COMPUTING SUMS AND THE AVERAGE VALUE OF THE DIVISOR FUNCTION Abstract. We itroduce a method for computig sums of the form f( where f( is ice. We apply this method to study the average value of d(, where d( is the umber of positive divisors of.. Itroductio: Computig sums Let be a iteger. Cosider the problem of summig + + 3 + + ( + = =. = We will compute this sum i two differet ways. First, we observe that if S =, the S = + + 3 + 4 + + ( + S = + ( + ( + ( 3 + + +. Summig the left had side ad the right had side, we have that Solvig for S, we have that As a eample, S = S = ( +. = ( + = +. 00 + + 3 + 4 + 5 + 6 + 7 + + 99 + 00 = i = i= 00 0 To describe the secod method of computig S, we will use the fact that S = = 5050. Date: March, 07.
is approimately equal to the area uder the curve f( = from = to =. Note that this area, which we write as t dt, is equal to. (Thik of the area uder f( = as a right isosceles triagle with the two equal sides of legth. We are almost there; we are just missig +. To recover this last bit, we itroduce the floor fuctio For eample, t = largest iteger such that t =,.999 =, π = 3. (We sometimes call the iteger part of. If we write A for the area uder the curve f(t = t from t = to t =, the we fid that S A = + t dt t dt = + ( dt. That is, S A equals the area uder the curve g(t = from t = up to t =. Now, ( dt = + i= i+ i (t i dt = + i= = + = +, which is the remaiig piece. This techique of relatig a sum to a itegral (i.e., area uder a curve is a very importat techique i moder mathematics because we have much better kowledge of how to compute itegrals tha we do sums. Theorem. (Euler s summatio formula. Suppose that f(t is a fuctio whose derivative f (t is cotiuous o the iterval t, where is a iteger. The f( + f( + f(3 + + f( + f( = f( equals f( + f(t dt + (f (t dt. Recall that f (t, the derivative of f, is the slope of f at the poit (t, f(t. Eercise.. Show that + + 3 + 4 + + ( + = = 3 3 + + 6 for ay iteger. (Hit: If f(t = t, the t dt = ( 3 /3 ad f (t = t o the iterval t.
Eercise.3. Show that 3 + 3 + 3 3 + 4 3 + + ( 3 + 3 = = 3 = 4 4 + 3 + 4 for ay iteger. (Hit: If f(t = t 3, the t3 dt = ( 4 /4 ad f (t = 3t o the iterval t.. A importat eample: The harmoic series Oe of the most importat sums i mathematics is the harmoic sum + + 3 + 4 + + + =. (Remember, we take to be a positive iteger here. We ca ow compute this sum eactly usig Theorem. ad the fact that dt = l, t where l is the atural logarithm of. Recall that log 0 ( = l, l 0 =.305850999405... l 0 Eercise.. Show that if is a iteger ad ˆ is the umber of digits i, the ˆ l. l 0 Usig Theorem., we have that (. = + t dt dt = l + dt. t t Eercise.. Let be a iteger. Usig the fact that t dt =, show that < dt <. t (Hit: Show that 0. Notatio.3. If g( > 0 for all a, we say that f( = O(g( = (read: f( is big oh of g( to mea that f(/g( is bouded for a. That is, there eists a costat M > 0 such that f( M g( for all a. A equatio of the form f( = h( + O(g( meas that f( h( = O(g(. O( g(t dt for a. a 3 If f(t = O(g(t for t a, the a f(t dt =
Eercise.4. Prove the followig: si( = O( for 0. log = O( for. + = O( for. = + O( for 0. Eercise. shows us that for all itegers. Thus < l <. is very-well approimated by l for all values of, but for all itegers, l. Eercise.5. Show that if, the dt = γ + O(, t where γ = 0.577566490... Thus ( = log + γ + O for all. Eercise.6. Show that as grows, the quatity l approaches γ. Eercise.7. Determie whether γ ca be epressed as the ratio of two itegers a/b, where a b 0 ad a has o prime factors i commo with b. 3. The average behavior of the divisor fuctio Let be a positive iteger. I the study of umbers, it is very importat to uderstad the umber of positive itegers that divide. To this ed, we defie τ( = the umber of distict positive itegers d such that d divides. For eample, τ( =, τ( = because ad divide, τ(3 = because ad 3 divide 3, τ(4 = 3 because ad ad 4 divide 4, ad so o. Let s eplore the behavior of τ(. Eercise 3.. Compute τ( for all itegers 00. Try to fid some patters. Eercise 3.. Suppose that p is prime. What is τ(p? τ(p 3? What is τ(p k for ay positive iteger k? 4
Let p =, p = 3, p 3 = 5, p 4 = 7, ad i geeral, let p deote the -th prime. Defie # = p p p 3 p p. Eercise 3.3. What is τ(3#? τ(4#? τ(5#? How large is τ(# for ay iteger? The precedig eercises show that τ( behaves pretty wildly. Notice that τ(p = for every prime p, but τ(# grows very quickly. This begs the questio: How do we study τ(? We ca take a cue from statistics ad study the mea value of τ(. That is, we study τ( istead of τ(. We ow itroduce the otatio d to say that d divides. Oe way that we ca write τ( is τ( = d That is, we sum for every distict divisor d of. Now, τ( =. We ow observe that divisors come i pairs: if d, the we ca write = dq for some iteger q. (This might be the same as d if, say, = d. Thus we ca rewrite the sum as (3.. d τ( =. This ca be iterpreted as a sum eteded over poits with iteger coordiates i the dqplae; we call such poits lattice poits. The lattice poits i dq = lie o a hyperbola, so the ier sum i (3. couts the umber of lattice poits which lie o the hyperbolas dq =, dq =, dq = 3,..., dq =. For each fied d, we ca cout first those lattice poits o the horizotal lie segmet q /d, ad the sum over all d. Therefore, (3. Now, so d,q dq τ( =. q /d d q /d = = d d + O(, = ( d + O( = ( d + O = d + O(. d q /d d d d d 5
But we already proved that Thus so We coclude d τ( = d ( d = log + γ + O. q /d = l + O(, τ( = l + O(. Theorem 3.4. The mea value of τ( is l. Eercise 3.5. Show that for, l = l t dt + dt t = log + + dt t = log + O(log. (Hit: Use the fact that l t dt = l + ad (l t = /t. Eercise 3.6. Show that as grows, approaches zero. Defie (τ( l! = 3 4 (. Eercise 3.7. Show that l(! = l + O(l. Eercise 3.8. Show that So, as grows,! approaches l(! = l π (. e l + l π ( + O. 6