A Bombieri-type theorem for convolution with application on number field

Bounded gaps between product of two primes in imaginary quadratic number fields

“…If arithmetic functions f and g both satisfy (3.9) and have level of distribution 1 2 then Motohashi [11] obtained that the Dirichlet convolution f * g also does so. In [2], we extend Motohashi's [11] result to arithmetic functions on imaginary quadratic number fields. As the proof can be carried forward for any level of distribution 0 < ϑ ≤ 1 2 , viewing β as a Dirichlet convolution of characteristic functions of P(Y , N b ) and P(N b , ∞), we get the following lemma.…”

“…As the proof can be carried forward for any level of distribution 0 < ϑ ≤ 1 2 , viewing β as a Dirichlet convolution of characteristic functions of P(Y , N b ) and P(N b , ∞), we get the following lemma. More precisely, it is a direct application of Cauchy-Schwarz inequality and Corollary 1.5 of [2].…”

“…Lemma 16 (Corollary 1.4, [2]) Let K be an imaginary quadratic field. Then product of two primes in O K have level of distribution…”

“…The case when K is imaginary is first considered by Vatwani [15]. In particular it is shown in [15] that there are infinitely many prime pairs (p 1 , p 2 ) ∈ Z[i] × Z[i] such that N (p 1 − p 2 ) < 246 2 , where N (•) denotes the norm on Q(i). The method of the proof can be generalized to cover any imaginary quadratic number field with class number 1.…”

An induction principle for the Bombieri-Vinogradov theorem over $\mathbb{F}_q[t]$ and a variant of the Titchmarsh divisor problem

“…Recently, Darbar and Mukhopadhyay [10] generalized Motohashi's result to imaginary quadratic fields. In this paper, we establish an analogous induction principle for equidistribution in arithmetic progressions, in the setting of F q [t].…”