On the weighted average number of subgroups of Z m × Z n with mn≤x
aa r X i v : . [ m a t h . N T ] A ug On the weighted average number ofsubgroups of Z m × Z n with mn ≤ x Isao Kiuchi and Sumaia Saad Eddin
Abstract
Let Z m be the additive group of residue classes modulo m . For anypositive integers m and n , let s ( m, n ) and c ( m, n ) denote the total numberof subgroups and cyclic subgroups of the group Z m × Z n , respectively.Define e D s ( x ) = X mn ≤ x s ( m, n ) log xmn and e D c ( x ) = X mn ≤ x c ( m, n ) log xmn . In this paper, we study the asymptotic behaviour of functions e D s ( x ) and e D c ( x ). Let Z m be the additive group of residue classes modulo m . Let µ , τ and φ be the M¨obius function, the divisor function and the Euler totient function,respectively. For any positive integers m and n , s ( m, n ) and c ( m, n ) denote thetotal number of subgroups and cyclic subgroups of Z m × Z n , respectively. Theproperties of the subgroups of the group Z m × Z n were studied by Hampejs,Holighaus, T´oth and Wiesmeyr in [1]. We recall that gcd( m, n ) is the greatestcommon divisor of m and n . The authors deduced formulas for s ( m, n ) and c ( m, n ), using a simple elementary method. They showed that s ( m, n ) = X a | m,b | n gcd( a, b ) = X d | gcd( m,n ) φ ( d ) τ (cid:16) md (cid:17) τ (cid:16) nd (cid:17) = X d | gcd( m,n ) dτ (cid:16) mnd (cid:17) , Mathematics Subject Classification 2010: 11A25, 11N37, 11Y60.Keywords: Number of subgroups; Number of cyclic subgroups; Dirichlet series; Divisor func-tion. c ( m, n ) = X a | m,b | n gcd( m/a,n/b )=1 gcd( a, b ) = X a | m,b | n φ (gcd( a, b ))= X d | gcd( m,n ) ( µ ∗ φ )( d ) τ (cid:16) md (cid:17) τ (cid:16) nd (cid:17) = X d | gcd( m,n ) φ ( d ) τ (cid:16) mnd (cid:17) . Here, as usual, the symbol ∗ denotes the Dirichlet convolution of two arithmeti-cal functions f and g defined by ( f ∗ g )( n ) = P d | n f ( d ) g ( n/d ), for every positiveinteger n . Suppose x > S (1) ( x ) := X m,n ≤ x s ( m, n ) , S (2) ( x ) := X m,n ≤ x gcd( m,n ) > s ( m, n ) S (3) ( x ) := X m,n ≤ x c ( m, n ) , S (4) ( x ) := X m,n ≤ x gcd( m,n ) > c ( m, n )The functions S (2) ( x ) and S (4) ( x ) represent the number of total subgroups, andcyclic subgroups of Z m × Z n , respectively, having rank two, with m, n ≤ x .W.G. Nowak and L. T´oth [5] studied the above functions and proved that S ( j ) ( x ) = x X r =0 A j,r (log x ) r ! + O (cid:16) x + ε (cid:17) , where A j,r (1 ≤ j ≤ , ≤ r ≤
3) are computable constants. Moreover, theyshowed that the double Dirichlet series of the functions s ( m, n ) and c ( m, n ) canbe represented by the Riemann zeta function. Later, the above error term hasbeen improved by T´oth and Zhai [8] to O (cid:16) x (log x ) (cid:17) . More recently, Sui and Liu [6] considered the sum of s ( m, n ) and of c ( m, n )in the Dirichlet region { ( m, n ) : m, n ≤ x } . Define D s ( x ) := X mn ≤ x s ( m, n ) and D c ( x ) := X mn ≤ x c ( m, n ) , the authors obtained two asymptotic formulas of D s ( x ) and D c ( x ) by using themethod of exponential sums. They proved that D s ( x ) = xP (log x ) + O (cid:16) x / (log x ) (cid:17) and D c ( x ) = xR (log x ) + O (cid:16) x / (log x ) (cid:17) , where P ( u ) and R ( u ) are polynomials in u of degree 4 with the leading coef-ficients 1 / (8 π ) and 3 / (4 π ), respectively. Put∆ s ( x ) := D s ( x ) − xP (log x ) , ∆ c ( x ) := D c ( x ) − xR (log x ) , s ( x )and ∆ c ( x ) and guessed that ∆ s ( x ) , ∆ c ( x ) ≪ x / ε hold on average. More-over, they conjectured that ∆ s ( x ) , ∆ c ( x ) ≪ x / ε . In this paper, we study the weighted average of s ( m, n ) and c ( m, n ) withweight concerning logarithms. Let e D s ( x ) = X mn ≤ x s ( m, n ) log xmn and e D c ( x ) = X mn ≤ x c ( m, n ) log xmn , then, we have the following results. Theorem 1.
Let the notation be as above. For any positive real number x > ,we have e D s ( x ) = x e P (log x ) + O (cid:16) x log x (cid:17) , (1) and e D c ( x ) = x e R (log x ) + O (cid:16) x log x (cid:17) , (2) where e P ( u ) and e R ( u ) are polynomials in u of degree with computable coeffi-cients. In order to prove our main result, we first show some necessary lemmas.
Lemma 1.
For every z, w ∈ C with ℜ ( z ) , ℜ ( w ) > , we have ∞ X m,n =1 s ( m, n ) m z n w = ζ ( z ) ζ ( w ) ζ ( z + w − ζ ( z + w ) , (3) and ∞ X m,n =1 c ( m, n ) m z n w = ζ ( z ) ζ ( w ) ζ ( z + w − ζ ( z + w ) . (4) Proof.
The proof can be found in [5, Theorem 1].
Lemma 2.
For t ≥ t > uniformly in σ , we have ζ ( σ + it ) ≪ ( t (3 − σ ) log t (cid:0) ≤ σ ≤ (cid:1) ,t (1 − σ ) log t (cid:0) ≤ σ ≤ (cid:1) . Moreover, for σ > we have ζ ( σ + it ) ≪ min (cid:18) σ − , log( | t | + 2) (cid:19) ζ − ( σ + it ) ≪ min (cid:18) σ − , log( | t | + 2) (cid:19) . roof. The first estimate follows immediately from [7, Theorem II.3.8]. Thesecond and third estimates can be found in [6].
Lemma 3.
We have Z T | ζ (1 / it ) | | ζ (1 + 2 it ) | dt ≪ T (log T ) . (5) Proof.
The proof of this result can be deduced from [3, Proposition 2] when k = 1. Our proof is similar in spirit to the proof of Theorem 1.1 in [6]. Both proofsare based on the residue theorem and the classical method to estimate theintegrals. Here, we only prove our theorem for the function e D s ( x ). The proofof the function e D c ( x ) is similar. Suppose that the double Dirichlet series of s ( m, n ) α ( s ) = ∞ X m,n =1 s ( m, n )( mn ) s , has abscissa of convergence σ c . Applying Riesz typical means, (see [4, Chapter5: (5.21) and (5.22)]), show that α ( s ) = s Z ∞ e D s ( x ) x − s − dx and that e D s ( x ) = 12 πi Z σ + i ∞ σ − i ∞ α ( s ) x s s ds, when x > σ > max(0 , σ c ) . Using Lemma 1 with σ = 1 + 1 / log x , wehave e D s ( x ) := X mn ≤ x s ( m, n ) log xmn = 12 πi Z σ + i ∞ σ − i ∞ ζ ( s ) ζ (2 s − ζ (2 s ) x s s ds. Let T ≥ e D s ( x ) = 12 πi Z σ + iTσ − iT ζ ( s ) ζ (2 s − ζ (2 s ) x s s ds + O (cid:18) x ε T (cid:19) , where we used Lemma 2 to estimate12 πi Z σ ± i ∞ σ ± iT ζ ( s ) ζ (2 s − ζ (2 s ) x s s ds ≪ Z + ∞ T (cid:12)(cid:12) ζ ( σ ± it ) (cid:12)(cid:12) | ζ (2 σ − ± i t ) || ζ (2 σ ± i t ) | x σ t dt ≪ x σ + ε Z + ∞ T dtt ≪ x σ + ε T .
4e consider a rectangle D in the s plane with vertices at the points 1 / − iT , σ − iT , σ + iT and 1 / iT , where T ≥
1. Notice that our function ζ ( s ) ζ (2 s − ζ (2 s ) x s s has a pole at s = 1 of order 5 . Then, by the residue theorem we find that R ( x, T ) = I ( x, T ) − I ( x, T ) − I ( x, T ) + I ( x, T ) , (6)where R ( x, T ) = Res s =1 ζ ( s ) ζ (2 s − ζ (2 s ) x s s ,I ( x, T ) = 12 πi Z σ + iTσ − iT ζ ( s ) ζ (2 s − ζ (2 s ) x s s ds,I ( x, T ) = 12 πi Z σ + iT + iT ζ ( s ) ζ (2 s − ζ (2 s ) x s s ds,I ( x, T ) = 12 πi Z + iT − iT ζ ( s ) ζ (2 s − ζ (2 s ) x s s ds,I ( x, T ) = 12 πi Z σ − iT − iT ζ ( s ) ζ (2 s − ζ (2 s ) x s s ds. For calculating R ( x, T ), we recall that the Laurent series expansion of the Rie-mann zeta function at s = 1 is given by ζ ( s ) = 1 s − γ + γ ( s −
1) + γ ( s − + γ ( s − + · · · , where the constants γ n are often called the Stieltjes constants or generalizedEuler constants (see [2]). In particular, γ = 0 . · · · is the well-known the Euler constant. Using the above expansion, the function f ( s ) = ζ ( s ) ζ (2 s −
1) can be written as f ( s ) := ζ ( s ) ζ (2 s −
1) = 1 / s − + 3 γ ( s − + 7 γ − γ ( s − + 8 γ − γ γ + 3 γ ( s − + γ − γ γ + 11 γ + 13 γ γ − γ s − (cid:18) γ − γ γ + 36 γ γ + 21 γ γ − γ γ − γ γ + 34 γ (cid:19) + O ( s − . g ( s, x ) = x s s − ζ − (2 s ), then, we have R ( x, T ) := Res s =1 f ( s ) g ( s, x ) = 12 × g (4) (1 , x ) + 44! × γ × g (3) (1 , x )+ 124! × (cid:0) γ − γ (cid:1) × g (2) (1 , x )+ 244! × (cid:0) γ − γ γ + 3 γ (cid:1) × g ′ (1 , x )+ 244! × (cid:18) γ − γ γ + 11 γ + 13 γ γ − γ (cid:19) × g (1 , x ) , where g ( i ) (1 , x ) denotes the i -th derivative of the function g ( s, x ) with respectto s at s = 1. By careful calculations, we find that g (4) (1 , x ) = 6 π x (log x ) − π ( π + 6 ζ ′ (2)) x (log x ) + 216 π (cid:0) − π ζ ′′ (2) + 48 ζ ′ (2) + 8 π ζ ′ (2) + π (cid:1) x (log x ) + 576 π (cid:0) − ζ ′ (2) + 72 π ζ ′ (2) ζ ′′ (2) − π ζ ′ (2) − π (cid:1) x log x + 576 π (cid:16) − ζ (3) (2) − ζ ′ (2) + 6 ζ ′′ (2) (cid:17) x log x + 144 π (cid:0) ζ ′ (2) + 1728 π ζ ′ (2) (2 ζ ′ (2) − ζ ′′ (2)) + 5 π (cid:1) x + 6912 π (cid:16) ζ ′′ (2) + 9 ζ ′ (2) − ζ ′ (2)(3 ζ ′′ (2) − ζ (3) (2)) (cid:17) x + 576 π (cid:16) − ζ (4) (2) + 4 ζ (3) (2) − ζ ′′ (2) + 12 ζ ′ (2) (cid:17) x,g (3) (1 , x ) = 6 π x (log x ) − π (cid:0) ζ ′ (2) + π (cid:1) x (log x ) + 108 π (cid:0) ζ ′ (2) + 4 π (2 ζ ′ (2) − ζ ′′ (2)) + π (cid:1) x log x − π (cid:0) ζ ′ (2) + 72 π ζ ′ (2) ( ζ ′ (2) − ζ ′′ (2)) + π (cid:1) x − π (cid:16) ζ ′ (2) − ζ ′′ (2) + 2 ζ (3) (2) (cid:17) x,g (2) (1 , x ) = 6 π x (log x ) − π (cid:0) ζ ′ (2) + π (cid:1) x log x + 36 π (cid:0) ζ ′ (2) + 4 π (2 ζ ′ (2) − ζ ′′ (2)) + π (cid:1) x and that g ′ (1 , x ) = 6 π x log x − π (6 ζ ′ (2) + π ) x, g (1 , x ) = xζ (2) . R ( x, T ) becomes R ( x, T ) = x X r =0 B r (log x ) r , (7)where B = 18 π ,B = 3 γ − π − π ζ ′ (2) ,B = − π ζ ′′ (2) + 216 π ζ ′ (2) + 36 π ( − γ + 1) ζ ′ (2) + 32 π (cid:0) γ − γ − γ (cid:1) ,B = − π ζ ′ (2) + 864 π ζ ′ (2) ( − ζ ′ (2) + 3 γ ζ ′ (2) + ζ ′′ (2)) − π (cid:16) γ ζ ′ (2) + ζ ′ (2) (9 − γ ) − ζ ′′ (2) + 18 γ ( − ζ ′ (2) + ζ ′′ (2)) + 2 ζ (3) (2) (cid:17) + 6 π (cid:0) γ − γ + 9 γ − γ γ + 8 γ + 3 γ − (cid:1) ,B = 1 π (cid:0) − γ + 27 γ − γ + 66 γ − γ ( − γ ) − γ (cid:1) + 1 π (6 γ ( −
12 + 36 γ + 13 γ ) − γ ) − π ζ ′ (2) ( − ζ ′ (2) + 6 γ ζ ′ (2) + 3 ζ ′′ (2)) − π (cid:0) γ ζ ′ (2) + 6 ( − γ + 3 γ ) ζ ′ (2) + 9 ζ ′′ (2) − γ ζ ′′ (2) + 42 γ ( − ζ ′ (2) + ζ ′′ (2)) (cid:1) + 12 π (cid:16) γ (cid:16) − γ ) ζ ′ (2) + 6 ζ ′′ (2) − ζ (3) (2) (cid:17) + 4 ζ (3) (2) − ζ (4) (2) (cid:17) + 144 π (cid:0) γ ζ ′ (2) + (9 − γ ) ζ ′ (2) − γ ζ ′ (2) ( ζ ′ (2) − ζ ′′ (2)) + 3 ζ ′′ (2) (cid:1) + 144 π (cid:16) ζ ′ (2) (cid:16) − ζ ′′ (2) + ζ (3) (2) (cid:17)(cid:17) + 62208 π ζ ′ (2) . Again, we use Lemma 2 to estimate the function I ( x, T ) I ( x, T ) = 12 πi Z + Z σ ! ζ ( σ + iT ) ζ (2 σ − iT ) ζ (2 σ + 2 iT )( σ + iT ) x σ + iT dσ ≪ log TT Z + Z σ ! | ζ ( σ + iT ) | | ζ (2 σ − iT ) | x σ dσ ≪ T (log T ) Z (cid:18) xT (cid:19) σ dσ + (log T ) Z σ (cid:16) xT (cid:17) σ dσ ≪ xT + (cid:16) xT (cid:17) + (cid:18) xT (cid:19) ! (log T ) . choosing T = x , we deduce that I ( x, T ) ≪ x − / (log x ) . A similar argumentshows that the function I ( x, T ) is estimated by x − / (log x ) . I ( x, T ). Using Lemmas 2 and 3 we find that I ( x, T ) = 12 π Z T − T ζ ( + it ) ζ (2 it ) ζ (1 + 2 it )( + it ) x + it dt ≪ x + x Z T | ζ ( + it ) | | ζ (1 + 2 it ) | | ζ (2 it ) | t dt ≪ x + x X k ≤ log T log 2 Z k k − | ζ ( + it ) | | ζ (1 + 2 it ) | | ζ (2 it ) | t dt ≪ x + x X k ≤ log T log 2 k ) Z k k − | ζ ( + it ) | | ζ (1 + 2 it ) | dt ≪ x + x X k ≤ log T log 2 k ) · k k ≪ x + x X k ≤ k k (2 ) k + x X k Acknowledgement The second author is supported by the Austrian Science Fund (FWF): ProjectsF5507-N26 and F5505-N26, which are part of the Special Research Program“Quasi Monte Carlo Methods: Theory and Applications”. References [1] M. Hampejs, N. Holighaus, L. T´oth and C. Wiesmeyr, Representing andcounting the subgroups of the group Z m × Z n , Journal of Numbers vol.2014 , Article ID 491428, 6 pages.[2] A. Ivi´c, The Riemann Zeta-Function, Theory and Applications , Dover pub-lications, Inc. Mineola, New York 1985.[3] C. Jia and A. Sankaranarayanan, The mean square of the divisor function, Acta Arith. (2014), 181-208.[4] H. L. Montgomery and R. C. Vaughan, Multiplicative Number Theory I.Classical Theory , Cambridge Studies in advanced mathematics Z m × Z n , Int. J. Number Theory (2014), 363-374.[6] Y. Sui and D. Liu, On the error term concerning the number of subgroups ofthe groups Z m × Z n with mn ≤ x , J. Number Theory (2020), 264-279.[7] G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory ,Graduate Studies in Mathematics, AMS, 2008.[8] L. T´oth and W. Zhai, On the error term concerning the number of subgroupsof the groups Z m × Z n with m, n ≤ x , Acta Arith. (2018), 285-299. Isao Kiuchi: Department of Mathematical Sciences, Faculty of Science, Yamaguchi University,Yoshida 1677-1, Yamaguchi 753-8512, Japan.e-mail: [email protected] Sumaia Saad Eddin: Institute of Financial Mathematics and Applied Number Theory, Jo-hannes Kepler University, Altenbergerstrasse 69, 4040 Linz, Austria.e-mail: sumaia.saad [email protected] [email protected]