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An asymptotic formula for the number of prime solutions for multivariate linear equations
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In this paper, we study the multivariate linear equations with arbitrary positive integral coefficients. Under the Generalized Riemann Hypothesis, we obtained the asymptotic formula for the linear equations with more than five prime variables. This asymptotic formula is composed of three parts, that is, the first main term, the explicit second main term and the error term. Among them, the first main term is similar with the former one, the explicit second main term is relative to the non-trivial zeros of Dirichlet L-functions, and our error term improves the former one.
Yafang KONG . An asymptotic formula for the number of prime solutions for multivariate linear equations[J]. Frontiers of Mathematics in China, 2022 , 17(6) : 1001 -1013 . DOI: 10.1007/s11464-022-1029-7
| 1 |
Friedlander J B, Goldston D A. Sums of three or more primes. Trans Amer Math Soc 1997; 349: 287–319
|
| 2 |
Languasco A, Zaccagnini A. Sums of many primes. J Number Theory 2012; 132: 1265–1283
|
| 3 |
Li W P, Zhou H G. Linear equations with three or more primes. Pure and Appl Math 2004; 20(4): 350–359
|
| 4 |
LiuM CTsang K M. Small prime solutions of linear equations. In: Théorie des Nombres (Quebec, PQ, 1987), Berlin, New York: de Gruyter, 1989
|
| 5 |
Montgomery H L, Vaughan R C. Error terms in additive prime number theory. Quart J Math 1973; 24(2): 207–216
|
| 6 |
PanC DPan C B. Basic Analytic Number Theory. Beijing: Science Press, 2016 (in Chinese)
|
| 7 |
Vaughan R C, Wooley T D. The asymptotic formula in Waring’s problem: Higher order expansions. J Reine Angrew Math 2018; 742: 17–46
|
| 8 |
Vinogradov I M. Representation of an odd number as a sum of three primes. Dokl Akad Nauk SSSR 1937; 15: 291–294
|
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