#C4.txt: Help:=proc(): print(` CheckI23(f,n,x) , CheckI27(s,L) , Dir(x), CheckI210(x) `) : end: with(numtheory): #CheckI23(f,n,x): the rhs (error term) of (2.3) with a=1 b is x #sum(f(n),n=1..x)-int(f,n=1..x), CheckI23:=proc(f,n,x) local n1,L1,R1: L1:=evalf(abs(add(subs(n=n1,f),n1=1..x)-int(f,n=1..x))): R1:=1/2*Int(abs(diff(f,n)),n=1..x)+ 1/2*abs(subs(n=x,f)-subs(n=1,f)): R1:=evalf(R1): [L1,R1]: end: #CheckI27(s1,L): Checks (2.7) for numeric s and pos. integer L CheckI27:=proc(s1,L) local L1,R1,C,s: if Re(s1)<=1 then RETURN(FAIL): fi: L1:=subs(s=s1,diff(Zeta(s),s$L))- (-1)^L*L!/(s1-1)^(L+1): L1:=evalf(L1): R1:=log(2*abs(s1))^(L+1): R1:=evalf(R1): C:=abs(L1/R1): end: #Dir(x): (2.9) sum(tau(n),n=1..x): Dir:=proc(x) local n: add(tau(n),n=1..x): end: #CheckI210(x): checks (2.10) CheckI210:=proc(x) local D1: D1:=Dir(x)-x*(log(x)+2*gamma-1): evalf(D1/sqrt(x)): end: