# zeta(a) is Riemann zeta function, 
= sum(1/(k+1)^a,k,0,inf)=sum(1/k^a,k,1,inf)
= integrate(t^(a-1)/(e^t-1),t,0,inf);
# zeta(a,b) is Hurwitz zeta function, 
= sum(1/(k+b)^a,k,0,inf);
# zeta(a,1,x) is incomplete Riemann zeta function of integral from 0 to x 
= integrate(t^(a-1)/(e^t-1),t,0,x);

zeta(a_,1,x_):= harmonic(a,x);
zeta(a_,b_,0):= 0;
zeta(a_,b_,inf):= zeta(a,b);
zeta(a_,b_, -inf):= -zeta(a,b);

zeta(a_, -1):=zeta(a)+1;
zeta(a_,0):= zeta(a);
zeta(a_,0.5):=(2^a-1)*zeta(a);
zeta(a_,1):= zeta(a);
zeta(a_,2):= zeta(a)-1;
zeta(a_,3):= zeta(a)-1/2^a-1;
zeta(a_,inf):= 0;

zeta(0,x_):=1/2-x;
#zeta(1,x_):= inf;
#zeta(2,x_):=psi(1,x);

zeta(-6):= 0;
zeta(-5):= -1/252;
zeta(-4):= 0;
zeta(-3):= 1/120;
zeta(-2):= 0;
zeta(-1):= -1/12;
zeta(0):= -1/2;
zeta(1):=inf;
zeta(2):=pi^2/6;
zeta(4):= pi^4/90;
zeta(inf):=1;