|
%I #38 Nov 07 2020 11:40:10
%S 0,0,-2,0,0,6,2,0,0,-10,0,-2,10,0,0,14,0,-10,0,0,-6,0,0,10,18,-2,0,0,
%T 6,-14,0,0,-22,0,14,0,22,0,0,-26,0,-18,0,-14,-2,0,0,0,0,30,26,0,-30,0,
%U 2,0,-26,0,-18,10,0,-34,0,0,26,22,0,18,0,-10,34,0,0,14,0,0,-34,38,2,-6,0,30,0,34,0,0,-14,42,38,0,0,0,0,0,0,0,-10,-22,0,-42
%N The p-defect p - N(p) of the congruence y^2 == x^3 + 4*x (mod p) for primes p, where N(p) is the number of solutions given by A276730.
%C This sequence gives also the p-defects for the congruences y^2 == x^3 - x (mod p), y^2 == x^3 - 11*x - 14 (mod p) and y^2 == x^3 - 11*x + 14 (mod p). See the Cremona link, Table 1, N = 32. - _Wolfdieter Lang_, Dec 22 2016
%C This elliptic curve y^2 = x^3 + 4*x appears as strong Weil curve for the weight 2 newform (eta(4*tau)*eta(8*tau))^2 of level N=32, with Dedekind's eta function. See the Martin-Ono link, Theorem 2, p. 3173, the row with Conductor 32. See also A002171 for the expansion of this newform in powers of q^4 (but with different offset). The also Nr. 49 of the Martin Table 1.
%C From this L-series of this elliptic curve one has:
%C a(n) = 0 if prime(n) == 2 or 3 (mod 4). (see the conjecture by _Robert Israel_, Sep 28 2016 in A276730).
%C If prime(n) == 1 (mod 4) = A002144(m) (for a unique m = m(n)) then prime(n) = A(m)^2 + B(m)^2 with the odd A(m) = A002972(m) and the even B(m) = 2*A002973(m). It turns out that 4*A002144(m) - a(m^2) = (2*B(m))^2 for m=m(n), and the sign s(m) of a(m) is + if A(m) + B(m) == 1 (mod 4) and - if A(m) + B(m) == 3 (mod 4). For the primes == 1 (mod 4) leading to sign + or - see A279392 or A279393, respectively. One has thus s(m) = (-1)^((A(m)-1)/2 + B(m)/2). See the Martin-Ono formula for a_{32}(p) in Theorem 3, p. 3175. This leads to the a(n) formula given below.
%H Seiichi Manyama, <a href="/A278720/b278720.txt">Table of n, a(n) for n = 1..10000</a>
%H George E. Andrews, <a href="https://archive.org/details/NumberTheory_862/page/n143/mode/2up">Number Theory</a>, see S(1) expression p. 135.
%H J. E. Cremona, <a href="https://homepages.warwick.ac.uk/staff/J.E.Cremona/book/fulltext/index.html">Algorithms for Modular Elliptic Curves</a>.
%H Y. Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
%H Yves Martin and Ken Ono, <a href="http://dx.doi.org/10.1090/S0002-9939-97-03928-2">Eta-Quotients and Elliptic Curves</a>, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176.
%F a(n) = 0 if prime(n) == 2 or 3 (mod 4) (this is conjecture II from above).
%F a(n) = s(m)*2*A(m) if prime(n) = A002144(m), with A(m) = A002972(m) and the sign s(m) = (-1)^((A(m)-1)/2 + B(m)/2).
%F a(n) = - Sum_{k=1..p-3} ((k*(k+1)*(k+2))/p) where (x/y) is the Kronecker symbol. - _Michel Marcus_, Nov 06 2020
%e a(1) = 0 because prime(1) = 2 == 2 (mod 4).
%e a(2) = 0 because prime(2) = 3 == 3 (mod 4).
%e a(3) = -2 because prime(3) = 5 = A002144(1) = A002972(1)^2 + (2*A002973(1))^2 = 1^2 + 2^2. Hence 2*A(1) = 2*A002972(1) = 2, and the sign s(1) = - because A(1) + B(1) = 1 + 2*1 = 3 == 3 (mod 4).
%e a(6) = +6 because prime(6) = 13 = A002144(2) = A(2)^2 + B(2)^2 = 3^2 + (2*1)^2. Hence 2*A(2) = 6 and the sign is + because A(2) + B(2) = 3 + 2 = 5 == 1 (mod 4).
%o (PARI) a(n) = my(p=prime(n)); -sum(k=1, p-3, kronecker(k*(k+1)*(k+2), p)); \\ _Michel Marcus_, Nov 06 2020
%Y Cf. A000040, A002144, A002171, A002972, A002973, A279392, A279393.
%K sign,easy
%O 1,3
%A _Wolfdieter Lang_, Dec 11 2016
|
