#March 22, 2018, Experimental Mathematics (RU) Spring 2018
Help:=proc(): print(` H(s), xi(s), G(s,g), Check185(s,g) `): end:

H:=proc(s)  1/2*s*(1-s)*Pi^(-s/2)*GAMMA(s/2): end:

xi:=proc(s) : Zeta(s)*H(s): end:

#diff1(f,x,j): the j-th derivative of f w.r.t. to
diff1:=proc(f,x,j)
if j=0 then
 f:
else
 diff(f,x$j):
fi:

end:
#G(s,g): Conrey's function G(s) defined by  Eq. (18.1) , where g is a finite list of [g1,g3,g5,...]
G:=proc(s,g) local k,yukun:


yukun:=xi(s)+add(g[k]*diff1(xi(s),s,(2*k-1)),k=1..nops(g)):
expand(yukun/(2*H(s))):
end:



#Check185(s,g): checks (18.5) for s and a list g
Check185:=proc(s,g) local G1,G2,k,j:
G1:=G(s,g):

G2:=Zeta(s)+ add(add( g[k]*binomial(2*k-1,j)*diff1(H(s),s,(2*k-1-j))/H(s)*diff1(Zeta(s),s,j),j=0..2*k-1),
          k=1..nops(g)):
G2:=G2/2:
G2-G1:
#simplify(G2-G1):

end: