪ k ≪ M 5/18 is to observe that when k ≪ M 5/18 , we have k 3/2 M 1/4 ≪ M 2/3 .…”

Section: And Letmentioning

confidence: 76%

“…In particular, these sums are Ω(k 1/2 M 1/4 ). For shorter sums we have the following theorem, which is similar to Theorem 6 in [50]. In particular, the sum in question is Ω(∆ 1/2 ) when ∆ ≫ M 2ϑ+ε .…”

Section: Logarithm Removal Near Rational Pointsmentioning

confidence: 77%

“…In particular, the sum in question is Ω(∆ 1/2 ) when ∆ ≫ M 2ϑ+ε . This follows for intance by following the proof of Theorem 6 in [50]: the main difference is that the error terms k 2 Ξ M ε + k Ξ ∆ 1/2 M ε are to be replaced by k 2 Ξ M 2ϑ+ε +k Ξ ∆ 1/2 M ϑ+ε . Except for the error term of the truncated Voronoi identity, the exponent ϑ never appears as the Fourier coefficients are estimated by the Rankin-Selberg estimate.…”

Section: Logarithm Removal Near Rational Pointsmentioning

confidence: 99%

See 3 more Smart Citations

Exponential sums related to Maass forms

Jääsaari

Vesalainen

2019

Acta Arith.

Self Cite

View full text Add to dashboard Cite

We estimate short exponential sums weighted by the Fourier coefficients of a Maass form. This requires working out a certain transformation formula for non-linear exponential sums, which is of independent interest. We also discuss how the results depend on the growth of the Fourier coefficients in question. As a byproduct of these considerations, we can slightly extend the range of validity of a short exponential sum estimate for holomorphic cusp forms.The short estimates allow us to reduce smoothing errors. In particular, we prove an analogue of an approximate functional equation previously proven for holomorphic cusp form coefficients.As an application of these, we remove the logarithm from the classical upper bound for long linear sums weighted by Fourier coefficients of Maass forms, the resulting estimate being the best possible. This also involves improving the upper bounds for long linear sums with rational additive twists, the gains again allowed by the estimates for the short sums. Finally, we shall use the approximate functional equation to bound somewhat longer short exponential sums.When ∆ = M 2/3 this gives the upper bound ≪ M ϑ/3+4/9+ε , and so splitting a longer sum into sums of this length and estimating the subsums separately gives the following bound for longer sums.Actually, Theorem 1 is valid for a slightly larger range of ∆ than Theorem 5.5 in [12] is. In fact, with a minor modification [11], the proof of Theorem 5.5 of [12] can be easily modified to give the analogous result for holomorphic cusp forms:Theorem 3. Let us consider a fixed holomorphic cusp form of weight κ ∈ Z + for the full modular group with the Fourier expansion ∞ n=1 a(n) n (κ−1)/2 e(nz) for z ∈ C with ℑz > 0. Also, let M ∈ [1, ∞[, ∆ ∈ [1, M ], and let α ∈ R. If ∆ ≪ M 2/3 , then M n M+∆ a(n) e(nα) ≪ ∆ 1/6 M 1/3+ε , where the implicit constant depends only on the underlying cusp forms and ε. Similarly, if M 2/3 ≪ ∆, then M n M+∆ a(n) e(nα) ≪ ∆ M −2/9+ε .The proof of Theorem 1 depends on an estimate for short non-linear sums, analogous to Theorem 4.1 in [12]. Fortunately, the proof in [12] works almost verbatim for Maass forms and we shall indicate the differences later. On the other hand, the proof of the non-linear estimate requires a transformation formula of a certain shape for smoothed exponential sums, and this particular result does not seem to have been worked out before yet. Thus, in Section 4, we will give an analogue of the relevant Theorem 3.4 of Jutila's monograph [30], which considers smooth sums with holomorphic cusp form coefficients, with full details for Maass forms. An analogue of Theorem 3.2 of [30] has been given by Meurman in [40].The following estimates provide a concrete example of how estimates for short sums allow one to reduce smoothing errors thereby leading to improved upper bounds.

show abstract

Section: The Results: Bounds For Short Exponential Sums With Applicatsupporting

confidence: 65%

Section: And Letmentioning

confidence: 76%

Section: Logarithm Removal Near Rational Pointsmentioning

confidence: 77%

Section: Logarithm Removal Near Rational Pointsmentioning

confidence: 99%

See 2 more Smart Citations

Exponential sums related to Maass forms

Jääsaari

Vesalainen

2019

Acta Arith.

Self Cite

View full text Add to dashboard Cite

show abstract

“…The average behaviour of short rationally additively twisted exponential sums weighted by Fourier coefficients of holomorphic cusp forms has been studied e.g. by Jutila [12], Ernvall-Hytönen [2,3], and Vesalainen [27]. In the higher rank case, the mean square of long rationally additively twisted sums involving Fourier coefficients of SL(3, Z) Maass cusp forms has been considered in [15].…”

Section: The Main Resultsmentioning

confidence: 99%