We investigate the distribution of αp modulo one in quadratic number fields 𝕂 with class number one, where p is restricted to prime elements in the ring of integers of 𝕂. Here we improve the relevant exponent 1/4 obtained by the first- and third-named authors for imaginary quadratic number fields [On the distribution of αp modulo one in imaginary quadratic number fields with class number one, J. Théor. Nombres Bordx. 32 (2020), no. 3, 719–760]) and by the first- and second-named authors for real quadratic number fields [Diophantine approximation with prime restriction in real quadratic number fields, Math. Z. (2021)] to 7/22. This generalizes a result of Harman [Diophantine approximation with Gaussian primes, Q. J. Math. 70 (2019), no. 4, 1505–1519] who obtained the same exponent 7/22 for ℚ (i) by extending his method which gave this exponent for ℚ [On the distribution of αp modulo one. II, Proc. London Math. Soc. 72, (1996), no. 3, 241–260]. Our proof is based on an extension of Harman’s sieve method to arbitrary number fields. Moreover, we need an asymptotic evaluation of certain smooth sums over prime ideals appearing in the above-mentioned work by the first- and second-named authors, for which we use analytic properties of Hecke L-functions with Größencharacters.
BAIER, S.—TECHNAU, M.: On the distribution ofαp modulo one in imaginary quadratic number fields with class number one, J. Théor. Nombres Bordeaux. 32 (2020), no. 3, 719–760.Search in Google Scholar
BAIER, S.—MAZUMDER, D.: Diophantine approximation with prime restriction in real quadratic number fields, Math. Z. 2021.http://link.springer.com/article/10.1007/s00209-021-02705-x10.1007/s00209-021-02705-xSearch in Google Scholar
COLEMAN, M. D.: . The Rosser-Iwaniec sieve in number fields, with an application, Acta Arith. 65 (1993), no. 1, 53–83.Search in Google Scholar
FOGELS, E.: On the zeros of Hecke’s L-functions. I, II, Acta Arith. 7 (1962) 87–106, 131–147.Search in Google Scholar
HARMAN, G.: On the distribution of αp modulo one. J. London Math. Soc. 27 (1983), no. 2, 9–18.Search in Google Scholar
HARMAN, G.: On the distribution of αp modulo one. II, Proc. London Math. Soc. 72 (1996), no. 3, 241–260.Search in Google Scholar
HARMAN, G.: Prime-detecting Sieves.In: London Mathematical Society Monographs Series, Vol. 33. Princeton University Press, Princeton, N J, 2007.Search in Google Scholar
HARMAN, G.: Diophantine approximation with Gaussian primes, Q. J. Math. 70 (2019), no. 4, 1505–1519.Search in Google Scholar
HECKE, E.: Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen. I, Math. Z. 1 (1918), 357–376.10.1007/BF01465095Search in Google Scholar
HECKE, E.: Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen. II, Math. Z. 6 (1920), 11–51.10.1007/BF01202991Search in Google Scholar
IWANIEC, H.: Rosser’s Sieve, Acta Arith. 36 (1980), 171–202.10.4064/aa-36-2-171-202Search in Google Scholar
IWANIEC, H.—KOWALSKI, E.: Analytic Number Theory.In: American Mathematical Society Colloquium Publications, Vol. 53. American Mathematical Society (AMS), Providence, RI, 2004.Search in Google Scholar
JIA, C.: On the distribution of αp modulo one, J. Number Theory 45 (1993), 241–253.10.1006/jnth.1993.1075Search in Google Scholar
MATOMÄKI, K.: The distribution of αp modulo one, Math. Proc. Camb. Philos. Soc. 147 (2009), no. 2, 267–283.Search in Google Scholar
MONTGOMERY, H. L.—VAUGHAN, R. C.: Multiplicative Number Theory. I. Classical Theory.In: Cambridge Studies in Advanced Mathematics, Vol. 97. Cambridge University Press, Cambridge, 2007.Search in Google Scholar
TENENBAUM G.: Introduction to Analytic and Probabilistic Number Theory.(3rd expanded ed.) In: Graduate Studies in Mathematics, Vol. 163. American Mathematical Society (AMS), Providence, RI, 2015.10.1090/gsm/163Search in Google Scholar
