Formulas for the number of gridlines

Ernvall-Hytönen, Anne-Maria; Matomäki, Kaisa; Haukkanen, Pentti; Merikoski, Jorma K.

doi:10.1007/s00605-010-0236-6

Cited by 3 publications

(14 citation statements)

References 5 publications

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“…In practical computation of all these quantities, recursive formulas [8,9,3,4] are useful. They have also been applied in computer experiments to find asymptotic formulas.…”

Section: Conclusion and Remarksmentioning

confidence: 99%

“…for all ε > 0. To prove (7) and ( 8), we simply note that t(n) and f (n) have by (2) the same asymptotic behavior and refer to [3,Lemma 6].…”

Section: Improved Formulasmentioning

confidence: 99%

“…Besides the number of threshold functions, there are several other quantities whose asymptotics can be studied in a similar way. For example, asymptotic formulas for the number of gridlines [3] and q-gridlines [5] are well-known in G(n). A simple modification of our above procedure gives such formulas in G(m, n).…”

Section: Conclusion and Remarksmentioning

confidence: 99%

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Asymptotics of the number of threshold functions on a two-dimensional rectangular grid

Haukkanen¹,

Merikoski²

2013

Discrete Applied Mathematics

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“…In practical computation of all these quantities, recursive formulas [8,9,3,4] are useful. They have also been applied in computer experiments to find asymptotic formulas.…”

Section: Conclusion and Remarksmentioning

confidence: 99%

“…for all ε > 0. To prove (7) and ( 8), we simply note that t(n) and f (n) have by (2) the same asymptotic behavior and refer to [3,Lemma 6].…”

Section: Improved Formulasmentioning

confidence: 99%

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Asymptotics of the number of threshold functions on a two-dimensional rectangular grid

Haukkanen¹,

Merikoski²

2013

Discrete Applied Mathematics

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“…(3.41)] (and only if[4]) RH holds. (For (18), see also[5, Lemma 4]. For (19) and (20), see also[5, Lemma 7].)…”

mentioning

confidence: 97%

“…(For (18), see also[5, Lemma 4]. For (19) and (20), see also[5, Lemma 7].) By (16), (18) and(19),B(m) = O max k≤m (m|E Φ (k)| + m|E R (k)|) = O max k≤m (mk log k + mk 2 exp(−A(log k)3 5 (log log k) − 1 5 )) = O(m 2 log m + m 3 exp(−A(log m) , we proceed similarly but use (20) instead of (19), so obtaining B(m) = O(n 5 2 +ε ).…”

mentioning

confidence: 99%

Asymptotics for Numbers of Line Segments and Lines in a Square Grid

Haukkanen

Merikoski

2012

Int. J. Number Theory

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Exponential sums related to Maass forms

Jääsaari

Vesalainen

2019

Acta Arith.

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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.

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Formulas for the number of gridlines

Cited by 3 publications

References 5 publications

Asymptotics of the number of threshold functions on a two-dimensional rectangular grid

Asymptotics of the number of threshold functions on a two-dimensional rectangular grid

Asymptotics for Numbers of Line Segments and Lines in a Square Grid

Exponential sums related to Maass forms

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