{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T00:36:10Z","timestamp":1759970170751,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,6]],"date-time":"2025-01-06T00:00:00Z","timestamp":1736121600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Beijing Municipal","award":["1242003","12471009"],"award-info":[{"award-number":["1242003","12471009"]}]},{"name":"National Natural Science Foundation of China","award":["1242003","12471009"],"award-info":[{"award-number":["1242003","12471009"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Let x be a large real number, d(n) be the Dirichlet divisor function and \u0394(x)=\u2211n\u2a7dxd(n)\u2212xlogx\u2212(2\u03b3\u22121)x. In this paper, we consider a weighted form of the three primes theorem: S(N;k1,k2,k3):=\u2211p1+p2+p3=N\u0394k1(p1)\u0394k2(p2)\u0394k3(p3), where p1,p2,p3 run over primes, k1,k2,k3\u2208{1,2,\u22ef,9}, and N is an large odd integer. For the case ki\u2208{2,3,\u22ef,9}(i=1,2,3), the others two kj\u2208{2,3,4} with j\u2260i, an asymptotic formula for S(N;k1,k2,k3) has been derived, along with a non-trivial upper bound estimate for S(N;k1,k2,k3) when min(k1,k2,k3)=1.<\/jats:p>","DOI":"10.3390\/sym17010076","type":"journal-article","created":{"date-parts":[[2025,1,6]],"date-time":"2025-01-06T10:40:52Z","timestamp":1736160052000},"page":"76","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On the Symmetric Form of the Three Primes Theorem Weighted by \u0394(x)"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0009-0007-1880-8170","authenticated-orcid":false,"given":"Zhen","family":"Guo","sequence":"first","affiliation":[{"name":"Department of Mathematics, China University of Mining and Technology, Beijing 100083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1112\/plms\/s3-66.2.279","article-title":"Exponential sums and lattice points. 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