Is it clear to anyone why
inf inf , ==== ==== \ \ 1 epsteinzeta(2 s) := > > ---------- / / 2 2 s ==== ==== (k + j ) j = - inf k = - inf
has the negative integers as real roots, and a proper superset of zeta's complex roots, all with realpart 1/2?
I.e. epzeta(2*s)/zeta(s) apparently has no poles. (And the complex 0s seem more evenly spaced.)
The first few epstein roots are 0.50000000000000 + 6.020948904697597 I, 0.50000000000000 + 10.243770304166555 I, 0.50000000000000 + 12.988098012312423 I, 0.50000000000000 + 14.13472514173469 I, Z 0.50000000000000 + 16.34260710458722 I, 0.50000000000000 + 18.29199319612353 I, 0.50000000000000 + 21.02203963877156 I, 0.50000000000000 + 21.45061134398345 I, Z 0.50000000000000 + 23.27837652045958 I, 0.50000000000000 + 25.01085758014479 I, Z
with those marked with the mark of Zorro matching Zeta's. Does Epstein[1-s] come out in Epstein[s]?
Empirically, it's merely Gamma(s) epzeta(2 s) s - 1 -------------------- = epzeta(2 - 2 s) Gamma(1 - s) pi . s pi http://www.stephenwolfram.com/publications/articles/physics/83-properties1/4... seems to give an enormous generalization of this.
--rwg
(No answers so far on:
(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 ) Last one tests out so far. d/ds the integral for some weird identities. Also, subtracting %e^- t * (sqrt(%pi/t) - 1) * t^s from the integrand extends convergence across the critical strip. Check out the zeta(0) and "zeta(1)" limits.
E.g., 1 - Sqrt[Pi/t] -t -1 + -------------- + EllipticTheta[3, 0, E ] t E Integrate[--------------------------------------------, Sqrt[t] {t, 0, Infinity}] == Sqrt[Pi] (1 + 2 EulerGamma - Log[4])