A329648 Let D = A014601(n) be the n-th positive integer congruent to 0 or 3 mod 4, then a(n) = b(D) := -Sum_{i=1..D} Kronecker(-D,i)*i, where Kronecker(-D,i) is the Kronecker symbol.
1, 2, 7, 8, 11, 8, 30, 8, 19, 40, 69, 48, 9, 0, 93, 32, 70, 36, 156, 80, 43, 88, 235, 32, 102, 104, 220, 224, 177, 0, 126, 32, 67, 272, 497, 0, 50, 152, 395, 160, 249, 336, 522, 176, 182, 0, 760, 192, 0, 0, 515, 624, 321, 72, 888, 0, 230, 696, 1190, 480, 246, 0, 635
Offset: 1
Keywords
Examples
For n = 7, D = 15, b(15) = -(1 + 2 + 4 - 7 + 8 - 11 - 13 - 14) = 30, which is equal to 15*h(-15). Note that the class number of Q[sqrt(-15)] is 2. For D < 100, b(D) = 0 for D = 28 = 7*2^2, 60 = 15*2^2, 72 = 8*3^2, 92 = 23*2^2, 99 = 11*3^2 and 100 = 4*5^2, where -7, -15, -8, -23, -11 and -4 are fundamental discriminants. Note that Kronecker(-7,2) = Kronecker(-15,2) = Kronecker(-8,3) = Kronecker(-23,2) = Kronecker(-11,3) = 1. On the other hand, for D = 213444 = 4*231^2, we have c(213444) = 2*h(-4)/w(-4) * (1-Kronecker(-4,3))*(1-Kronecker(-4,7))*(1-Kronecker(-4,11)) = 4 and b(213444) = 213444*4 = 853776.
Links
- Jianing Song, Table of n, a(n) for n = 1..10000
- Jianing Song, Notes on the value of -(1/D)*(Sum_{i=1..D} Kronecker(-D,i)*i)
- Eric Weisstein's World of Mathematics, Class Number
- Eric Weisstein's World of Mathematics, Dirichlet L-Series
Programs
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Mathematica
b[n_] = -Sum[KroneckerSymbol[n, i]*i, {i, 1, n}]; a[n_] = b[2 n + Mod[n, 2]] -
PARI
b(n) = -sum(i=1, n, kronecker(-n,i)*i) a(n) = b(2*n + (n%2))
Formula
Let c(D) = b(D)/D = -(1/D)*(Sum_{i=1..D} Kronecker(-D,i)*i). Let -d be the unique fundamental discriminant (i.e., d is in A003657) such that D/d is a square, then c(D) = 2*h(-d)/w(-d) * Product_{primes p|D} (1-Kronecker(-d,p)), where h(-d) is the class number of K = Q[sqrt(-d)], w(-d) is the number of elements in K whose norms are 1 (w(-d) = 6 if d = 3, 4 if d = 4 and 2 if d > 4). This can be seen as the generalization of the well known class number formula: if -d is a fundamental discriminant then c(d) = 2*h(-d)/w(-d). See my notes in the Links section.
b(D) = 0 if and only if there exists a prime p being a factor of D such that if we write D = p^e * s, gcd(p,s) = 1, then e is even and Kronecker(-s,p) = 1; if p = 2, then s == 7 (mod 8).
If -d is a fundamental discriminant, then Sum_{k>=1} Kronecker(-d,k)/k = 2*Pi*h(-d)/(sqrt(d)*w(-d)) = Pi*c(d)/sqrt(d) = Pi*b(d)/d^(3/2). Here Sum_{k>=1} Kronecker(-d,k)/k is the value of the Dirichlet L-series of a non-principal character modulo d at s=1.
Comments