Sign-changes of the Thue-Morse fractal function and Dirichlet L-series

Drmota, Michael; Skałba, Mariusz

doi:10.1007/bf02568009

Cited by 11 publications

(8 citation statements)

References 12 publications

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“…The first part of Theorem 2 generalizes a result by the authors [2], where it is shown that 3 and 5 are the exceptional primes of P 1 and 17 and possibly 41 those of P 2 . (In fact, p = 41 is not exceptional, see section 3.…”

Section: Theoremsupporting

confidence: 71%

“…2 We will prove Theorems 1 and 2 in sections 4 and 5. The negative part of Theorem 3 is proved at the end of section 3 and the positive part at the end of section 4.…”

Section: Theoremmentioning

confidence: 98%

“…In [2] it is shown that p = 3 and p = 5 are the only exceptional primes in P 1 , and p = 17 and possibly p = 41 those of P 2 . (We will treat the case p = 41 in a moment.)…”

Section: Lemma 3 There Exist Infinitely Many N ≥ 0 Such Thatmentioning

confidence: 99%

“…If p ∈ P 1 and p ≥ 17, this concept can be used to prove a sign change of ψ p (x) near x = 1 2 (see [2]). However, if t > 1 we do not know a general concept to decide whether (43) or (44) are satisfied or not.…”

Section: Proof (33) Follows From Lemma 7 and The Fact Thatmentioning

confidence: 99%

“…It is also an interesting problem to determine other zeroes and sign changes of ψ p (x). In [2] it is shown that for almost all primes p ∈ P 1 the fractal function ψ p (x) has a zero near x = 1/2. Furthermore, a similar result may be expected for P 2 .…”

Section: Proof Of Theoremmentioning

confidence: 99%

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Rarified sums of the Thue-Morse sequence

Drmota

Skałba

1999

Trans. Amer. Math. Soc.

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Abstract. Let q be an odd number and S q,0 (n) the difference between the number of k < n, k ≡ 0 mod q, with an even binary digit sum and the corresponding number of k < n, k ≡ 0 mod q, with an odd binary digit sum. A remarkable theorem of Newman says that S 3,0 (n) > 0 for all n. In this paper it is proved that the same assertion holds if q is divisible by 3 or q = 4 N + 1. On the other hand, it is shown that the number of primes q ≤ x with this property is o(x/ log x). Finally, analoga for "higher parities" are provided.

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Section: Theoremsupporting

confidence: 71%

“…2 We will prove Theorems 1 and 2 in sections 4 and 5. The negative part of Theorem 3 is proved at the end of section 3 and the positive part at the end of section 4.…”

Section: Theoremmentioning

confidence: 98%

“…In [2] it is shown that p = 3 and p = 5 are the only exceptional primes in P 1 , and p = 17 and possibly p = 41 those of P 2 . (We will treat the case p = 41 in a moment.)…”

Section: Lemma 3 There Exist Infinitely Many N ≥ 0 Such Thatmentioning

confidence: 99%

Section: Proof (33) Follows From Lemma 7 and The Fact Thatmentioning

confidence: 99%

Section: Proof Of Theoremmentioning

confidence: 99%

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