A097916 - OEIS

%I #17 Feb 29 2024 08:53:56

%S 1,1,1,1,4,7,7,1,10,4,2,19,2,23,20,25,1,34,40,2,46,38,5,41,52,8,18,21,

%T 74,56,26,7,92,14,33,85,11,28,16,112,41,4,134,116,22,41,4,46,56,54,43,

%U 6,155,52,26,206,6,212,172,34,19,206,76,12,87,197,9,206,244,12,88,278,277,248

%N Numerator of 2*zeta_K(-1) where K is the totally real field Q(sqrt(n)), as n runs through the squarefree numbers.

%D F. Hirzebruch, Hilbert modular surfaces, Ges. Abh. II, 225-323.

%H F. Hirzebruch, <a href="http://www.maths.ed.ac.uk/~v1ranick/papers/hirzhilb.pdf">Hilbert modular surfaces</a>, L'Enseignement Math., 19 (1973), 183-281. See p. 200.

%e 1/6, 1/3, 1/15, 1, 4/3, 7/3, 7/3, 1/3, 10/3, 4, ...

%o (Sage) [(round(60*QuadraticField(d).zeta_function(100)(-1).real())/30).numerator() for d in range(2, 100) if Integer(d).is_squarefree()]  # _Robin Visser_, Feb 28 2024

%o (PARI)

%o z(d) = -(1/2)*bernfrac(2)*d*sum(k=1, d-1, kronecker(d, k)*subst(bernpol(2), x, k/d)*(-1/2))

%o {v=[]; for(k=2, 100, if(issquarefree(k), my(d=k); if(k%4 <> 1, d = 4*k); v=concat(v, numerator(2*z(d)) ))); v} \\ _Thomas Scheuerle_, Feb 28 2024

%Y Cf. A097917.

%K nonn,frac

%O 1,5

%A _N. J. A. Sloane_, Sep 04 2004

%E More terms from _Robin Visser_, Feb 28 2024