MDIV. Multiple divisor fuctios The fuctios τ k For k, defie τ k ( to be the umber of (ordered factorisatios of ito k factors, i other words, the umber of ordered k-tuples (j, j 2,..., j k with j j 2... j k =. So τ ( = ad τ 2 is the usual divisor fuctio τ. These otes presuppose familiarity with the elemetary facts about τ. For a prime p, we have τ k (p = k: the k-tuples are (,..., p,...,, with a sigle p. We also assume familiarity with covolutios. Deote by u the uit fuctio defied by u( = for all. The τ = u u. Also, τ(/ = (h h(, where h( =. MDIV. We have τ k ( = j τ k (j. ( Hece τ k = τ k u = u u u (i which u is repeated k times. Proof. For a fixed divisor j of, the umber of k-tuples with j j 2... j k j = is the umber of (k -tuples j j 2... j k = /j, that is, τ k (/j. Whe j rus through the divisors of, so does /j. The statemet follows. This implies the Dirichlet series idetity τ k ( = ζ(s k, s = but we will ot use this idetity i these otes. Similarly, τ k( = (h h h(. MDIV2. The fuctio τ k is multiplicative. For prime p, ( ( a + k a + k τ k (p a = =. (2 k a So if has prime factorisatio m j= pa j j, the m ( aj + k τ k ( =. j= Proof. The fuctio τ k is multiplicative, sice it is a covolutio of multiplicative fuctios. Alteratively, a direct proof is as follows. If has prime factorisatio m j= pa j j, the k-fold factorisatios of are obtaied by combiig k-fold factorisatios of each p a j j. a j
Now k-fold factorisatios of p a correspod to k-tuples (a, a 2,..., a k with a j 0 for each j ad a + a 2 + + a k = a. Cosider sequeces composed of a s ad k zeros. Clearly, the umber of such sequeces is ( a+k k. I such a sequece, the zeros divide the s ito a ordered sequece of k subsets, with a total of a elemets. So such sequeces correspod to k-tuples (a, a 2,..., a k as above. Alterative proof of (2. I the power series idetity ( x k = ( + x + x 2 + k, the coefficiet of x is k(k +... (k +! ( k + =. But the right-had side shows that this coefficiet is the umber of o-egative k-tuples (, 2,..., k with + 2 + + k =. I particular, τ 3 (p a = (a + (a + 2, 2 τ 4 (p a = (a + (a + 2(a + 3. 6 Yet aother proof of (2 is by iductio o a. This is very simple i the case k = 3: τ 2 (p a = a τ 2 (p b = b=0 a (b + = (a + (a + 2. 2 b=0 We ow give two iequalities for τ k (. MDIV3. We have τ k (m τ k (mτ k ( for all m,. Proof. The statemet is trivial if m or is. Sice τ k is multiplicative, it is eough to prove this for m = p a, = p b, where p is prime ad a ad b. Now so τ k (p a = (a + r, (k! r= [(k!] 2 τ k (p a τ k (p b = (a + r(b + r, r= [(k!] 2 τ k (p a+b = (k! (a + b + r = r= r(a + b + r. The statemet follows, sice (a + r(b + r = r(a + b + r + ab > r(a + b + r. r= 2
MDIV4. For all j, k,, we have τ j (τ k ( τ jk (. so Proof. It is sufficiet to prove this for = p a, where a. Now The statemet follows, sice τ j (p a = a (j + r, a! a (a! 2 τ j (p a τ k (p a = (j + r(k + r, a (a! 2 τ jk (p a = a! (jk + r = a (r + (jk + r. (r + (jk + r (j + r(k + r = rjk r(j + k + r = r(j (k 0. Recallig that τ 2 = τ, we deduce at oce by iductio: MDIV5 COROLLARY. We have τ( k τ K (, where K = 2 k. Summary of some results assumed (A (Basic itegral estimatio, [Jam,.4.2]: Let f(t be a decreasig, o-egative fuctio for t. Write S(x = f( ad I(x = x f(t dt. The for all x, I(x S(x I(x + f(. (A2 (Variat of itegral estimatio, proved by a slight extesio of the same method [Nath, p. 206 208]: Defie S(x, I(x as i (A. Suppose that f(t is o-egative, icreasig for t x 0 ad decreasig for x x 0, with maximum value f(x 0 = M. The I(x M S(x I(x + M. (A3 (Abel s summatio formula [Jam, Prop..3.6]: Let A(x = a( ad let f have cotiuous derivative o [, x]. The a(f( = A(xf(x x A(tf (t dt. 3
(A4 (Partial sums of covolutios [Jam,.8.4]: If A(x = a( ad B(x = b(, the (a b( = j x a(jb ( x = A b(k. j k k x Write H(x =. By (A, applied to f(t = /t, we have A more accurate estimatio of H(x [Jam,.4.] is: where γ is Euler s costat ad q(x /x. log x H(x log x +. (3 H(x = log x + γ + q(x, (4 We also eed the followig estimatio, derived from (A2. Write log x for (log x. The fuctio f(t = (log r t/t icreases for t e r f(e r = (re r, so we deduce log r ad decreases for t e r, with = r + logr+ x + q r (x, (5 where q r (x (re r. Of course, q 0 (x = H(x x = γ + O(/x. I fact, q r(x = γ r + O(x log r x, where γ r is the rth Stieltjes costat, but we wo t use this. O[g(x]. Where coveiet, we use the otatio f(x g(x to mea the same as f(x = Summatio fuctios We write T k (x = τ k (, also S k (x = τ k (. Clearly, T k (x is the umber of k-tuples with j j 2... j k x, ad S k (x is the sum of the reciprocals of these products. Note that T (x = [x] ad S (x = H(x. 4
It is elemetary that T 2 (x = [x/]. This, together with (3, gives the simple estimatio x log x x T 2 (x x log x + x. (6 A more accurate estimatio of T 2 (x is give by the famous theorem of Dirichlet ([HWr, p. 264], [Jam, Prop. 2.5.] ad may other books: T 2 (x = x log x + (2γ x + 2 (x, (7 where 2 (x = O(x /2. The true order of magitude of 2 (x has bee the subject of a great deal of study. Deote by θ 0 the ifimum of umbers θ such that (x is O(x θ. It was already show by Vorooi i 903 that θ 0 [Te, sect..6.4]. The curret best estimate, 3 due to M. N. Huxley, is θ 0 3 0.3490. 46 We will give estimatios of T k (x ad S k (x correspodig to these two levels of accuracy (the more accurate oes are ot eeded for the later applicatio to τ( 2. The startig poit is the followig: MDIV6 PROPOSITION. We have T k (x = T k = [ x τ k ( ] (8 (9 ad S k (x = = S k τ k (H (0. ( Proof. These idetities all follow from (A4. Direct reasoig is also easy, as follows. Fix x. The umber of k-tuples (j, j 2,..., j k, with j j 2... j k x is the umber of (k -tuples with j j 2... j k x/, that is, T k (x/. Hece T k (x = T k (x/. Also, there are τ k ( ways to express as j j 2... j k. There are the [x/] choices of j k such that j k x. Hece T k (x = τ k ([x/]. (0 ad ( are proved i the same way, addig reciprocals of terms istead of coutig them. I the case k = 2, both (8 ad (9 equate to [x/], ad both (0 ad ( to H. 5
MDIV7 COROLLARY. We have xs k (x T k (x T k (x xs k (x. (2 Proof. By (9, τ k ( T k (x τ k. This equates to (2. We shall derive estimatios of S k (x ad T k (x (i that order from estimatios of A k (x, where A k (x = (log x log k. (3 At the first level of accuracy, we have: MDIV8. We have k + logk+ x A k (x k + logk+ x + log k x. (4 Proof. Let f(t = t (log x log tk for t x (also f(t = 0 for t > x. The f(t is decreasig ad o-egative, ad x [ f(t dt = ] x (log x log tk+ = logk+ (x k + k +. The statemet follows, by (A. MDIV9 THEOREM. For all k, S k (x = logk x k! + O(log k x. (5 Also, S k (x logk x. (6 k! Proof. Iductio o k. The case k = is (3. Assume (5 for k, with error term deoted by q k (x. The by (0, S k+ (x = S k = I(x + Q(x, 6
where Q(x = I(x = k! x k( q x logk = k! A k(x, = x logk = A k (x. By (4, I(x = (k +! logk+ x + O(log k x, ad Q(x log k x. Hece (5 holds for k +. Also, S (x = H(x > log x, ad (6 follows easily, usig the left-had side of (4. MDIV0 THEOREM. For all k 2, T k (x = (k! x logk x + O(x log k 2 x, (7 Proof. The case k = 2 is (6. Assume (7 for k. By (2 ad (5, we the have T k+ (x = xs k (x + O[T k (x] = x k! logk x + O(x log k x. Alteratively, oe ca prove (7 directly, without referece to S k (x, i the same way as MDIV9, usig (8 istead of (0. Aother alterative is to prove (5 ad (7 simultaeously by iductio. Istead of usig (4, oe deduces (5 from (7 by Abel summatio; as i MDIV0, the pair of statemets for k the implies (7 for k +. There is o simple lower boud correspodig to (6 for T k (x; i fact, T 2 (x < x log x for some values of x, for example o itervals to the left of the itegers to 5. By MDIV5 ad (7, we have: MDIV PROPOSITION. Write 2 k = K. For some costat C k, τ( k (K! x logk x + C k x log K 2 x. (8 We ow move to the secod level of accuracy. We cosider A (x separately, sice it is much simpler tha the geeral case. MDIV2 LEMMA. We have A (x = 2 log2 x + γ log x + O(. (9 7
Proof. By (4 ad (5, we have A (x = H(x log x log = log x[log x + γ + O(/x] 2 log2 x q (x = 2 log2 x + γ log x + O(. Sice q (x = γ +O(log x/x, the term O( i (9 ca be replaced by γ +O(log x/x for still greater accuracy. We derive the estimates for S 2 (x ad T 3 (x. MDIV3 PROPOSITION. We have S 2 (x = 2 log2 x + 2γ log x + O(. (20 Proof. With q(x as i (4, S 2 (x = H = [ log x log + γ + q ] = A (x + γh(x + q (x, where q (x = q(x/. Sice q(x/ /x, we have q (x. Also, γh(x = γ log x + O(. With (9, we obtai (20. With more effort, oe ca establish the followig estimate, at a third level of accuracy: S 2 (x = 2 log2 x + 2γ log x + c 0 + O(x /2 log x, where c 0 = γ 2 2γ. See DIVSUMS. MDIV4 PROPOSITION. We have T 3 (x = x 2 log2 x + (3γ x log x + O(x. (2 Proof. By (8 ad (7, T 3 (x = T 2 = x (log x log + (2γ x + r(x = xa (x + (2γ xh(x + r(x, 8
where r(x = 2 x /2. /2 By itegral estimatio, (//2 2x /2, so r(x = O(x. (This estimate would ot be improved by usig (x = O(x θ. Also, by (6, xa (x = 2 x log2 x + γx log x + O(x, ad by (4, (2γ xh(x = (2γ x log x + O(x. We ow tur to the geeral case. The estimatio of A k (x requires rather more work. MDIV5 LEMMA. We have k ( r r + ( k = r k +. (22 Hece Proof. Clearly, (k + k k + r + ( r r + ( k = r ( k r ( k +. r + k ( k + = ( r r + k ( k + = ( s s s= = ( k+ =. MDIV6 PROPOSITION. We have Proof. By the biomial theorem, A k (x = k + logk+ x + γ log k x + O(log k x. (23 A k (x = k ( k ( r r k ( k = ( r r 9 log k r x log r log k r x logr.
Usig (5 to substitute for logr, we have A k (x = B k (x + Q k (x, where By (8, Also B k (x = Q k (x = k ( k ( r log k r x logr+ x r r +, B k (x = log k+ x k ( k ( r log k r xq r (x. r k ( r r + ( k = r k + logk+ x. Q k (x = q 0 (x log k x + O(log k x = γ log k x + O(log k x. Remark. This proof is give i [Nath, Theorem 6.6, p. 209], but oly i order to state (4, which, as we have see, ca be proved much more simply. MDIV7 THEOREM. For all k 2, S k (x = k! logk x + c k log k x + O(log k 2 x, (24 where c k = kγ (k!. Proof. Iductio o k. The case k = 2 is (2. Assume (24 for k >, with error term deoted by δ k (x. By (0, we the have S k+ (x = I (x + I 2 (x + P (x, where By (23, By (4, P (x = k x I (x = I (x = k! I 2 (x = c k δ k logk x = k! A k(x, x logk = c ka k (x, k x x logk 2 = A k 2(x. (k +! logk+ x + γ k! logk x + O(log k x. I 2 (x = c k k logk x + O(log k x, ad P (x log k x. Together, these estimatios give the statemet for k +, with c k+ = c k k + γ k!. 0
By (20, c 2 = 2γ, ad it is easily verified by iductio that c k = kγ/(k!. This time, we caot simply use (2 to deduce the correspodig result for T k (x. Istead, we prove it separately, i similar style. MDIV8 THEOREM. For k 3, T k (x = x (k! logk x + d k x log k 2 x + O(x log k 3 x. (25 where d k = kγ (k 2!. Proof. Iductio o k. The case k = 3 is (2. Assume (25 for k, with error term deoted by k (x. By (8, we the have T k+ (x = J (x + J 2 (x + Q(x, where By (23, By (4, J (x = x x (k! logk = J 2 (x = d k x Q(x = k J (x = k! x logk x + J 2 (x = x (k! A k (x, logk 2 x = d kxa k 2 (x, x logk 3 x = xa k 3(x. γ (k! x logk x + O(x log k 2 x. c k k x logk x + O(x log k 2 x ad Q(x x log k 2 x. Together, these estimatios give the statemet for k +, with d k+ = d k k + γ (k!. By (2, d 3 = 3γ, ad it is easily verified by iductio that d k = (kγ /(k 2!. A alterative for (24 is to deduce it from (25 by Abel summatio, but this is ot really ay shorter tha the direct proof give. Summatio of τ( 2 By MDIV9, we have τ( 2 x 6 log3 x + r(x,
where r(x x log 2 x. This, of course, is oly a upper boud estimate. We ow derive a asymptotic estimate. It was origially obtaied by Ramauja. We deote the Möbius fuctio by µ(, ad write M 2 (x = µ(2. We ow use the well-kow estimate where q(x = O(x /2 [HWr, p. 270], [Jam, 2.5.5]. MDIV9. We have τ 2 = τ 3 µ 2. M 2 (x = 6 x + q(x, (26 π2 Proof. Both sides are multiplicative, so it is sufficiet to cosider the values at a prime power p k. We have (τ 3 µ 2 (p k = τ 3 (p k + τ 3 (p k = 2 (k + (k + 2 + 2 k(k + = (k + 2 = τ(p k 2. MDIV20 LEMMA. We have τ 3 ( /2 x/2 log 2 x. (27 Proof. By Abel summatio, τ 3 ( /2 = T x 3(x x + T 3 (t /2 2 t dt. 3/2 Usig oly the fact that T 3 (x x log 2 x, we have T 3 (x x /2 x /2 log 2 x ad x T 3 (t t 3/2 x dt log2 x t dt < /2 2x/2 log 2 x. From (5, oe ca show that the sum i (27 is actually x /2 log 2 x + O(x /2 log x. MDIV2 THEOREM. We have τ( 2 = π x 2 log3 x + O(x log 2 x, (28 τ( 2 = 4π 2 log4 x + O(log 3 x. (29 2
Proof. By MDIV9 ad (A4, with q(x as i (26, we have τ( 2 = τ 3 (M 2 = ( 6 x τ 3 ( π 2 + q. By (5, ad by (27 This establishes (28. x τ 3( = xs 3 (x = 6 x log3 x + O(x log 2 x, τ 3 (q x /2 τ 3 ( x /2 log2 x. We deduce (29 by Abel summatio. Write τ(2 = A(x, ad deote the error term i (28 by r(x. By (A3, where where J 2 = x τ( 2 = J + J 2, J = A(x log 3 x, x ( log 3 t + r(t dt = π 2 t t 2 4π 2 log4 x + R(x, R(x x log 2 t t dt = 3 log3 x. A alterative method [Nath, sect. 7.2] is as follows. Defie { µ(k if = k µ S ( = 2, 0 if is ot a square. Oe shows that τ 2 = µ S τ 4, either directly or by combiig MDIV9 with the easily show idetity µ S u = µ 2. It follows that τ( 2 = µ S (T 4 = µ(kt 4. k 2 k 2 x Now use the estimate (7 for T 4 (x (istead of M 2 (x, together with k y [µ(k/k2 ] = 6/π 2 + O(/y, to deduce (28. This method (more readily tha the first oe ca be refied to give a secod-level estimate. Usig the more accurate estimate (25 for T 4 (x, together with the idetity µ( log = ζ (2 2 ζ(2, 2 = 3
oe fids that where τ( 2 = π x 2 log3 x + C 2 x log 2 x + O(x log x, C 2 = 4γ 2ζ(2 ζ (2 ζ(2 2. Refereces [HWr] G. H. Hardy ad E. M. Wright, Itroductio to the Theory of Numbers, 5th ed., Oxford Uiv. Press (979. [Jam] G. J. O. Jameso, The Prime Number Theorem, Cambridge Uiv. Press (2003. [Nath] Melvy B. Nathaso, Elemetary Methods i Number Theory, Spriger (2000. [Te] Gerald Teebaum, Itroductio to Aalytic ad Probabilistic Number Theory, Cambridge Uiv. Press (995. 4