Das asymptotische Verhalten von Summen �ber multiplikative Funktionen

Abstract. Define the Liouville function for A, a subset of the primes P , by λ A (n) = (−1) Ω A (n) , where Ω A (n) is the number of prime factors of n coming from A counting multiplicity. For the traditional Liouville function, A is the set of all primes. Denoten . It is known that for eachGiven certain restrictions on the sifting density of A, asymptotic estimates for n≤x λ A (n) can be given. With further restrictions, more can be said. For an odd prime p, define the character-like function λ p as λ p (pk + i) = (i/p) for i = 1, . . . , p − 1 and k ≥ 0, and λ p (p) = 1, where (i/p) is the Legendre symbol (for example, λ 3 is defined by λ 3 (3k + 1) = 1, λ 3 (3k + 2) = −1 (k ≥ 0) and λ 3 (3) = 1). For the partial sums of character-like functions we give exact values and asymptotics; in particular, we prove the following theorem. Theorem. If p is an odd prime, thenThis result is related to a question of Erdős concerning the existence of bounds for number-theoretic functions. Within the course of discussion, the ratio φ(n)/σ(n) is considered.

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“…In fact, Wirsing in [20] showed more generally that every real multiplicative function f with |f (n)| ≤ 1 has a mean value; i.e., the limit…”

Section: Corollary 1 Ifmentioning

confidence: 99%

Completely multiplicative functions taking values in ${-1,1}$

Borwein

Choi

Coons

2010

Trans. Amer. Math. Soc.

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“…Our principal tool for the proof of Theorem 2 is a well known theorem of Wirsing [10], which may be formulated as follows:…”

Section: Thus We Obtain (9)mentioning

confidence: 99%

Roughly squarefree values of the Euler and Carmichael functions

Banks

Luca

2005

Acta Arith.

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“…The proof of Theorem 1 occupies §1, and relies on a refinement of the classical Wirsing method [15,16] for certain non-negative multiplicative functions, due to H. Halberstam. In §1 we axiomatise the processes involved in proving Theorem 1, since the method may turn out to be applicable to other problems.…”

Section: Here Given χ T → M (χ T) Is a Unique Compactly Supported mentioning

confidence: 99%

Solution of a generalised version of a problem of Rankin on sums of powers of cusp-form coefficients

Odoni

2002

Acta Arith.

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In memory of Robert Alexander Rankin, 27.X.1915Rankin, 27.X. -27.I.2001 0. Introduction. In [5] we gave a partial solution to a problem of Rankin [8] on the behaviour of the Fourier coefficients of a special family of cusp-forms introduced by Hecke. Our purpose in this paper is to extend and refine the methods of [5]. One consequence of this will be a complete solution of Rankin's problem. To explain our new results we adhere to the notation and terminology of [5]. Let K be a number field, [K : Q] = κ, 2 ≤ κ < ∞, and let χ be a normalised Grössencharakter of K. For n ∈ N let T (n, χ) = χ(a), the sum being over all integral ideals a in K with absolute norm N a = n. In the present context, the most important result in [5] is the asymptotic formulaHere β is any fixed positive number, while A(χ, β), c(χ, β) and γ(χ, β) are positive constants depending only on χ and β. In fact, we showed that (0.1) holds with γ(χ, β) ≥ min{1, β/2}; in §1 this will be improved to γ(χ, β) ≥ min{1, β}. Unfortunately we were not able in [5] to give an explicit expression for c(χ, β), although we were able to prove the existence of a unique upper-semicontinuous probability distribution functionHere M (0−, χ) = 0 and M (κ 2 , χ) = 1. In particular M (t, χ) is compactly supported, and so, by classical Hausdorff moment theory [2], it is uniquely determined if, for example, we know the values c(χ, k) for all k ∈ N, or even just for a subset of k ∈ N with k −1 = ∞.

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Das asymptotische Verhalten von Summen �ber multiplikative Funktionen

Cited by 164 publications

References 12 publications

Completely multiplicative functions taking values in ${-1,1}$

Completely multiplicative functions taking values in ${-1,1}$

Roughly squarefree values of the Euler and Carmichael functions

Solution of a generalised version of a problem of Rankin on sums of powers of cusp-form coefficients

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