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Multiplicative Functions on Arithmetic Progressions. VII: Large Moduli
Author(s) -
Elliott P. D. T. A.
Publication year - 2002
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610702003228
Subject(s) - multiplicative function , integer (computer science) , moduli , mathematics , combinatorics , prime factor , unit (ring theory) , prime (order theory) , function (biology) , arithmetic progression , arithmetic , discrete mathematics , physics , mathematical analysis , quantum mechanics , mathematics education , evolutionary biology , computer science , biology , programming language
A complex valued function g, defined on the positive integers, is multiplicative if it satisfies g ( ab ) = g ( a ) g ( b ) whenever the integers a and b are mutually prime. THEOREM 1. Let D be an integer , 2 ⩽ D ⩽ x , ε > 0. Let g be a multiplicative function with values in the complex unit disc . There is a character χ 1(mod D ) , real if g is real, such that when 0 < γ < 1,∑n ⩽ y n ≡ a ( mod D )g ( n ) − 1 ϕ ( D )∑n ⩽ y( n , D ) = 1g ( n ) −χ 1 ( a )ϕ ( D )*¯ ∑ n ⩽ yg ( n ) χ 1 ( n ) ≪ y ϕ ( D )(log D log y)1 / 4 − ɛuniformly for ( a , D ) = 1, D ⩽ y , x γ ⩽ y ⩽ x , the implied constant depending at most upon ε, γ.