Conditional asymptotic formulae for a class of arithmetic functions
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- by Werner Georg Nowak and Michael Schmeier PDF
- Proc. Amer. Math. Soc. 103 (1988), 713-717 Request permission
Abstract:
Under the assumption of Riemann’s Hypothesis, a general asymptotic formula for sums $\sum \nolimits _{n \leq x} {a\left ( n \right )}$ is established which applies (e.g.) to arithmetic functions $a\left ( n \right )$ defined by a kind of convolution of the Möbius function with some divisor function.References
-
A. Axer, Über einige Grenzwertsätze, Sitzungsber. Österr. Akad. Wiss., Abt. II, 120 (1911), 1253-1298.
H. Iwaniec and C. J. Mozzochi, On the circle and divisor problems (to appear).
- G. Kolesnik, On the estimation of multiple exponential sums, Recent progress in analytic number theory, Vol. 1 (Durham, 1979) Academic Press, London-New York, 1981, pp. 231–246. MR 637349
- Ekkehard Krätzel, Teilerprobleme in drei Dimensionen, Math. Nachr. 42 (1969), 275–288 (German). MR 262184, DOI 10.1002/mana.19690420408
- H. L. Montgomery and R. C. Vaughan, The distribution of squarefree numbers, Recent progress in analytic number theory, Vol. 1 (Durham, 1979) Academic Press, London-New York, 1981, pp. 247–256. MR 637350
- B. Z. Moroz, On the number of primitive lattice points in plane domains, Monatsh. Math. 99 (1985), no. 1, 37–42. MR 778169, DOI 10.1007/BF01300737
- Bahman Saffari, Sur le nombre de diviseurs “$r$-free” d’un entier. Cas $r=2$ et $r=3$, C. R. Acad. Sci. Paris Sér. A-B 265 (1967), A705–A707 (French). MR 220687 —, Sur le nombre de diviseurs "$r$-libres" d’un entier, et sur les points à coordenées entières dans certains règions du plan, C. R. Acad. Sci. Paris 266 (1968), 601-603.
- Peter Georg Schmidt, Zur Anzahl quadratvoller Zahlen in kurzen Intervallen, Acta Arith. 46 (1986), no. 2, 159–164 (German). MR 842942, DOI 10.4064/aa-46-2-159-164
- M. V. Subbarao and D. Suryanarayana, The divisor problem for $(k,\,r)$-integers, J. Austral. Math. Soc. 15 (1973), 430–440. MR 0327696
- D. Suryanarayana, The divisor problem for $(k,r)$-integers. II, J. Reine Angew. Math. 295 (1977), 49–56. MR 457383, DOI 10.1515/crll.1977.295.49
- R. Sitaramachandra Rao and D. Suryanarayana, The number of pairs of integers with $\textrm {L.C.M.}\leq x$, Arch. Math. (Basel) 21 (1970), 490–497. MR 274406, DOI 10.1007/BF01220953
- D. Suryanarayana and R. Sitaramachandra Rao, The distribution of square-full integers, Ark. Mat. 11 (1973), 195–201. MR 344208, DOI 10.1007/BF02388516
- D. Suryanarayana and V. Siva Rama Prasad, The number of $k$-free divisors of an integer, Acta Arith. 17 (1970/71), 345–354. MR 327697, DOI 10.4064/aa-17-4-345-354
- D. Suryanarayana and V. Siva Rama Prasad, The number of $k$-free divisors of an integer. II, J. Reine Angew. Math. 276 (1975), 200–205. MR 374058
- E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford, at the Clarendon Press, 1951. MR 0046485
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 103 (1988), 713-717
- MSC: Primary 11N37; Secondary 11M26, 11M41
- DOI: https://doi.org/10.1090/S0002-9939-1988-0947644-0
- MathSciNet review: 947644