Mean square estimate for relatively short exponential sums involving Fourier coefficients of cusp forms

Ernvall-Hytönen, Anne-Maria

doi:10.5186/aasfm.2015.4019

Cited by 4 publications

(5 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The second estimate, which has already been obtained without twists by Wu and Zhai [30], is proved following Heath-Brown and Tsang as in the proof of [7, lemma 2], and following the proof of [13, theorem 2]. We would also like to mention that recently Ernvall-Hytönen [4] has considered the mean square of short exponential sums for which is larger than M 1/2 .…”

Section: Moments and Oscillations Of Exponential Sums Related To Cuspmentioning

confidence: 74%

“…The second estimate, which has already been obtained without twists by Wu and Zhai [30], is proved following Heath-Brown and Tsang as in the proof of Lemma 2 of [7], and following the proof of Theorem 2 in [13]. We would also like to mention that recently Ernvall-Hytönen [4] has considered the mean square of short exponential sums for which ∆ is larger than M 1/2 . Our final Theorem 8 will be an analogue of the main theorem of Heath-Brown and Tsang in [7], and it is the original motivation for the various moment estimates of this paper.…”

Section: What We Do In This Papermentioning

confidence: 78%

See 1 more Smart Citation

Moments and oscillations of exponential sums related to cusp forms

Vesalainen

2016

Math. Proc. Camb. Phil. Soc.

View full text Add to dashboard Cite

We consider large values of long linear exponential sums involving Fourier coefficients of holomorphic cusp forms. The sums we consider involve rational linear twists e(nh/k) with sufficiently small denominators. We prove both pointwise upper bounds and bounds for the frequency of large values. In particular, the k-aspect is treated. As an application we obtain upper bounds for all the moments of the sums in question. We also give the asymptotics with the right main term for fourth moments.We also consider the mean square of very short sums, proving that on average short linear sums with rational additive twists exhibit square root cancellation. This result is also proved in a slightly sharper form.Finally, the consideration of moment estimates for both long and short exponential sums culminates in a result concerning the oscillation of the long linear sums. Essentially, this result says that for a positive proportion of time, such a sum stays in fairly long intervals, where its order of magnitude does not drop below the average order of magnitude and where its argument is in a given interval of length 3π/2 + ε and so can not wind around the origin.

show abstract

Section: Moments and Oscillations Of Exponential Sums Related To Cuspmentioning

confidence: 74%

Section: What We Do In This Papermentioning

confidence: 78%

Moments and oscillations of exponential sums related to cusp forms

Vesalainen

2016

Math. Proc. Camb. Phil. Soc.

View full text Add to dashboard Cite

show abstract

“…The paper [28] actually considered the behaviour of the error terms in the Dirichlet divisor problem and the second moment for the Riemann ζ-function in short intervals, but the proof for the divisor function carries through fairly easily for Fourier coefficients of holomorphic cusp forms or Maass forms. In the last section, we will discuss the second moments with more details, and add here only that for holomorphic cusp forms second moments of rationally additively twisted short sums have been considered in the works [9,10,50]. We would like to emphasize that there are reasons to believe that even if the best possible upper bounds conform to the above Ω-results, they are likely to be very difficult to prove.…”

Section: ω-Resultsmentioning

confidence: 98%

“…For shorter sums very little is known in the higher rank setting. The moment estimates in the GL 2 situation [25,5,6,39], the square-root-cancellation heuristics, and the shape of the truncated Voronoi identity for rationally additively twisted sums related to SL 3 (Z) Maass cusp forms derived in [22] give rise to the conjectural bound x⩽m⩽x+∆ A(m, 1)e mh k ≪ ε min ∆ 1/2 x ε , k 1/2 x 1/3+ε . (1.2) Our aim in this paper is to investigate what limitations there are for the extent of cancellation in rationally additively twisted sums attached to Maass cusp forms for the group SL 3 (Z) by giving stronger evidence towards the conjectural bounds (1.2).…”

Section: Introductionmentioning

confidence: 99%

Exponential sums related to Maass forms

Jääsaari

Vesalainen

2019

Acta Arith.

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

“…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%