2019/01/15 by Zhi‐Wei Sun, Sun, Zhi-Wei
Mathematics · #11A15 #11C20 #15A99 #33B10 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1901.04837
openalex publication_date 2019/01/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let p be an odd prime and let a,b∈\mathbb Z with p\nmid ab. In this paper we mainly evaluate Tp(δ)(a,b,x):=det[x+tanπ\fracaj2+bk2p]δ≤ j,k≤ (p-1)/2 (δ=0,1). For example, in the case p≡3\pmod4 we show that Tp(1)(a,b,0)=0 and Tp(0)(a,b,x)=\begincases 2(p-1)/2p(p+1)/4amp;if (\fracabp)=1,
p(p+1)/4amp;if (\fracabp)=-1,\endcases where (\frac⋅p) is the Legendre symbol. When (\frac-abp)=-1, we also evaluate the determinant det[x+\cotπ\fracaj2+bk2p]1≤ j,k≤(p-1)/2. In addition, we pose several conjectures one of which states that for any prime p≡3\pmod4 there is an integer xp≡1\pmod p such that det[\sec2π\frac(j-k)2p]0≤ j,k≤ p-1=-p(p+3)/2xp2.