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

1,2

Comments

Note that {Kronecker(D,i)} is a Dirichlet character mod |D| if and only if D == 0, 1 (mod 4).

We have the identity: -Sum_{i=1..D} Kronecker(-D,i)*i^2 = D*b(D). Proof: -Sum_{i=1..D} Kronecker(-D,i)*i^2 = -(1/2)*Sum_{i=1..D} (Kronecker(-D,i)*i^2+Kronecker(-D,D-i)*(D-i)^2) = -(1/2)*Sum_{i=1..D} (Kronecker(-D,i)*(i^2-(D-i)^2)) = -(1/2)*Sum_{i=1..D} (Kronecker(-D,i)*(2*D*i-D^2) = D*b(D) + (D^2/2)*(Sum_{i=1..D} Kronecker(-D,i)) = D*b(D).

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

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.

Example

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.

Mathematica

b[n_] = -Sum[KroneckerSymbol[n, i]*i, {i, 1, n}];
a[n_] = b[2 n + Mod[n, 2]]

Prog

(PARI) b(n) = -sum(i=1, n, kronecker(-n, i)*i)
a(n) = b(2*n + (n%2))

Crossrefs

Cf. A014601, A003657, A001221.

Sequence in context: A258823 A226612 A005538 * A206331 A053013 A070044

Adjacent sequences: A329645 A329646 A329647 * A329649 A329650 A329651

Keyword

nonn

Author

Jianing Song, Nov 18 2019

Status

approved