# [seqfan] A071961: Can we characterize zeroes?

Alonso Del Arte alonso.delarte at gmail.com
Wed Sep 2 00:49:39 CEST 2020

```As I try to make sense of positive prime numbers *p* such that *h*(*Q*(*√*
*p*)) > 1, I thought I might get an insight from Kronecker symbol sums,
which are given by A071961. The indices that are of particular interest to
me are 4*p* and *p* one more than a square. Looking at the whole sequence,
though, certain questions arise that don't directly pertain to the field
class number question, but which might interest y'all.

- Can we characterize the zeroes of this function?
- What about the negative values?

For context, here's my local Scala REPL session:

C:\Users\AL\Documents\NetBeansProjects\AlgebraicIntegerCalculator\build>scala
-cp classes
Welcome to Scala 2.13.1 (Java HotSpot(TM) 64-Bit Server VM, Java 1.8.0_261).
Type in expressions for evaluation. Or try :help.

scala> def kronSum(n: Int): Int = (1 to n -
1).map(calculators.NumberTheoreticFunctionsCalculator.symbolKronecker(n,
_)).sum
kronSum: (n: Int)Int

scala> kronSum(2) // Just to check an easy value
res0: Int = 1

scala> kronSum(3) // Still checking...
res1: Int = 0

scala> (1 to 50).map(kronSum) // This matches A071961 except for the
initial term
res2: IndexedSeq[Int] = Vector(0, 1, 0, 2, 0, 2, 4, 0, 6, 2, 2, 0, 0, 4, 6,
8, 0, 0, 2, 0, 0, 2, 6, 0, 20, 6, 0, 0, 0, 4, 12, 0, 0, 4, 4, 12, 0, 6, 12,
0, 0, 4, 0, 0, 0, 4, 10, 0, 42, 2)

scala> (1 to 100).filter(kronSum(_) == 0) // No OEIS results even after
omitting initial 1
res3: IndexedSeq[Int] = Vector(1, 3, 5, 8, 12, 13, 17, 18, 20, 21, 24, 27,
28, 29, 32, 33, 37, 40, 41, 43, 44, 45, 48, 52, 53, 56, 57, 60, 61, 65, 68,
69, 72, 73, 76, 77, 80, 84, 85, 88, 89, 92, 93, 96, 97)

scala> (1 to 1000).filter(kronSum(_) < 0)
res4: IndexedSeq[Int] = Vector(67, 99, 115, 147, 163, 169, 187, 256, 267,
275, 283, 307, 315, 355, 379, 400, 403, 427, 441, 475, 484, 507, 523, 529,
531, 576, 595, 619, 625, 643, 675, 729, 747, 763, 787, 883, 900, 907, 931,
947, 955, 961, 975)

--
Alonso del Arte
Author at SmashWords.com
<https://www.smashwords.com/profile/view/AlonsoDelarte>
Musician at ReverbNation.com <http://www.reverbnation.com/alonsodelarte>

```