Diophantine approximation with arithmetic functions, I
HTML articles powered by AMS MathViewer
- by Emre Alkan, Kevin Ford and Alexandru Zaharescu PDF
- Trans. Amer. Math. Soc. 361 (2009), 2263-2275 Request permission
Abstract:
We prove a strong simultaneous Diophantine approximation theorem for values of additive and multiplicative functions provided that the functions have certain regularity on the primes.References
- Emre Alkan, Glyn Harman, and Alexandru Zaharescu, Diophantine approximation with mild divisibility constraints, J. Number Theory 118 (2006), no. 1, 1–14. MR 2220258, DOI 10.1016/j.jnt.2005.08.001
- R. C. Baker, Diophantine inequalities, London Mathematical Society Monographs. New Series, vol. 1, The Clarendon Press, Oxford University Press, New York, 1986. Oxford Science Publications. MR 865981
- R. C. Baker, G. Harman, and J. Pintz, The difference between consecutive primes. II, Proc. London Math. Soc. (3) 83 (2001), no. 3, 532–562. MR 1851081, DOI 10.1112/plms/83.3.532
- Harold Davenport, Multiplicative number theory, 3rd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York, 2000. Revised and with a preface by Hugh L. Montgomery. MR 1790423
- H. Diamond, H. Halberstam, and H.-E. Richert, Combinatorial sieves of dimension exceeding one, J. Number Theory 28 (1988), no. 3, 306–346. MR 932379, DOI 10.1016/0022-314X(88)90046-7
- H. G. Diamond, H. Halberstam, and H.-E. Richert, Combinatorial sieves of dimension exceeding one. II, Analytic number theory, Vol. 1 (Allerton Park, IL, 1995) Progr. Math., vol. 138, Birkhäuser Boston, Boston, MA, 1996, pp. 265–308. MR 1399343
- P. Erdős, Some remarks on Euler’s $\varphi$ function, Acta Arith. 4 (1958), 10–19. MR 110664, DOI 10.4064/aa-4-1-10-19
- P. Erdős and A. Schinzel, Distributions of the values of some arithmetical functions, Acta Arith. 6 (1960/61), 473–485. MR 126410, DOI 10.4064/aa-6-4-473-485
- Paul Erdös and Aurel Wintner, Additive arithmetical functions and statistical independence, Amer. J. Math. 61 (1939), 713–721. MR 247, DOI 10.2307/2371326
- John B. Friedlander, Fractional parts of sequences, Théorie des nombres (Quebec, PQ, 1987) de Gruyter, Berlin, 1989, pp. 220–226. MR 1024564
- S. W. Graham, Jeffrey J. Holt, and Carl Pomerance, On the solutions to $\phi (n)=\phi (n+k)$, Number theory in progress, Vol. 2 (Zakopane-Kościelisko, 1997) de Gruyter, Berlin, 1999, pp. 867–882. MR 1689549
- H. Halberstam and H.-E. Richert, Sieve methods, London Mathematical Society Monographs, No. 4, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1974. MR 0424730
- G. Harman, A. Kumchev, and P. A. Lewis, The distribution of prime ideals of imaginary quadratic fields, Trans. Amer. Math. Soc. 356 (2004), no. 2, 599–620. MR 2022713, DOI 10.1090/S0002-9947-03-03104-0
- Florian Luca and Igor E. Shparlinski, Approximating positive reals by ratios of kernels of consecutive integers, Diophantine analysis and related fields 2006, Sem. Math. Sci., vol. 35, Keio Univ., Yokohama, 2006, pp. 141–149. MR 2331654
- A. Schinzel, On functions $\varphi (n)$ and $\sigma (n)$, Bull. Acad. Polon. Sci. Cl. III. 3 (1955), 415–419. MR 0073625
- Isac Schoenberg, Über die asymptotische Verteilung reeller Zahlen mod 1, Math. Z. 28 (1928), no. 1, 171–199 (German). MR 1544950, DOI 10.1007/BF01181156
- Dieter Wolke, Eine Bemerkung über die Werte der Funktion $\sigma (n)$, Monatsh. Math. 83 (1977), no. 2, 163–166 (German, with English summary). MR 441891, DOI 10.1007/BF01534638
Additional Information
- Emre Alkan
- Affiliation: Department of Mathematics, Koc University, Rumelifeneri Yolu, 34450, Sariyer, Istanbul, Turkey
- Email: ealkan@ku.edu.tr
- Kevin Ford
- Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, Illinois 61801
- MR Author ID: 325647
- ORCID: 0000-0001-9650-725X
- Email: ford@math.uiuc.edu
- Alexandru Zaharescu
- Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, Illinois 61801
- MR Author ID: 186235
- Email: zaharesc@math.uiuc.edu
- Received by editor(s): June 6, 2006
- Published electronically: December 10, 2008
- Additional Notes: The second author was supported in part by the National Science Foundation Grant DMS-0555367.
The third author was supported in part by the National Science Foundation Grant DMS-0456615. - © Copyright 2008 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 361 (2009), 2263-2275
- MSC (2000): Primary 11N64, 11N36, 11K60
- DOI: https://doi.org/10.1090/S0002-9947-08-04822-8
- MathSciNet review: 2471917