Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-25T07:09:30.351Z Has data issue: false hasContentIssue false
  • English
  • Français

NONNEGATIVE MULTIPLICATIVE FUNCTIONS ON SIFTED SETS, AND THE SQUARE ROOTS OF −1 MODULO SHIFTED PRIMES

Part of: Multiplicative number theory Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  20 February 2019

PAUL POLLACK
PAUL POLLACK*
Affiliation:
Department of Mathematics, University of Georgia, Athens, GA 30602, USA e-mail: pollack@uga.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

An oft-cited result of Peter Shiu bounds the mean value of a nonnegative multiplicative function over a coprime arithmetic progression. We prove a variant where the arithmetic progression is replaced by a sifted set. As an application, we show that the normalized square roots of −1 (mod m) are equidistributed (mod 1) as m runs through the shifted primes q − 1.

MSC classification

Primary: 11N37: Asymptotic results on arithmetic functions
Secondary: 11N36: Applications of sieve methods 11J71: Distribution modulo one
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 62 , Issue 1 , January 2020 , pp. 187 - 199
Copyright
Copyright © Glasgow Mathematical Journal Trust 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barban, M. B., Multiplicative functions of ΣR–equidistributed sequences, Izv. Akad. Nauk UzSSR Ser. Fiz.–Mat. Nauk. 1964(6) (1964), 13–19.Google Scholar
Barban, M. B., On the number of divisors of “translations” of the prime number-twins, Acta Math. Acad. Sci. Hungar. 15 (1964), 285–288.Google Scholar
Barban, M. B. and Levin, B. V., Multiplicative functions on “shifted” prime numbers, Dokl. Akad. Nauk SSSR 181 (1968), 778–780.Google Scholar
Barban, M. B. and Vehov, P. P., Summation of multiplicative functions of polynomials, Mat. Zametki 5 (1969), 669–680.Google Scholar
Duke, W., Friedlander, J. B., and Iwaniec, H., Equidistribution of roots of a quadratic congruence to prime moduli, Ann. of Math. 141(2) (1995), 423–441.CrossRefGoogle Scholar
Elliott, P. D. T. A. and Halberstam, H., Some applications of Bombieri’s theorem, Mathematika 13 (1966), 196–203.CrossRefGoogle Scholar
Erdős, P., On the sum $\sum^x_{k=1} d(\kern1.7pt f(k))$, J. London Math. Soc. 27 (1952), 7–15.Google Scholar
Hall, R. R. and Tenenbaum, G., Divisors, Cambridge Tracts in Mathematics, vol. 90 (Cambridge University Press, Cambridge, 1988).CrossRefGoogle Scholar
Hooley, C., On the representation of a number as the sum of two squares and a prime, Acta Math. 97 (1957), 189–210.CrossRefGoogle Scholar
Hooley, C., On the distribution of the roots of polynomial congruences, Mathematika 11 (1964), 39–49.CrossRefGoogle Scholar
Kátai, I., A note on a sieve method, Publ. Math. Debrecen 15 (1968), 69–73.Google Scholar
Kátai, I., On an application of the large sieve: Shifted prime numbers, which have no prime divisors from a given arithmetical progression, Acta Math. Acad. Sci. Hungar 21 (1970), 151–173.CrossRefGoogle Scholar
Linnik, Yu. V., An asymptotic formula in the Hardy–Littlewood additive problem, Izv. Akad. Nauk SSSR, Ser. Mat. 24 (1960), 629–706.Google Scholar
Montgomery, H. L. and Vaughan, R. C., Multiplicative number theory. I. Classical theory, Cambridge Studies in Advanced Mathematics, vol. 97 (Cambridge University Press, Cambridge, 2007).Google Scholar
Shiu, P., A Brun-Titchmarsh theorem for multiplicative functions, J. Reine Angew. Math. 313 (1980), 161–170.Google Scholar
Vinogradov, A. I. and Linnik, Yu. V., Estimate of the sum of the number of divisors in a short segment of an arithmetic progression, Uspehi Mat. Nauk (N.S.) 12(4(76)) (1957), 277–280.Google Scholar
Wolke, D., Multiplikative Funktionen auf schnell wachsenden Folgen, J. Reine Angew. Math. 251 (1971), 54–67.Google Scholar