Gál-type GCD sums beyond the critical line
aa r X i v : . [ m a t h . N T ] A p r GÁL-TYPE GCD SUMS BEYOND THE CRITICAL LINE
ANDRIY BONDARENKO, TITUS HILBERDINK, AND KRISTIAN SEIPA
BSTRACT . We prove that N X k , ℓ = ( n k , n ℓ ) α ( n k n ℓ ) α ≪ N − α (log N ) b ( α ) holds for arbitrary integers 1 ≤ n < ··· < n N and 0 < α < b ( α ). This estimate complementsrecent results for 1/2 ≤ α ≤ < α <
1. I
NTRODUCTION
The study of greatest common divisor (GCD) sums of the form(1) N X k , ℓ = ( n k , n ℓ ) α ( n k n ℓ ) α begins with Gál’s theorem [6] which asserts that when α = C N (log log N ) is an optimalupper bound for (1), with C an absolute constant independent of N and the distinct positiveintegers n , ..., n N (the best possible value for C is 6 e γ / π , where γ is Euler’s constant, asshown recently by Lewko and Radziwiłł [8]). Dyer and Harman [5], motivated by applicationsin the metric theory of diophantine approximation, obtained the first estimates for the range1/2 ≤ α <
1. Recent work of Aistleitner, Berkes, and Seip [2] for 1/2 < α < α = N X k , ℓ = ( n k , n ℓ ) α ( n k n ℓ ) α ≪ N exp ³ c ( α ) (log N ) − α (loglog N ) α ´ , 1/2 < α < N exp ³ A q log N logloglog N loglog N ´ , α = Mathematics Subject Classification. which are optimal, up to the precise values of the constants c ( α ) and A ; the asymptotic be-havior of c ( α ) has been clarified both when α ց α ր ζ ( s ). This line of research was initiated in work of Soundararajan[11] and Hilberdink [7] and later pursued by Aistleitner [1] who was the first to make the linkto Gál-type estimates. Recently, using Soundararajan’s resonance method [11] and a certainlarge Gál-type sum for α = c , 0 < c < p β , 0 < β <
1, such that the maximum of | ζ (1/2 + i t ) | on the interval T β ≤ t ≤ T exceeds exp ¡ c p log T log log log T / log log T ¢ for all T large enough.These developments have led us to look more closely at the “phase transition” at α = < α < ζ ( σ + i t ) beyond the critical line σ = α is replaced by 1 − α , as one mighthave expected from the functional equation for ζ ( s ).To state our main result, we let M denote an arbitray finite set of positive integers andintroduce the quantity Γ α ( N ) : = max | M |= N N X m , n ∈ M ( m , n ) α ( mn ) α . Theorem 1.
For every α , < α < , there exist positive constants a ( α ) and b ( α ) such that (3) N − α (log N ) a ( α ) ≤ Γ α ( N ) ≤ N − α (log N ) b ( α ) for sufficiently large N . Before giving the proof of this theorem, we will in the next section set the stage by consid-ering the simpler but closely related question of finding the largest eigenvalue of the positivedefinite matrix ( m , n ) α /( mn ) α , 1 ≤ m , n ≤ N : ÁL-TYPE GCD SUMS BEYOND THE CRITICAL LINE 3
Theorem 2.
For every α , < α < , there exists a constant C α such that (4) max | a | +···+| a N | = X m , n ≤ N a m ¯ a n ( m , n ) α ( mn ) α ≤ C α N − α .We refer to [7] for further information and for the precise asymptotics of the maximum in(4) in the range 1/2 ≤ α ≤ N ). One may notice that it would not be possible toconstruct a similar example if we required the set to consist only of square-free numbers.Hence it remains an open problem to prove the analogue of Theorem 1 in the square-freecase. More specifically, we may ask whether the logarithmic power can be discarded in thiscase as well. 2. P ROOF OF T HEOREM S ( a ) : = N X d = X m , n ≤ N ( m , n ) = d | a m a n | d α ( mn ) α ≤ N X d = Ã X m , n ≤ N / d | a md | m α ! .We introduce the multiplicative function g ( m ) : = P d | m d − + α . By the Cauchy–Schwarz in-equality,(5) S ( a ) ≤ N X d = Ã X m ≤ N / d | a md | m α ! ≤ N X d = X m ≤ N / d | a md | g ( m ) X m ≤ N / d g ( m ) m α .To estimate the sum P m ≤ N / d g ( m ) m α , we notice that X n ≤ x g ( n ) n α = X n ≤ x n α X d | n d − α = X d ≤ x d + α X n ≤ x / d n α ≪ X d ≤ x d + α ³ xd ´ − α ≪ x − α . ANDRIY BONDARENKO, TITUS HILBERDINK, AND KRISTIAN SEIP
Hence by (5) we have S ( a ) ≪ N − α N X d = X m ≤ N / d | a md | d − α g ( m ) = N − α N X n = | a n | X d | n d α − g ( n / d ) .So to finish the proof of Theorem 2, it is sufficient to show that(6) h ( n ) : = X d | n d α − g ( n / d )is a bounded arithmetic function. We observe that h ( n ) is a multiplicative function, whichmeans that it suffices to consider h ( p m ) = m X ℓ = p (2 α − ℓ g ( p m − ℓ ) ≤ g ( p m − r ) r X ℓ = p (2 α − ℓ + m X ℓ = r + p (2 α − ℓ for any r . Taking r = [ m /3] and using 1/ g ( p m ) → − p α − as m → ∞ shows that h ( p m ) ≤ p m sufficiently large. We infer from this that h ( n ) is a bounded arithmetic function.As far as the numerical value of the constant C α in Theorem 2 is concerned, we have con-fined ourselves to the following special case which seems to be of independent interest:(7) 1 N N X m , n = ( m , n ) α ( mn ) α = ζ (2 − α ) ζ (2)(1 − α ) N − α + O (1). Proof of (7) . Write F α ( N ) for the sum on the left and put S α ( x ) : = P m , n ≤ x ( m , n ) = mn ) α . Then F α ( N ) = X d ≤ N X m , n ≤ N ( m , n ) = d ( m , n ) α ( mn ) α = X d ≤ N S α ( N / d ).Also let T α ( x ) = P n ≤ x n α = − α x − α + O (1). Then T α ( x ) = X m , n ≤ x mn ) α = X d ≤ x d α X m , n ≤ x / d ( m , n ) = mn ) α = X d ≤ x d α S α ³ xd ´ .By Möbius inversion, S α ( x ) = P d ≤ x µ ( d ) d α T α ( xd ) and so F α ( N ) = X d ≤ N β ( n ) T α ³ Nn ´ ,where β ( n ) = P d | n µ ( d ) d α . We note that 0 < β ( n ) ≤ n . Thus F α ( N ) = − α ) X n ≤ N β ( n ) ³³ Nn ´ − α + O ³ Nn ´ − α ´ = N − α (1 − α ) X n ≤ N β ( n ) n − α + O ³ N − α X n ≤ N n − α ´ . ÁL-TYPE GCD SUMS BEYOND THE CRITICAL LINE 5
The final term is O ( N ), while P n > N β ( n ) n − α ≤ P n > N n − α ≪ N α − . Finally ∞ X n = β ( n ) n − α = ζ (2 − α ) ζ (2) ,giving the result. ■
3. P
ROOF OF THE BOUND FROM ABOVE IN T HEOREM ω ( n ) denotes the number of distinct prime factors in n and d ( n ) is thedivisor function.We begin by stating the main auxiliary result used to prove the upper bound in (3). Lemma 1.
For every finite set M of positive integers there exists a divisor closed set M ′ of posi-tive integers with | M ′ | = | M | such that X m , n ∈ M ( m , n ) α ( mn ) α ≤ X m , n ∈ M ′ ( m , n ) α ( mn ) α ω ( mn /( m , n ) ) . Proof.
Following the proof of [2, Lemma 2], we transform M into M ′ by means of the follow-ing algorithm. Fix a prime p such that p divides some number in M . Then there exist distinctnumbers m j , j =
1, ..., ℓ with ℓ ≤ | M | such that we may write M = ℓ [ j = M j ,where M j consists of those m in M such that m / m j is a power of p . We then replace the num-bers in M j by the numbers m j , m j p , ..., m j p | M j |− . This transformation is then performed forevery prime dividing some number in M . A close inspection of the largest possible change inthe GCD sum in each step of this series of transformations (carried out in detail in the proofof [2, Lemma 2]) gives the desired estimate. ■ We will also need the following two lemmas.
Lemma 2.
Suppose that < α < and that β is a real number. Then for every β ′ > β /(2 α ) there exists a positive constant C with the following property. If K is a set of positive integers ANDRIY BONDARENKO, TITUS HILBERDINK, AND KRISTIAN SEIP with | K | = K , then (8) X m ∈ K d ( m ) β m α ≤ C K − α [log K ] β ′ − . Proof.
We begin by observing that X m ∈ K d ( m ) β m α ≤ K X m = d ( m ) β m α + ∞ X ℓ = X ℓ K < m ≤ ℓ + Kd ( m ) β > αℓ d ( m ) β m α .Now X ℓ K < m ≤ ℓ + Kd ( m ) β > αℓ d ( m ) β m α ≤ − ( β ′ / β − αℓ X ℓ K < m ≤ ℓ + K d ( m ) β ′ m α ≪ − ( β ′ / β − αℓ · (1 − α ) ℓ K − α (log 2 ℓ K ) β ′ − ,where we used the classical formula X n ≤ x d ( n ) β ′ = B x (log x ) β ′ − ¡ + O ((log x ) − ) ¢ which holds with B an absolute constant [12, 10]. It follows that the sum over ℓ is dominatedby a convergent geometric series if β ′ > β /(2 α ). ■ We mention without proof that a more careful analysis shows that the exponent 2 β ′ − α (2 β ′ −
1) with the same requirement that β ′ > β /(2 α ). Using results on the distribution of ‘large’ values of d ( n ) (see [9]), we can show thatthis is optimal in the sense that the inequality fails with any exponent less than 2 α (2 β /(2 α ) − Lemma 3. If M is a divisor closed set of square-free numbers, then | p M | ≤ | M | for everyprime p in M .Proof. Suppose that | p M | = ℓ and write p M = { m , . . . , m ℓ }. Then M contains pm , . . . , pm ℓ and hence also m , . . . , m M , since it is divisor closed. As M is square-free, these numbers areall distinct, and so it follows that | M | ≥ ℓ . ■ ÁL-TYPE GCD SUMS BEYOND THE CRITICAL LINE 7
We are now prepared to prove the bound from above in (3). To begin with, we define e Γ α ( N ) : = max M divisor closed, | M |= N N X m , n ∈ M ( m , n ) α ( mn ) α ω ( mn /( m , n ) ) .By Lemma 1, we have Γ α ( N ) ≤ e Γ α ( N ), which means that it suffices to estimate e Γ α ( N ). Hencewe assume that the set M is divisor closed and estimate instead the sum e S : = X m , n ∈ M ( m , n ) α ( mn ) α ω ( mn /( m , n ) ) .In what follows, M ∗ will denote the subset of M consisting of the square-free numbers in M .In addition, given m in M ∗ , we let M ( m ) denote the subset of M consisting of those numbers n in M such that p | n if and only if p | m . Hence M = [ m ∈ M ∗ M ( m ) and X m ∈ M ∗ | M ( m ) | = N .Now suppose that k and ℓ are in M ∗ and that | M ( k ) | ≥ | M ( ℓ ) | . We then find that X m ∈ M ( k ), n ∈ M ( ℓ ) ( m , n ) α ( mn ) α ω ( mn /( m , n ) ) ≤ ( k , ℓ ) α ( k ℓ ) α ω ( k ℓ /( k , ℓ ) ) X n ∈ M ( ℓ ) Y p | k ¡ + ∞ X ν = p − ν ¢ ≪ ( k , ℓ ) α ( k ℓ ) α ω ( k ℓ /( k , ℓ ) ) | M ( ℓ ) | d ( k ) ε ,where the implicit constant in the latter relation only depends on α and ε . Here ε can be anypositive number, but in what follows we will require that 0 < ε < − α . We infer from thelatter relation that e S ≪ X m , n ∈ M ∗ | M ( m ) | d ( m ) ε | M ( n ) | d ( n ) ε ( m , n ) α ( mn ) α ω ( mn /( m , n ) ) .This leads to the bound e S ≪ X k ∈ M ∗ X m ∈ k M ∗ | M ( mk ) | d ( mk ) ε d ( m ) + ε m α .By the Cauchy–Schwarz inequality, we obtain from this that e S ≪ X k ∈ M ∗ d ( k ) ε X n ∈ k M ∗ | M ( nk ) | d ( n ) β X m ∈ k M ∗ d ( m ) β + + ε m α , ANDRIY BONDARENKO, TITUS HILBERDINK, AND KRISTIAN SEIP where β is a positive parameter to be chosen later. Using Lemma 2 and the estimate | k M | ≤ N − ω ( k ) ,which we get from Lemma 3, we therefore get e S ≪ N − α (log N ) β ′ − X k ∈ M ∗ d ( k ) α + ε − X n ∈ k M ∗ | M ( nk ) | d ( n ) β = N − α (log N ) β ′ − X m ∈ M ∗ | M ( m ) | X k | m (1 − α + ε )) ω ( k ) + βω ( n / k ) with β ′ > ( β + + ε )/(2 α ). Since n X d | n (1 − α + ε )) ω ( d ) + βω ( n / d ) is a multiplicative function, and n is squarefree, it suffices to make sure that12 − α + ε ) + β ≤ β ≥ log − α + ε ) − log 2to obtain the uniform bound X k | m (1 − α + ε )) ω ( k ) + βω ( n / k ) ≤ e S ≪ N − α (log N ) β ′ − X m ∈ M ∗ | M ( m ) | = N − α (log N ) β ′ − ,which in turn leads to the desired conclusion.4. P ROOF OF THE BOUND FROM BELOW IN T HEOREM φ ( n ). We will also need anadditional multiplicative function, namely f ( n ) : = Y p | n µ p α − ³ − p ´ α + ³ − p ´¶µ p ³ − p ´ + ³ − p ´ − α ¶ . ÁL-TYPE GCD SUMS BEYOND THE CRITICAL LINE 9
We now fix a positive number M and set k : = Q p ≤ M p . We will need the following lemma. Lemma 4.
For every c such that c | k, we have (9) 1 k X d | k ( c , d ) α ³ φ ¡ kc ¢ φ ¡ kd ¢´ α ¡ φ ( c ) φ ( d ) ¢ − α = Y p | k µ p µ − p ¶ + µ − p ¶ − α ¶ ck f ¡ kc ¢ . Moreover, we have (10)1 k X c , d | k ( c , d ) α ³ φ ¡ kc ¢ φ ¡ kd ¢´ α ¡ φ ( c ) φ ( d ) ¢ − α = Y p | k µ p α − µ − p ¶ α + p µ − p ¶ + µ − p ¶ − α ¶ . Proof.
Since every divisor d of k has a unique representation d = d d , where d | c and d | kc we find that the left-hand side LH S of (9) is
LH S = k φ ( k / c ) α φ ( c ) − α X d | c X d | kc d α φ ¡ cd ¢ α φ ¡ k / cd ¢ α φ ( d ) − α φ ( d ) − α = k φ ( k / c ) α φ ( c ) − α à X d | c d α φ ¡ cd ¢ α φ ( d ) − α ! X d | kc φ ¡ k / cd ¢ α φ ( d ) − α .(11)Using the formula φ ( d ) = d Q p | d ³ − p ´ and the fact that the respective sums represent mul-tiplicative functions, we find that X d | c d α φ ¡ cd ¢ α φ ( d ) − α = Y p | c µ p α µ − p ¶ α + p + α µ − p ¶ − α ¶ ,and X d | kc φ ¡ k / cd ¢ α φ ( d ) − α = Y p | kc µ p α µ − p ¶ α + p − α µ − p ¶ − α ¶ .Returning to (11) and using again that φ ( d ) = d Q p | d ³ − p ´ , we therefore obtain LH S = k X d | k ( c , d ) α ³ φ ¡ kc ¢ φ ¡ kd ¢´ α ¡ φ ( c ) φ ( d ) ¢ − α = ck Y p | c µ p µ − p ¶ + µ − p ¶ − α ¶ Y p | kc µ p α − µ − p ¶ α + µ − p ¶¶ = Y p | k µ p µ − p ¶ + µ − p ¶ − α ¶ ck f ¡ kc ¢ , where the last expression is the right-hand side of (9). Finally, we get (10) from (9) by usingthat n P d | n d f ( d ) is a multiplicative function. ■ In addition to the identities of the preceding lemma, we need following quantitative esti-mate.
Lemma 5.
For every α , < α < , there exists a positive constant c α such that Y p ≤ M µ p α − µ − p ¶ α + p µ − p ¶ + µ − p ¶ − α ¶ ≥ c α (log M ) α . Proof.
The result follows from the fact that p α − µ − p ¶ α + p µ − p ¶ + µ − p ¶ − α = + α p + O ( p α − ), p → ∞ ,along with Mertens’s third theorem, i.e., the fact that Q p ≤ M (1 − p ) ∼ e γ / log M when M → ∞ . ■ The following theorem yields the bound from below in Theorem 1.
Theorem 3.
For every α , < α < , there exists a positive constant c α such that if N is apositive integer, then there exists a set of integers M of cardinality N such that X m , n ∈ M ( m , n ) α ( mn ) α ≥ c α N − α (log N ) α . Proof.
Fix α , 0 < α < N be positive integer. Set M = N δ , where δ , 0 < δ <
1, is aconstant depending only on α to be chosen later, k = Q p ≤ M p . Let A be the set of the first[ N ] M -smooth square-free numbers and D be the set of integers of the form k / a with a in A . For every number d in D denote by S d the set of the first [ N φ ( d )/ k ] integers s such that( s , k / d ) =
1, and by s d the maximal number in S d . Also, let d S d be the set of integers of theform d s , where s ∈ S d and d ∈ D . Finally, set M : = S d ∈ D d S d .It is clear that all numbers d in D are square-free and also that the sets d S d are pairwisedisjoint. Moreover, since P d | k φ ( d ) = k we have that | M | < N . Also,(12) | S d | ≥ N N ÁL-TYPE GCD SUMS BEYOND THE CRITICAL LINE 11 for sufficiently large N and every d in D ; this follows from the formula φ ( d ) = d Q p | d ³ − p ´ and an application Mertens’s third theorem. We can use this to get an upper bound for s d .Indeed, each set S d is just a set of numbers of the form a mod ( k / d ), where a is from a set ofcardinality φ ( k / d ). Therefore if d is in D , then s d ≤ N φ ( d ) d φ ( k / d ) .Here we used that | S d | ≥ k / d which follows from (12). Now we have, using (12) in the last step, X m , n ∈ M ( m , n ) α ( mn ) α ≥ X c , d ∈ D ( c , d ) α ( cd ) α X m ∈ S c m α X n ∈ S d n α ≥ X c , d ∈ D ( c , d ) α ( cd ) α | S c || S d | s α c s α d ≥ c α N − α k X c , d ∈ D ( c , d ) α ( φ ( k / c ) φ ( k / d )) α ( φ ( c ) φ ( d )) − α for some positive constant c α depending only on α . In view of (10) of Lemma 4 and Lemma 5,the proof will be complete if we can prove that X c , d ∈ D ( c , d ) α ( φ ( k / c ) φ ( k / d )) α ( φ ( c ) φ ( d )) − α ≥ X c , d | k ( c , d ) α ( φ ( k / c ) φ ( k / d )) α ( φ ( c ) φ ( d )) − α .The latter inequality will follow from the bound X c D , d | k ( c , d ) α ( φ ( k / c ) φ ( k / d )) α ( φ ( c ) φ ( d )) − α ≤ X c , d | k ( c , d ) α ( φ ( k / c ) φ ( k / d )) α ( φ ( c ) φ ( d )) − α .By (9), this is equivalent to(13) X n ∈ F f ( n ) n ≤ Y p ≤ M µ + f ( p ) p ¶ ,where F is the set of all M -smooth square-free numbers larger than [ N ]. By Rankin’s trick,we have X n ∈ F f ( n ) n ≤ N − δ log N Y p ≤ M µ + f ( p ) p p δ log N ¶ ≤ e − δ Y p ≤ M µ + f ( p ) p ¶ Y p ≤ M µ + f ( p ) p ³ p δ log N − ´¶ . Now we note that the second product is bounded by a constant depending only on α (not on δ ). Indeed, X p ≤ M f ( p ) p ³ p δ log N − ´ ≤ X p ≤ M f ( p ) − p + X p ≤ M log pp log M .The first sum is bounded because f ( p ) = + p α − + o ( p α − ) as p → ∞ , and the second sumis bounded since P p ≤ M (log p )/ p − log M ≤ δ sufficiently small (depending only on α ), we get (13). Theorem 3 is proved. ■ R EFERENCES [1] C. Aistleitner,
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Proofs of some formulae enunciated by Ramanujan , Proc. London Math. Soc. (1922),235–255.D EPARTMENT OF M ATHEMATICAL A NALYSIS , T
ARAS S HEVCHENKO N ATIONAL U NIVERSITY OF K YIV , V
OLODY - MYRSKA
64, 01033 K
YIV , U
KRAINE
ÁL-TYPE GCD SUMS BEYOND THE CRITICAL LINE 13 D EPARTMENT OF M ATHEMATICAL S CIENCES , N
ORWEGIAN U NIVERSITY OF S CIENCE AND T ECHNOLOGY , NO-7491 T
RONDHEIM , N
ORWAY
E-mail address : [email protected] T ITUS H ILBERDINK , D
EPARTMENT OF M ATHEMATICS AND S TATISTICS , U
NIVERSITY OF R EADING , R
EADING
RG6 6AX, UK
E-mail address : [email protected] K RISTIAN S EIP , D
EPARTMENT OF M ATHEMATICAL S CIENCES , N
ORWEGIAN U NIVERSITY OF S CIENCE AND T ECH - NOLOGY , NO-7491 T
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