Baustian, Falko; Bobkov, Vladimir On asymptotic behavior of Dirichlet inverse. (English) Zbl 1457.11010 Int. J. Number Theory 16, No. 6, 1337-1354 (2020). Let \(f\) be an arithmetic function with \(f(1)\ne 0\), and let \(f^{-1}\) denote its Dirichlet inverse. The authors find upper bounds for \(\vert f^{-1}(n)\vert\) assuming that \(\vert f(n)\vert\) grows or decays with at most polynomial or exponential speed, that is, \(\vert f(n)\vert\le Cn^{\gamma}\) or \(\vert f(n)\vert\le Ac^{n}\) for some \(C>0\), \(\gamma\in\mathbb{R}\) and \(A, c>0\). The examination is divided into multiplicative functions and the general case. As a by-product, upper bounds for the number of ordered factorizations of \(n\) into \(k\) factors are obtained. Reviewer: Pentti Haukkanen (Tampere) Cited in 1 Document MSC: 11A25 Arithmetic functions; related numbers; inversion formulas 11N37 Asymptotic results on arithmetic functions 11N56 Rate of growth of arithmetic functions 05A16 Asymptotic enumeration 05A17 Combinatorial aspects of partitions of integers Keywords:Dirichlet inverse; multiplicative function; inequality; asymptotics; ordered factorizations PDFBibTeX XMLCite \textit{F. Baustian} and \textit{V. Bobkov}, Int. J. Number Theory 16, No. 6, 1337--1354 (2020; Zbl 1457.11010) Full Text: DOI arXiv Online Encyclopedia of Integer Sequences: Decimal expansion of the positive real solution to (1 - 1/2^x) * zeta(x) = 2. References: [1] Apostol, T. M., Introduction to Analytic Number Theory (Springer-Verlag, New York, 1976). · Zbl 0335.10001 [2] Binding, P., Boulton, L., Čepička, J., Drábek, P. and Girg, P., Basis properties of eigenfunctions of the \(p\)-Laplacian,Proc. Amer. Math. Soc.134(12) (2006) 3487-3494. · Zbl 1119.34064 [3] Chor, B., Lemke, P. and Mador, Z., On the number of ordered factorizations of natural numbers, Discrete Math.214(1-3) (2000) 123-133. · Zbl 0970.11036 [4] Coppersmith, D. and Lewenstein, M., Constructive bounds on ordered factorizations, SIAM J. Discrete Math.19(2) (2005) 301-303. · Zbl 1090.05004 [5] Haukkanen, P., On the real powers of completely multiplicative arithmetical functions, Nieuw Arch. Wiskd.15(1-2) (1997) 73-78. · Zbl 0928.11005 [6] Haukkanen, P., Expressions for the Dirichlet inverse of arithmetical functions, Notes Number Theory Discrete Math.6(4) (2000) 118-124. [7] Hedenmalm, H., Lindqvist, P. and Seip, K., A Hilbert space of Dirichlet series and systems of dilated functions in \(L^2(0,1)\), Duke Math. J.86(1) (1997) 1-37. · Zbl 0887.46008 [8] Hedenmalm, H., Lindqvist, P. and Seip, K., Addendum to “A Hilbert space of Dirichlet series and systems of dilated functions in \(L^2(0,1)\)”, Duke Math. J.99(1) (1999) 175-178. · Zbl 0953.46015 [9] Hille, E., A problem in “Factorisatio Numerorum”, Acta Arith.2(1) (1936) 134-144. · JFM 62.1149.01 [10] Hwang, H. K. and Janson, S., A central limit theorem for random ordered factorizations of integers, Electron. J. Probab.16 (2011) 347-361. · Zbl 1267.11103 [11] Ingham, A. E., Some Tauberian theorems connected with the prime number theorem, J. London Math. Soc.1(3) (1945) 171-180. · Zbl 0061.12802 [12] Jakimczuk, R., Sum of prime factors in the prime factorization of an integer, Int. Math. Forum7(53-56) (2012) 2617-2621. · Zbl 1255.11009 [13] Jukes, K. A., On the Ingham and \((D,h(n))\) summation methods, J. London Math. Soc.2(4) (1971) 699-710. · Zbl 0213.08302 [14] Kaczorowski, J. and Perelli, A., On the prime number theorem for the Selberg class, Arch. Math.80(3) (2003) 255-263. · Zbl 1126.11334 [15] Kalmár, L., A “factorisatio numerorum” problémájáról, Mat. Fiz. Lapok38 (1931) 1-15. · JFM 57.1366.01 [16] Knopfmacher, A. and Mays, M. E., A survey of factorization counting functions, Int. J. Number Theory1(4) (2005) 563-581. · Zbl 1084.11003 [17] Kuhn, H. W. and Tucker, A. W., Nonlinear Programming, in Proc. Second Berkeley Symp. Mathematical Statistics and Probability, 1950 (University of California Press, Berkeley, California, 1951), pp. 481-492. · Zbl 0044.05903 [18] Segal, S. L., Summability by Dirichlet convolutions, Proc. Cambridge Philos. Soc.63(2) (1967) 393-400, Erratum 65(1) (1969) 369. · Zbl 0145.29003 [19] Segal, S. L., Prime number theorem analogues without primes, J. Reine Angew. Math.265 (1974) 1-22. · Zbl 0275.10026 [20] Sowa, A., The Dirichlet ring and unconditional bases in \(L_2[0,2\pi]\), Funct. Anal. Appl.47(3) (2013) 227-232. · Zbl 1315.46015 [21] J. Sprittulla, Ordered factorizations with \(k\) factors, preprint (2016), arXiv:1610.04826 (2016). · Zbl 1471.11018 [22] Wintner, A., Eratosthenian Averages (Waverly Press, Baltimore, 1943). · Zbl 0060.10503 [23] Wintner, A., On Töpler’s wave analysis, Amer. J. Math.69(4) (1947) 758-768. · Zbl 0034.04202 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.