On Mahler’s Transcendence Measure for e

Ernvall-Hytönen, Anne-Maria; Matala-aho, Tapani; Seppälä, Louna

doi:10.1007/s00365-018-9429-3

Cited by 5 publications

(9 citation statements)

References 10 publications

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“…Proof of Theorem 3. These estimates follow from the proofs of Theorems 5.5 and 5.7 in [12], except that Theorem 5.1 should be modified a little [11]. The term k −1 ∆ |η| −1/2 M −1/2+ε only appears in the case in which k 2 η 2 M ≫ 1, in which case the relevant estimate (on p. 27 of [12]) is actually Furthermore, if we select X = 1/2, we can forget the first term.…”

Section: And Letmentioning

confidence: 84%

“…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.…”

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confidence: 99%

“…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:…”

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confidence: 99%

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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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Section: And Letmentioning

confidence: 84%

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confidence: 99%

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Exponential sums related to Maass forms

Jääsaari

Vesalainen

2019

Acta Arith.

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“…Another little lemma from [7] gives a useful upper estimate: Lemma 8.3. [7] If y ≥ re r , where r ≥ e, then z(y) ≤ 1 + log r r y log y .…”

Section: Lemma 82 [9]mentioning

confidence: 99%

Euler's factorial series at algebraic integer points

Seppälä

2020

Journal of Number Theory

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We study a linear form in the values of Euler's series F (t) = ∞ n=0 n!t n at algebraic integer points α 1 , . . . , α m ∈ Z K belonging to a number field K. Let v|p be a non-Archimedean valuation of K. Two types of non-vanishing results for the linear form Λare derived, the second of them containing a lower bound for the v-adic absolute value of Λ v . The first non-vanishing result is also extended to the case of primes in residue classes. On the way to the main results, we present explicit Padé approximations to the generalised factorial series ∞ n=0 n−1 k=0 P (k) t n , where P (x) is a polynomial of degree one.

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“…Due to the lack of explicit twin approximations (2), the works considering Baker-type lower bounds for linear forms have relied on Siegel's lemma; see Baker [1], Mahler [15], and, for a generalised transcendence measure of e, Ernvall-Hytönen et al [6]. In these Baker-type lower bounds the error terms are weaker than the corresponding ones in those transcendence measures that depend on the maximum height only (see Hata [11] and Ernvall-Hytönen et al [7]).…”

Section: Introductionmentioning

confidence: 99%

Hermite–Thue equation: Padé approximations and Siegel's lemma

Matala-aho

Seppälä

2018

Journal of Number Theory

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Padé approximations and Siegel's lemma are widely used tools in Diophantine approximation theory. This work has evolved from the attempts to improve Baker-type linear independence measures, either by using the Bombieri-Vaaler version of Siegel's lemma to sharpen the estimates of Padé-type approximations, or by finding completely explicit expressions for the yet unknown 'twin type' Hermite-Padé approximations. The appropriate homogeneous matrix equation representing both methods has an M × (L + 1) coefficient matrix, where M ≤ L. The homogeneous solution vectors of this matrix equation give candidates for the Padé polynomials. Due to the Bombieri-Vaaler version of Siegel's lemma, the upper bound of the minimal non-zero solution of the matrix equation can be improved by finding the gcd of all the M × M minors of the coefficient matrix. In this paper we consider the exponential function and prove that there indeed exists a big common factor of the M × M minors, giving a possibility to apply the Bombieri-Vaaler version of Siegel's lemma. Further, in the case M = L, the existence of this common factor is a step towards understanding the nature of the 'twin type' Hermite-Padé approximations to the exponential function. 1 approximations). Diagonal Hermite-Padé approximations of the generalised hypergeometric series are quite well established; see de Bruin [5], Huttner [13], Matala-aho [16], Nesterenko [18], and Nikišin [19] for more details. In this work we shall not pursue further this general Padé problem but focus on the Padé approximations to the exponential function instead.1.2. The twin problem. The problem of finding explicit type II Hermite-Padé approximations to the exponential series in the case where the degrees of the polynomials are free parameters and the orders of the remainders are equal was, as mentioned, resolved already by Hermite. But the twin problem (as we call it) of finding explicit type II Hermite-Padé approximations in the case where the degrees of the polynomials are the same but the orders of the remainders are free parameters is still open.Let now l 1 , . . . , l m be positive integers and let α 1 , . . . , α m be distinct variables. Denote α = (α 1 , . . . , α m ) T , l = (l 1 , . . . , l m ) T , and L := l 1 + . . . + l m . Then we may state the twin problem as follows: Find an explicit polynomial B l,0 (t, α), polynomials B l,j (t, α) and remainders S l,j (t, α), j = 1, . . . , m, satisfying the equations

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On Mahler’s Transcendence Measure for e

Cited by 5 publications

References 10 publications

Exponential sums related to Maass forms

Exponential sums related to Maass forms

Euler's factorial series at algebraic integer points

Hermite–Thue equation: Padé approximations and Siegel's lemma

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