scholarly journals The representation function for sums of three squares along arithmetic progressions

2016 ◽  
Vol 92 (8) ◽  
pp. 96-99 ◽  
Author(s):  
Paul Pollack
Keyword(s):  
Arithmetic Progressions ◽  
Representation Function ◽  
Sums Of Three Squares

scholarly journals Resonance between the Representation Function and Exponential Functions over Arithemetic Progression

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Li Ma ◽  
Xiaofei Yan
Keyword(s):  
Positive Integer ◽  
Exponential Sum ◽  
Upper Bounds ◽  
Arithmetic Progressions ◽  
Large Parameter ◽  
Exponential Functions ◽  
Representation Function

Let r n denote the number of representations of a positive integer n as a sum of two squares, i.e., n = x 1 2 + x 2 2 , where x 1 and x 2 are integers. We study the behavior of the exponential sum twisted by r n over the arithmetic progressions ∑ n ∼ X n ≡ l mod q r n e α n β , where 0 ≠ α ∈ ℝ , 0 < β < 1 , e x = e 2 π i x , and n ∼ X means X < n ≤ 2 X . Here, X > 1 is a large parameter, 1 ≤ l ≤ q are integers, and l , q = 1 . We obtain the upper bounds in different situations.

scholarly journals A mean value of the representation function for the sum of two primes in arithmetic progressions

2017 ◽  
Vol 13 (04) ◽  
pp. 977-990 ◽  
Author(s):  
Yuta Suzuki
Keyword(s):  
Asymptotic Formula ◽  
Riemann Hypothesis ◽  
Mean Value ◽  
Arithmetic Progressions ◽  
Representation Function ◽  
Real Zeros ◽  
Generalized Riemann Hypothesis ◽  
Goldbach Problem ◽  
Primes In Arithmetic Progressions ◽  
The Mean

In this paper, assuming a variant of the Generalized Riemann Hypothesis, which does not exclude the existence of real zeros, we prove an asymptotic formula for the mean value of the representation function for the sum of two primes in arithmetic progressions. This is an improvement of the result of F. Rüppel in 2009, and a generalization of the result of A. Languasco and A. Zaccagnini concerning the ordinary Goldbach problem in 2012.

scholarly journals Colorings with only rainbow arithmetic progressions

2020 ◽  
Vol 161 (2) ◽  
pp. 507-515
Author(s):  
J. Pach ◽  
I. Tomon
Keyword(s):  
Arithmetic Progressions
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