Multiplicative functions on arithmetic progressions
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- by Adolf Hildebrand PDF
- Proc. Amer. Math. Soc. 108 (1990), 307-318 Request permission
Abstract:
Let $f$ be a multiplicative arithmetic function satisfying $\left | f \right | \leq 1$, let $x \geq 10$ and $2 \leq Q \leq {x^{1/3}}$. It Is shown that, with suitable integers ${q_1} \geq 2$ and ${q_2} \geq 2$, the estimate \[ \sum \limits _{\begin {array}{*{20}{c}} {n \leq x} \\ {n \equiv a\bmod q} \\ \end {array} } {f(n) = \frac {1}{{\varphi (q)}}} \sum \limits _{\begin {array}{*{20}{c}} {n \leq x} \\ {(n,q) = 1} \\ \end {array} } {f(n) + O\left ( {\frac {x}{q}{{\left ( {\log \frac {{\log x}}{{\log Q}}} \right )}^{ - 1/2}}} \right )} \] holds uniformly for $\left ( {a,q} \right ) = 1$ and all moduli $q \leq Q$ that are not multiples of ${q_1}$ or ${q_2}$.References
- Enrico Bombieri, Le grand crible dans la théorie analytique des nombres, Astérisque 18 (1987), 103 (French, with English summary). MR 891718
- E. Bombieri, J. B. Friedlander, and H. Iwaniec, Primes in arithmetic progressions to large moduli, Acta Math. 156 (1986), no. 3-4, 203–251. MR 834613, DOI 10.1007/BF02399204
- P. D. T. A. Elliott, Arithmetic functions and integer products, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 272, Springer-Verlag, New York, 1985. MR 766558, DOI 10.1007/978-1-4613-8548-6
- P. D. T. A. Elliott, Additive arithmetic functions on arithmetic progressions, Proc. London Math. Soc. (3) 54 (1987), no. 1, 15–37. MR 872248, DOI 10.1112/plms/s3-54.1.15
- P. D. T. A. Elliott, Multiplicative functions on arithmetic progressions, Mathematika 34 (1987), no. 2, 199–206. MR 933499, DOI 10.1112/S0025579300013450
- P. D. T. A. Elliott, Multiplicative functions on arithmetic progressions. II, Mathematika 35 (1988), no. 1, 38–50. MR 962733, DOI 10.1112/S0025579300006252
- P. X. Gallagher, A large sieve density estimate near $\sigma =1$, Invent. Math. 11 (1970), 329–339. MR 279049, DOI 10.1007/BF01403187
- Adolf Hildebrand, Additive functions on arithmetic progressions, J. London Math. Soc. (2) 34 (1986), no. 3, 394–402. MR 864442, DOI 10.1112/jlms/s2-34.3.394
- Yoichi Motohashi, An induction principle for the generalization of Bombieri’s prime number theorem, Proc. Japan Acad. 52 (1976), no. 6, 273–275. MR 422179
- P. Shiu, A Brun-Titchmarsh theorem for multiplicative functions, J. Reine Angew. Math. 313 (1980), 161–170. MR 552470, DOI 10.1515/crll.1980.313.161
- E. Wirsing, Additive functions with restricted growth on the numbers of the form $p+1$, Acta Arith. 37 (1980), 345–357. MR 598887, DOI 10.4064/aa-37-1-345-357
- Dieter Wolke, Über die mittlere Verteilung der Werte zahlentheoretischer Funktionen auf Restklassen. I, Math. Ann. 202 (1973), 1–25 (German). MR 327688, DOI 10.1007/BF01351202
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 108 (1990), 307-318
- MSC: Primary 11N64; Secondary 11N37
- DOI: https://doi.org/10.1090/S0002-9939-1990-0991697-X
- MathSciNet review: 991697