(c6) ZETA(S) = ('INTEGRATE((THETA[3](0,%E^-T)^1-1)*T^(S/2-1),T,0,INF))/GAMMA(S/2)/2 inf / [ - t s/2 - 1 I (theta (0, %e ) - 1) t dt ] 3 / 0 (d6) zeta(s) = --------------------------------------- s 2 gamma(-) 2 (c8) DFLOAT(EVAL(SUBST([S = 2*%PI,NOUNIFY('INTEGRATE) = QUAD_INF],D6))) (d8) 1.01407286015004d0 = 1.01407286015011d0 inf inf ==== ==== \ \ 1 EZ(s) := 2 zeta(s) + 2 > > ------------ / / 2 2 s/2 ==== ==== (k + j ) k = 1 j =-inf inf / [ 2 - t s/2 - 1 I (theta (0, e ) - 1) t dt ] 3 / 0 = --------------------------------------. s Gamma(-) 2 E.g., In:= N[List @@ % /. s -> Pi] Out= {EpsteinZeta[3.14159], 8.27511, 8.27511}. (c12) \e\z(S)+2*SUM(SUM(SUM((I^2+J^2+K^2)^-(S/2),I,-INF,INF),J,-INF,INF),K,1,INF) = ('INTEGRATE((THETA[3](0,%E^-T)^3-1)*T^(S/2-1),T,0,INF))/GAMMA(S/2); inf inf inf ==== ==== ==== \ \ \ 1 (d12) EZ(s) + 2 > > > ----------------- = / / / 2 2 2 s/2 ==== ==== ==== (k + j + i ) k = 1 j = - inf i = - inf inf / [ 3 - t s/2 - 1 I (theta (0, %e ) - 1) t dt ] 3 / 0 ---------------------------------------, s gamma(-) 2 etc. (First two proved only for even integer s. Last one(s) completely untested.) --rwg