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On asymptotic behaviour of Dirichlet inverse
Abstract
Let $f(n)$ be an arithmetic function with $f(1)\neq0$ and let $f^{-1}(n)$ be its reciprocal with respect to the Dirichlet convolution. We study the asymptotic behaviour of $|f^{-1}(n)|$ with regard to the asymptotic behaviour of $|f(n)|$ assuming that the latter one grows or decays with at most polynomial or exponential speed. As a by-product, we obtain simple but constructive upper bounds for the number of ordered factorizations of $n$ into $k$ factors.
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We express the values of the Dirichlet inverse f −1 in terms of the values of f without using the values of f −1. We use a method based on representing f −1 * f = δ as a system of linear equations. Jagannathan has given many of the results of this paper without proof starting from the basic recurrence relation for the values of f −1 .
In this article we obtain some results on the sequence c(n), where c(n) is the sum of the prime factors in the prime factorization of n.
The general field of additive number theory considers questions concerning representations of a given positive integer n as a sum of other integers. In particular, partitions treat the sums as unordered combinatorial objects, and compositions treat the sums as ordered. Sometimes the sums are restricted, so that, for example, the summands are distinct, or relatively prime, or all congruent to ±1 modulo 5. In this paper we review work on analogous problems concerning representations of n as a product of positive integers. We survey techniques for enumerating product representations both in the unrestricted case and in the case when the factors are required to be distinct, and both when the product representations are considered as ordered objects and when they are unordered. We offer some new identities and observations for these and related counting functions and derive some new recursive algorithms to generate lists of factorizations with restrictions of various types.
For p ≥ 12/11, the eigenfunctions of the non-linear eigenvalue problem for the p-Laplacian on the interval (0,1) are shown to form a Riesz basis of L2(0, 1) and a Schauder basis of Lq(0, 1) whenever 1 < q < ∞.
The number of ways to factor a natural number into an ordered product of integers, each factor greater than one, is called the ordered factorization of n and is denoted H(n). We show upper and lower bounds on H(n) with explicit constructions.
For a function $\varphi$ in $L^2(0,1)$, extended to the whole real line as an
odd periodic function of period 2, we ask when the collection of dilates
$\varphi(nx)$, $n=1,2,3,\ldots$, constitutes a Riesz basis or a complete
sequence in $L^2(0,1)$. The problem translates into a question concerning
multipliers and cyclic vectors in the Hilbert space $\cal H$ of Dirichlet
series $f(s)=\sum_n a_nn^{-s}$, where the coefficients $a_n$ are square
summable. It proves useful to model $\cal H$ as the $H^2$ space of the
infinite-dimensional polydisk, or, which is the same, the $H^2$ space of the
character space, where a character is a multiplicative homomorphism from the
positive integers to the unit circle. For given $f$ in $\cal H$ and characters
$\chi$, $f_\chi(s)=\sum_na_n\chi(n)n^{-s}$ is a vertical limit function of $f$.
We study certain probabilistic properties of these vertical limit functions.
For p ⩾ 12 11 p\geqslant \frac {12}{11} , the eigenfunctions of the non-linear eigenvalue problem for the p p -Laplacian on the interval ( 0 , 1 ) (0,1) are shown to form a Riesz basis of L 2 ( 0 , 1 ) L_2(0,1) and a Schauder basis of L q ( 0 , 1 ) L_q(0,1) whenever 1 > q > ∞ 1>q>\infty .
1: The Fundamental Theorem of Arithmetic. 2: Arithmetical Functions and Dirichlet Multiplication. 3: Averages of Arithmetical Function. 4: Some Elementary Theorems on the Distribution of Prime Numbers. 5: Congruences. 6: Finite Abelian Groups and Their Characters. 7: Cirichlet's Theorem on Primes in Arithmetic Progressions. 8: Periodic Arithmetical Functions and Gauss Sums. 9: Quadratic Residues and the Quadratic Reciprocity Law. 10: Primitive Roots. 11: Dirichlet Series and Euler Products. 12: The Functions. 13: Analytic Proof of the Prime Number Theorem. 14: Partitions.
It is observed that the Dirichlet ring admits a representation in an infinite-dimensional matrix algebra. The resulting matrices are subsequently used in the construction of nonorthogonal Riesz bases in a separable Hilbert space. This framework enables custom design of a plethora of bases with interesting features. Remarkably, the representation of signals in any one of these bases is numerically implementable via fast algorithms.
We show that the prime number theorem is equivalent with the non-vanishing on the 1-line, in the general setting of the Selberg class S \mathcal{S} of L \mathcal{L} -functions. The proof is based on a weak zero-density estimate near the 1-line and on a simple almost periodicity argument. We also give a conditional proof of the non-vanishing on the 1-line for every L \mathcal{L} -function in S \mathcal{S} , assuming a certain normality conjecture.
Abstract We study the number of ways to factor a natural number n into an ordered product of integers, each factor greater than one, denoted by H(n). This counting function from number theory was shown by Newberg and Naor (Adv. Appl. Math. 14 (1993) 172{183) to be a lower bound on the number of solutions to the so-called probed partial digest problem, which arises in the analysis of data from experiments in molecular biology. Hille (Acta Arith. 2 (1) (1936) 134{144) established a relation between H(n) and the Riemann zeta function . This relation was used by Hille to prove tight asymptotic upper and lower bounds on H(n). In particular, Hille showed an existential lower bound on H(n): For any t,i, where d,1:6. c 2000 Elsevier Science B.V. All rights reserved.
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Ordered factorizations with k factors
- J Sprittulla
J. Sprittulla, Ordered factorizations with k factors, preprint (2016), arXiv:1610.04826
(2016).