A Bombieri-type theorem for convolution with application on number field
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A BOMBIERI-TYPE THEOREM FOR CONVOLUTION WITH APPLICATION ON NUMBER FIELD P. DARBAR1,∗ and A. MUKHOPADHYAY2,3 1
2 3
Indian Statistical Institute, Kolkata, West Bengal 700108, India e-mail: [email protected]
Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India
Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India e-mail: [email protected] (Received January 13, 2020; revised August 12, 2020; accepted August 22, 2020)
Abstract. Let K be an imaginary quadratic number field and OK be its ring of integers. We show that, if the arithmetic functions f, g : OK → C both have level of distribution ϑ for some 0 < ϑ ≤ 1/2 then the Dirichlet convolution f ∗ g also has level of distribution ϑ. As an application we also obtain an analogue of the Titchmarsh divisor problem for product of two primes in imaginary quadratic fields.
1. Introduction 1.1. A Bombieri-type theorem for convolution on integers. Let Λ(n) be the usual Von Mangoldt function. For x > 1, the Siegel–Walfisz theorem states that x χ(n)Λ(n) = O (log x)D n≤x
holds for all D > 0 and for any non-principal character χ (mod q) with q (log x)D . Definition 1. An arithmetic function f is said to have level of distribution ϑ for 0 < ϑ ≤ 1 if for any A > 0 there exists a constant B = B(A) ∗ Corresponding
author. Key words and phrases: number field, multiplicative function, convolution, distribution of prime ideals, Titchmarsh divisor problem. Mathematics Subject Classification: 11R44, 11R04, 11A25.
0236-5294/$20.00 © 2020 Akade ´miai Kiado ´, Budapest, Hungary
2
P. P. DARBAR DARBAR and A. MUKHOPADHYAY
such that (1.1) ϑ
q≤ (logNN )B
max max M ≤N (a,q)=1
N 1 f (n) − f (n) A . ϕ(q) (log N )A
n≤M n≡a (mod q)
n≤M (n,q)=1
The Bombieri–Vinogradov theorem states that the indicator function of primes has level of distribution 21 and the Elliott–Halberstam conjecture predicts the level of distribution to be 1. Definition 2. Let τ (n) be the number of divisors of a natural number n. Assume that x ≥ 2. A complex valued arithmetic function f : N → C is said to satisfy the Siegel–Walfisz condition if there exists a positive constant C such that x , f (n)χ(n) = O (1.2) f (n) = O(τ (n)C ) and (log x)3D n≤x
holds for all D > 0 and for any non-principal Dirichlet character χ (mod q) with q (log x)D . In [11], Motohashi proved the following: Theorem 1.1 [11]. If the arithmetic functions f and g both satisfy (1.2) and have level of distribution 12 then so does the Dirichlet convolution f ∗ g . Using Theorem 1.1 one can get a simple proof of the following analogue of Titchmarsh divisor problem. Theorem 1.2 (Fujii [5, Theorem 2]). We have τ (p1 p2 − 1) = 315π −4ζ(3)x log log x + O(x), p1 p2 ≤x
where the sum is over primes p1 and p2 .
1.2. A Bombieri-type theorem for convolution on number fields. Our aim is to take up the study [11] and explore the corresponding questions in number fields. Let K be a number field and let OK be its ring of integers. For α ∈ OK ,
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