On a question of A. Schinzel: Omega estimates for a special type of arithmetic functions

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1 Cent. Eur. J. Math. 11(3) DOI: 1.478/s Central European Journal of Mathematics On a question of A. Schinzel: Omega estimates for a special type of arithmetic functions Research Article Manfred Kühleitner 1, Werner Georg Nowak 1 1 Institute of Mathematics, Department of Integrative Biology, BOKU Wien, 118 Vienna, Austria Received 4 January 1; accepted 5 April 1 Abstract: he paper deals with lower bounds for the remainder term in asymptotics for a certain class of arithmetic functions. ypically, these are generated by a Dirichlet series of the form ζ (s)ζ(s 1)ζ M (s)h(s), where M is an arbitrary integer and H(s) has an Euler product which converges absolutely for Rs > σ, with some fixed σ < 1/. MSC: 11N37, 11A5, 11A5, 11L7 Keywords: Arithmetic functions Asymptotic formulas Omega estimates Versita Sp. z o.o. o Professor Andrzej Schinzel on his 75th birthday 1. Introduction 1.1. his article is concerned with a special class C of arithmetic functions f H with a generating Dirichlet series 1 F H (s) = f H (n) n s = ζ (s)ζ(s 1)ζ M (s)h(s), Rs > 1, (1) where M is an integer and H(s) has an Euler product which converges absolutely for Rs > σ, where σ < 1/ is arbitrary but fixed. kleitner@boku.ac.at nowak@boku.ac.at 1 he dependance of f H on the integer M will be suppressed in notation throughout. 477

2 On a question of A. Schinzel: Omega estimates for a special type of arithmetic functions We mention some examples of special arithmetic interest: Firstly, the function f (n) = ( gcd m, n ), m m n see Sloane [17], in a way quantifies the property of n to be not square-free, i.e., to possess non-unitary divisors. (For n square-free, f (n) coincides with the number-of-divisors function d(n).) he function f (n) is generated by the Dirichlet series f (n) = ζ (s)ζ(s 1), Rs > 1. n s ζ(s) his is (1) with M = 1, H(s) = 1 identically. Secondly, consider f 1 (n) = [ ( σ gcd m, n )], m m n where σ denotes the sum-of-divisors function, cf. again Sloane [19]. he corresponding generating function simply reads F 1 (s) = f 1 (n) n s = ζ (s)ζ(s 1), Rs > 1. () his is (1) with M =, H(s) = 1 identically. As a third example, we mention the modified Pillai s function, Sloane [18], his is generated by P (n) = 1 n n gcd(k, n). k=1 P (n) = ζ (s)ζ(s 1) H (s), Rs > 1, n s ζ (s) where H (s) has an Euler product absolutely convergent for Rs > 1/ he class of functions C has been dealt with in detail in a recent paper by Krätzel, Nowak, and óth [7]. In that article, the emphasis was on upper bounds for the error term R fh (x) in the asymptotic formula n x Since F H (s) has a triple pole at s = 1, explicitly f H (n) = Res s=1 Res s=1 ) (F H (s) xs + R fh (x). (3) s ) (F H (s) xs = xp H (log x), s where p H is a quadratic polynomial whose coefficients depend on H and M. Using contour integration and properties of the Riemann zeta-function, it has been proved in [7] that R fh (x) = O ( x /3 log 16/9 x ). Employing Krätzel s method [6], which involves fractional part sums and the theory of (classic) exponent pairs, the slight refinement R fh (x) = O ( x 95/139) 478

3 M. Kühleitner, W.G. Nowak has been obtained (95/139 = ). Finally, bringing in Martin Huxley s deep and new technique [ 4] ( Discrete Hardy Littlewood method ), the authors of [7] deduced the further improvement R fh (x) = O ( x 547/83 log 6947/83 x ), = For the context of the class C within the frame of the theory of arithmetic functions, as well as for a wealth of enlightening related results, see also the recent papers by óth [3, 4] he results of [7] have been presented at the th Czech and Slovak International Conference on Number heory in Stará Lesná, September 11 [8]. At the end of that talk, Professor Andrzej Schinzel raised the following question: What can be said concerning Omega-estimates for the remainder term? he authors of the present paper are very grateful for this valuable stimulation of further research and are pleased to be able to provide the following answer. heorem 1. For any arithmetic function f H C with a generating function F H (s) according to (1), it holds true that, as x, n x f H (n) = Res s=1 ) ( (F H (s) xs x log ) x + Ω. s log M+1 log x 1.4. Remarks. For the simplest case (), it is immediate that f 1 (n) = Ω( n), hence also R f1 (x) = Ω( x). he achievement of the elaborate analysis to come is thus only an improvement by a logarithmic factor. On the other hand, it is easy to see that, e.g. for the first example mentioned, it follows f (n) n log 3 log n, hence our Ω-bound cannot be deduced by consideration of the individual values of the arithmetic function involved. he situation may be compared with the sphere problem in R 3 : If r 3 (n) denotes the number of ways to write the positive integer n as a sum of three squares, then r 3 (n) = 4π 3 x3/ + Ω ( (x log x) 1/) n x is the best Omega-result known to date [1], while, by the very same asymptotics, r 3 (n) = Ω( n). he method of proof which turned out to be appropriate in this problem goes back to ideas due to Ramachandra [14] and Balasubramanian, Ramachandra & Subbarao [1]. hey have been worked out in papers by Schinzel [16], Kühleitner [9], and the authors [1]. However, in the present situation certain adaptions are necessary: On the one hand, f H (n) is not as small as O(n ε ). On the other hand, in the last step it will be advantageous to use special properties of the Riemann zeta-function, instead of a general Ramachandra s theorem on Dirichlet series [14]. It should be mentioned that also in the case M =, when the generating function does not contain any factor involving ζ(s), the present approach seems to give a better result than Soundararajan s method [] which up to date was most successful in the divisor and circle problems.. Preliminaries.1. First of all, we can restrict our analysis to the case H(s) = 1 identically, i.e., to F(s) = f(n) n s = ζ (s)ζ(s 1)ζ M (s), Rs > 1. (4) 479

4 On a question of A. Schinzel: Omega estimates for a special type of arithmetic functions In fact, assume that for some f H C and arbitrarily small c >, Let R fh (x) c x log x log M+1 log x, x x. 1 H(s) = h(n)n s, Rs > σ, where the series converges absolutely for Rs > σ, with σ < 1/ fixed. Since f = f H h, where denotes the convolution of arithmetic functions, it readily follows that, for x large, f(n) = h(k) H (m) = n x k x m x/kf k x ) = Res (F(s) xs + h(k) x s=1 s k p H k>x ( ( x h(k) k p H log x ) ( )) x + R fh k k ( log x k ) + ( ) x h(k)r fh. k k x From this it is immediate that R f (x) c x log x log M+1 log x, which yields a contradiction for c sufficiently small, provided that the theorem has been established for the case of (4)... he assertion will be an easy consequence of the following integral mean result. Proposition. here exist positive constants B and C with the property that R f (u) u e u/ B du C log 5 log M+ log (5) for all sufficiently large. he conclusion from this result to heorem 1 is easy and has many analogues in the literature; see [1, 9, 1]. Nevertheless, we supply the details for convenience of the reader. Assume that for any arbitrarily small constant c >, there exists u such that R f (u) c u log u for all u u log M+1. log u hen, for B as in the proposition, and sufficiently large, R f (u) e u/ B du c 1 u u e u/ B log 4 B u log M+ log u du = c + c log4 log M+ log For c sufficiently small this contradicts (5). B du u + c 1 B e v log 4 ( B v) log M+ log( B v) dv v c log5 log M+ log. It is clear by the asymptotics with O-terms cited, that the main term in this calculation amounts to Res s=1 (F(s)xs /s). 48

5 M. Kühleitner, W.G. Nowak.3. One special feature of the present situation as opposed to the cases considered in [1, 9, 1, 16] is that the function f(n) is not small, i.e., not O(n ε ) for each ε >. However, if f is generated by ζ (s)ζ(s 1)ζ M (s), call f the arithmetic function generated by ζ (s)ζ(s 1)ζ M (s). hen, f(n) f(n) = m 3 n 1/+ε. m 1 m (m 3 k 1 k M ) =n Hence, using a trivial version of (3), applied to f, and summation by parts, it follows that n β f (n) X 3/+β+ε (6) n X for each fixed β > 3/, large X, and any ε >..4. he following auxiliary result is classic and provides some information that the factor of F(s) involving ζ(s) is not too harmful close to the critical line. Lemma 3. Let ε > be a sufficiently small constant. hen, for a sufficiently large real parameter, there exists a set A( ) [, ] with the following properties: (i) A( ) is the union of at most O( ε ) open intervals, with a total length of O( ε ). (ii) sup t [, ]\A( ) ζ(1 +it) ±1 log log. (iii) here exist a real number δ(ε ) > and a certain constant C such that ζ(s) ±1 C uniformly in Rs 1/ δ(ε ), Is [, ] \ A( ). Proof. his result is contained in [14, heorem 1] and [15, Lemma 3.]. An extension to Dedekind zeta-functions, along with a neat proof, was given in [11, Lemma and formula (.6)]..5. We conclude this section by quoting a deep and celebrated result due to Montgomery and Vaughan which provides a mean-square bound for Dirichlet polynomials. Lemma 4. For an arbitrary sequence of complex numbers (γ n ) with the property that n γ n converges, and a large real parameter X, X γ n (n +u) dt it = γ n (X + O(n)), uniformly in 1 u 1, and X n= n= γ n n dt it = γ n (X + O(n)). he proof is an immediate consequence of Montgomery and Vaughan [1, Corollary, formula (1.9)]. 481

6 On a question of A. Schinzel: Omega estimates for a special type of arithmetic functions 3. Proof of Proposition 3.1. For positive real sufficiently large, we construct a set U( ) on the real line as follows: Let J = [log /(1 log )], then J 9/1. For j = 1,..., J, set j = j. For some appropriate small ε >, consider the sets A( j ) furnished by Lemma 3. Let A( j ) = ] (j) a i, b (j) [ i i I j be the decomposition of A( j ) into # I j = O( ε ) open intervals of total length O( ε ). hen we define U( ) = J ] j=1 i I j a (j) i [ log, b (j) i + log. By construction, it is clear that U( ) consists of O( ε ) open intervals of total length O( ε ). 3.. We set y = B, with a suitably large positive constant B, for throughout what follows. It suffices to consider those values of for which R f (u) u e u/y du log 6. (7) (Otherwise the assertion of Proposition is obvious.) In this subsection, our aim is to deduce the asymptotic representation f(n) F(s) = n s e n/y + O(1), (8) for which will be assumed throughout the sequel. By a version of Perron s formula, see, e.g., [13, p. 38], We use Stirling s formula in the crude form s = 1 + it, t [ J, ] \ U( ), (9) f(n) n s e n/y = 1 πi +i i Γ(σ + iτ) exp ( π ) τ τ σ 1/, τ, F(s + w)y w Γ(w) dw. (1) which holds uniformly in every strip σ 1 σ σ. From this it is an immediate consequence that, for every fixed k Z +, Γ (k) (σ + iτ) exp ( π4 ) τ, τ, (11) again uniformly in any strip of this kind. 48

7 M. Kühleitner, W.G. Nowak It is easy to see that we may break off the part of the integration line in (1) corresponding to w > log, with an error of only O(1), ±i F(s + w)y w Γ(w) dw B ( w 3/ e (π/)w dw B exp π ) 4 log 1. ±i log log Next, we replace the remaining line of integration by a broken line segment L which joins (in this order) i log, δ(ε ) i log, δ(ε ) + i log, and + i log, where δ(ε ) has the meaning as in Lemma 3. By Lemma 3 (iii), and known upper bounds for the zeta-function, F(s + w) C, with some positive constant C, as long as w lies on L and s satisfies (9). herefore, δ(ε )+i log F(s + w)y w Γ(w) dw C Bδ(ε ) Γ( δ(ε ) + iu) du 1 δ(ε ) i log if we have chosen B C /δ(ε ), and ±i log δ(ε )±i log ( F(s + w)y w Γ(w) dw C +B log 3 exp π ) log 1. Since these integrals are bounded, the main contribution to the right-hand side of (1) comes from the residue of the integrand F(s + w)y w Γ(w) at w =, which amounts to F(s). his readily yields (8) It is an easy consequence of (7) that there exists some [, ], not an integer, for which R f ( ) e /y log 3, 1 y R f (u) u e u/y du log 3. (1) his can be readily verified following the example of [1, p. 111, Lemma 4]. With this choice of, we will split up the series on the right-hand side of (8). In this subsection, we shall handle f(n) n s e n/y = n> u s e u/y d f(n) = n u u s e u/y d(up H (log u)) + where we have used Stieltjes integrals. Integrating by parts, we obtain u s e u/y dr f (u) = I 1 + I, (13) I = R f (u)u s e u/y u= u= +s u s 1 e u/y 1 R f (u) du + u s e u/y R f (u) du y +1 = s (n + v) s 1 e (n+v)/y R f (n + v) dv + O(log 3 ). n> (14) 483

8 On a question of A. Schinzel: Omega estimates for a special type of arithmetic functions Here = [ ] 1, and the bounds (1) have been used. o estimate I 1, we have to deal with integrals of the form u s log r ue u/y du, with r {, 1, } and s satisfying (9). We write this as u s log r ue u/y du u s log r u du + u s log r u(1 e u/y ) du = J 1 J + J 3. By the aylor expansion, J 3 1 y Integrating by parts r times, we infer that u 1/ log r u du B+3/+ε 1. J 1/ log r 1 s 1/ log r 9/1 1. Finally, by (11), J 1 = y 1 s u s log r (uy)e u du = y 1 s Altogether this shows that I 1 1, hence, together with (8), (13) and (14), for s satisfying (9). F(s) = n f(n) n s e n/y + s +1 r ρ= ( ) r log r ρ yγ (ρ) (1 s) 1. ρ R f (n + u) (n + u) s+1 e (n+u)/y du + O(log 3 ) (15) n> 3.4. he next step is to prove that [ J, ]\U( ) F(1/ + it) 1/ + it dt 1 + R f (u) u e u/y du. (16) By (15), the left-hand side of (16) is 9/5 J f(n) n n 1/+it e n/y dt + J +1 R f (n + u) (n + u) 3/+it e (n+u)/y du dt + O ( 9/1 log 6 ). (17) n> o bound the first integral here, we use Lemma 4. In this way, 9/5 J n f(n) n 1/+it e n/y dt 9/5 (f(n)) ( + O(n)) 3/1+ε, (18) n n 484

9 M. Kühleitner, W.G. Nowak for any ε >, by an appeal to (6). Similarly, using Cauchy s inequality and Lemma 4 again, we see that the second term of (17) is +1 R f (n + u) (n + u) n> 3/+it e (n+u)/y dt du +1( ) R f (n + u) e (n+u)/y ( + O(n)) du (n + u) 3 n> +1 ( ) R f (n + u) e (n+u)/y du (n + u) n> R f (u) e u/y du. u ogether with (17) and (18), this readily yields (16) In order to complete the proof of Proposition, it remains to show that [ J, ]\U( ) F(1/ + it) 1/ + it dt log 5 log M+ log. (19) In fact, by the functional equation of the zeta-function, see [, p. 95], ( ) ( ( 1 F + it = 1 )ζ( it)ζ ζ + it M (1 + it) 1 ζ )ζ + it M+1 (1 + it) t 1/ for t large. Further, by Lemma 3 (ii) and the construction of U( ) in subsection 3.1, it follows that ζ(1 + it) ±1 log log for t [ J, ] \ U( ). herefore, [ J, ]\U( ) F(1/ + it) 1/ + it dt log M+1 log [ J, ]\U( ) ( ) 1 4 ζ + it dt t. () For t U( ), we use the classic pointwise bound ζ(1/ + it) t 1/6+ε, see [, heorem 5.5]. Since the total length of U( ) is O( ε ), it follows that ( ) 1 4 ζ + it dt 3/1+3ε. (1) t U( ) On the other hand, by the known asymptotics for the fourth moment of the zeta-function, see [5, p. 19], J ( ) 1 4 ζ + it dt t log 5. () Combining (), (1), and (), we readily establish (19). his completes the proof of Proposition and, by the observation in subsection., also that of heorem 1. Acknowledgements he authors gratefully acknowledge support from the Austrian Science Fund (FWF) under project Nr. P847-N

10 On a question of A. Schinzel: Omega estimates for a special type of arithmetic functions References [1] Balasubramanian R., Ramachandra K., Subbarao M.V., On the error function in the asymptotic formula for the counting function of k-full numbers, Acta Arith., 1988, 5(), [] Huxley M.N., Area, Lattice Points, and Exponential Sums, London Math. Soc. Monogr. (N.S.), 13, Oxford University Press, New York, 1996 [3] Huxley M.N., Exponential sums and lattice points. III, Proc. London Math. Soc., 3, 87(3), [4] Huxley M.N., Exponential sums and the Riemann zeta-function. V, Proc. London Math. Soc., 5, 9(1), 1 41 [5] Ivić A., he Riemann Zeta-Function, Wiley-Intersci. Publ., John Wiley & Sons, New York, 1985 [6] Krätzel E., Lattice Points, Math. Appl. (East European Ser.), 33, Kluwer, Dordrecht, 1988 [7] Krätzel E., Nowak W.G., óth L., On certain arithmetic functions involving the greatest common divisor, Cent. Eur. J. Math., 1, 1(), [8] Krätzel E., Nowak W.G., óth L., On a class of arithmetic functions connected with a certain asymmetric divisor problem, In: th Czech and Slovak International Conference on Number heory, Stará Lesná, September 5 9, 11 (abstracts), Slovak Academy of Sciences, Bratislava, [9] Kühleitner M., An Omega theorem on Pythagorean triples, Abh. Math. Sem. Univ. Hamburg, 1993, 63, [1] Kühleitner M., Nowak W.G., An Omega theorem for a class of arithmetic functions, Math. Nachr., 1994, 165, [11] Kühleitner M., Nowak W.G., he average number of solutions of the Diophantine equation U + V = W 3 and related arithmetic functions, Acta Math. Hungar., 4, 14(3), 5 4 [1] Montgomery H.L., Vaughan R.C., Hilbert s inequality, J. London Math. Soc., 1974, 8, 73 8 [13] Prachar K., Primzahlverteilung, Springer, Berlin Göttingen Heidelberg, 1957 [14] Ramachandra K., A large value theorem for ζ(s), Hardy Ramanujan J., 1995, 18, 1 9 [15] Ramachandra K., Sankaranarayanan A., On an asymptotic formula of Srinivasa Ramanujan, Acta Arith., 3, 19(4), [16] Schinzel A., On an analytic problem considered by Sierpiński and Ramanujan, In: New rends in Probability and Statistics,, Palanga, 1991, VSP, Utrecht, 199, [17] Sloane N., On-Line Encyclopedia of Integer Sequences, # A55155, [18] Sloane N., On-Line Encyclopedia of Integer Sequences, # A7843, [19] Sloane N., On-Line Encyclopedia of Integer Sequences, # A14316, [] Soundararajan K., Omega results for the divisor and circle problems, Int. Math. Res. Notices, 3, 36, [1] Szegö G., Beiträge zur heorie der Laguerreschen Polynome. II: Zahlentheoretische Anwendungen, Math. Z., 196, 5, [] itchmarsh E.C., he heory of the Riemann Zeta-Function, Clarendon Press, Oxford University Press, Oxford, 1986 [3] óth L., Menon s identity and arithmetical sums representing functions of several variables, Rend. Sem. Mat. Univ. Politec. orino, 11, 69(1), [4] óth L., Weighted gcd-sum functions, J. Integer Seq., 11, 14(7), #