2021/10/18 by Shane Chern, Chern, Shane
Mathematics · #11A07 #11A41 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2110.09629
openalex publication_date 2021/10/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In 2003, Zhao discovered a curious congruence involving harmonic series and Bernoulli numbers: for any odd prime p, ∑_\substacki,j,k≥ 1
gcd(ijk,p)=1\ı+j+k=p(1)/(ijk)≡ -2Bp-3 \pmodp, where Bn is the n-th Bernoulli number. This congruence was generalized by Wang and Cai in 2014, and Cai, Shen and Jia in 2017 by replacing the odd prime p in the summation and modulus with an odd prime power, and a product of two odd prime powers, respectively. In particular, Cai, Shen and Jia proposed a conjectural congruence: for any positive integer n with an odd prime factor p such that pr ∥ n where r≥ 1, ∑_\substacki,j,k≥ 1
gcd(ijk,n)=1\ı+j+k=n(1)/(ijk)≡ -2Bp-3⋅ (n)/(p)⋅ ∏_\substackprime q| n
q≠ p(1-(2)/(q))(1-(1)/(q3)) \pmodpr. In this paper, we establish the following generalization of their conjecture: for any positive integer n with an odd prime factor p such that pr ∥ n where r≥ 1, \beginaligned ∑_\substacki,j,k≥ 1
gcd(ijk,n)=1
a1 i+a2 j+a3 k=An(1)/(ijk)amp;≡ -2Bp-3⋅ (n)/(p)⋅ (Ag3)/(3)((1)/(a12 g12)+(1)/(a22 g22)+(1)/(a32 g32))
amp; × ∏_\substackprime q| n
q≠ p(1-(2)/(q))(1-(1)/(q3)) \pmodpr, \endaligned where a1, a2 and a3 are positive integers coprime to p, and A is a positive common multiple of a1, a2 and a3. Also, g1=gcd(a2,a3), g2=gcd(a3,a1), g3=gcd(a1,a2) and g=gcd(a1,a2,a3).