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On Popov's explicit formula and the Davenport expansion

Quan YangJay MehtaShigeru Kanemitsu — 2023

Czechoslovak Mathematical Journal

We shall establish an explicit formula for the Davenport series in terms of trivial zeros of the Riemann zeta-function, where by the Davenport series we mean an infinite series involving a PNT (Prime Number Theorem) related to arithmetic function a n with the periodic Bernoulli polynomial weight B ¯ ϰ ( n x ) and PNT arithmetic functions include the von Mangoldt function, Möbius function and Liouville function, etc. The Riesz sum of order 0 or 1 gives the well-known explicit formula for respectively the partial...

Representation functions with different weights

Quan-Hui Yang — 2014

Colloquium Mathematicae

For any given positive integer k, and any set A of nonnegative integers, let r 1 , k ( A , n ) denote the number of solutions of the equation n = a₁ + ka₂ with a₁,a₂ ∈ A. We prove that if k,l are multiplicatively independent integers, i.e., log k/log l is irrational, then there does not exist any set A ⊆ ℕ such that both r 1 , k ( A , n ) = r 1 , k ( A , n ) and r 1 , l ( A , n ) = r 1 , l ( A , n ) hold for all n ≥ n₀. We also pose a conjecture and two problems for further research.

Sum and difference sets containing integer powers

Quan-Hui YangJian-Dong Wu — 2012

Czechoslovak Mathematical Journal

Let n > m 2 be positive integers and n = ( m + 1 ) + r , where 0 r m . Let C be a subset of { 0 , 1 , , n } . We prove that if | C | > n / 2 + 1 if m is odd , m / 2 + δ if m is even , where x denotes the largest integer less than or equal to x and δ denotes the cardinality of even numbers in the interval [ 0 , min { r , m - 2 } ] , then C - C contains a power of m . We also show that these lower bounds are best possible.

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