Extreme values of derivatives of the Riemann zeta function

Abstract It is proved that if T is sufficiently large, then uniformly for all positive integers ℓ⩽(logT)/(log2T), we have maxT⩽t⩽2Tζ(ℓ)1+it⩾eγ·ℓℓ·(ℓ+1)−(ℓ+1)·log2T−log3T+O(1)ℓ+1, where γ is the Euler constant. We also establish lower bounds for maximum of |ζ(ℓ)(σ+it)| when ℓ∈N and σ∈[1/2,1) are fixed.

He also mentioned that his methods do not provide any non-trivial results about the domains of the form ⩾ ( ) with ( ) = 1 − (log log log ∕ log log ). Note that Kalmynin did not obtain Ω-results for | ( ) ( + )| when ∈ ℕ and ∈ [1∕2, 1) are given.
It is still uncertain whether the methods of [3,14,18,23] are able to establish the result in our Theorem 1, since those methods basically rely on the fact that the -divisors function ( ) is multiplicative and/or the fact that the Riemann zeta function has an Euler product: ( ) = ∏ (1 − − ) −1 , ℜ( ) > 1. Note that the function ( ) ∶= (log ) is not multiplicative and the derivative ( ) ( ) does not have an Euler product.
We also emphasize that the key points in Theorem 1 are the range ⩽ (log )∕(log 2 ) and the constant in front of (log 2 ) +1 . In fact, one can use the method of Bohr-Landau to prove a much weaker result, that is, ( ) (1 + ) = Ω((log 2 ) +1 ) when ∈ ℕ is fixed. See Section 7 for such a short proof.
We will use Soundararajan's original resonance method [22] to prove Theorem 1. The new ingredient for the proof is the following Proposition 1.

Proposition 1. If is sufficiently large, then uniformly for all positive numbers , we have
where the supremum is taken over all functions ∶ ℕ → ℂ satisfying that the denominator is not equal to zero, when the parameter is given.
The following Proposition 2 will not be used to prove our theorems. However, it is closely related to Proposition 1 and can be viewed as a "log-type" greatest common divisor (GCD) sum, so we list it here for independent interest. where the supremum is taken over all subsets  ⊂ ℕ with size .
Remark 3. Actually we can also use Proposition 2 and Hilberdink's version of the resonance method [15] to prove a similar result to the one in Theorem 1. But the constant in front of (log 2 ) +1 will be much worse.
Soundararajan introduced his resonance method in [22] and proved that which improved earlier results of Montgomery and Balasubramanian-Ramachandra. Montgomery [21] proved it under RH and with the constant 1∕20 instead of 1 + (1) in Soundararajan's result. Balasubramanian-Ramachandra [4] proved the result unconditionally but also with a smaller constant compared to Soundararajan's result.
By constructing large GCD sums, Aistleitner [1] used a modified version of Soundararajan's resonance method to establish lower bounds for maximum of | ( + )| when ∈ (1∕2, 1) is fixed. His results improved early results of Voronin [24] and Hilberdink [15] via resonance methods. He proved that for large , and one can take = 0.18(2 − 1) 1− . The same result had been proved by Montgomery in [21] with a smaller value for . In [9], Bondarenko and Seip improved the value in Aistleitner's result. By constructing large GCD sums, using a convolution formula for in the resonance method, Bondarenko and Seip [7,8] proved the following surprising result: After optimizing the GCD sums, de la Bretèche and Tenenbaum [11] improved the factor from (1 + (1)) to ( √ 2 + (1)) in the above result. Following the work of Bondarenko-Seip and de la Bretèche-Tenenbaum, we use their modified versions of resonance methods to prove Theorem 2. The new ingredient is our convolution formula for 1 + 2 − + (−1) ( ) ( ). Throughout the paper, define the function ( ) as follows: Throughout the paper, also define the sequence { ( )} ∞ =1 as (1) = 1, (2) = 1 + (log 2) , and ( ) = (log ) for ⩾ 3. Then we have the following identity and the Dirichlet series converge absolutely: The reason why we add the part 1 + 2 − is that we want to make ( ) ⩾ 1 for all ⩾ 1. Since when ∈ [1∕2, 1), the factor (log ) has very small influence on the log-type GCD sums compared to the case = 1, we will simply use the fact that ( ) ⩾ 1 and then come to the situation of optimizing GCD sums.
Let ∈ (0, 1] be given and let  ⊂ ℕ be a finite set. The GCDs sums () of  are defined as follows: where ( , ) denotes the GCD of and and [ , ] denotes the least common multiple of and . The case = 1 was studied by Gál [13], who proved that The asymptotically sharp constant in (3) is 6 2 −2 . This fact was proved by Lewko and Radziwiłł in [19].
Bondarenko and Seip [6,7] proved the following result for GCD sums when Later, based on constructions of [6,7], de la Bretèche and Tenenbaum [11] optimized the result of Bondarenko-Seip and obtained the following: Aistleitner, Berkes, and Seip [2] proved the following essentially optimal result for GCD sums when ∈ ( 1 2 , 1), where and are positive constants only depending on ∶ Moreover, in [2, p. 1526], they also gave an example (following ideas of [13]) for the lower bound when ∈ ( 1 2 , 1). Let = 2 and let  be the set of all square-free integers composed of the first primes. Then for some positive constant˜. For simplicity, in our proof we will use this construction.
where the implied constant in big (⋅) only depends on 0 .
Proof. It follows from classical convex estimates for ( ) and Cauchy's integral formula. □ In the following, we will derive a "double version" convolution formula, similar to Lemma of 5.3 of de la Bretèche and Tenenbaum [11]. The proof is same as the proof of "single version" convolution formulas in Lemma 1 of Bondarenko and Seip [8].

PROOF OF PROPOSITION 1
Proof. We will use the construction of Bondarenko and Seip in [9]. Let = ⋅ ( + 1) −1 . Given a positive number and a positive integer , define We will choose a number and an integer later to make ( , ) ⩽ √ . Let  be the set of divisors of ( , ) and  be the set of divisors of ( , ). Let  be the complement of  in . Note that both  and  are divisor-closed which means | , ∈  ⇒ ∈  and | , ∈  ⇒ ∈  . Define the function ∶ ℕ → {0, 1} to be the characteristic function of , then As showed in [9], .
Also in [9, p. 129, lines [3][4][5][6][7][8][9], it is proved that Next, we split the sum into the following two parts: We will prove the following identity: To see this, let be the largest integer such that ⩽ and let be the largest integer such that ⩽ ( denotes the th prime). Then we have .
Since ⩽ 1 2 , for the off-diagonal terms ≠ we haveΦ( log( ∕ )) ≪ −2 , by the rapid decay ofΦ (see [22, p. 471]). Thus the contribution of the off-diagonal terms ≠ to the above summands can be bounded by Again, by = [ By Lemma 1, we have the following approximation formula and the implied constant in the big (⋅) term is absolute: In the integral of 2 ( , ), the big (⋅) term above contributes at most Combining this with (19), we have Finally, the above formula together with (18) gives that Now let = (log 2 ) −1 . By Stirling's formula, if is sufficiently large, then for all positive integers ⩽ (log )(log 2 ) −1 , we have ! ⋅ − ⋅ −1+ ⩽ (log 2 ) . Other big (⋅) terms can be easily bounded. Together with Proposition 1, we finish the proof of Theorem 1. □

Constructing the resonator
Given a set  of positive integers and a parameter , we will construct a resonator ( ), following ideas from [1], [7], and [11]. Define Let  be the set of integers such that  ≠ ∅ and let be the minimum of  for ∈  . We then set for every in  ′ . Then the resonator ( ) is defined as follows: By Cauchy's inequality, one has the following trivial estimates [11]: As in [7] , set Φ( ) ∶= − 2 ∕2 . Its Fourier transform satisfiesΦ( ) = √ 2 Φ( ).
We compute the integral1( ) by expanding the product of the resonator and the infinite series of ( ), and then integrate term by term, as in [7, p. 1699]. Using the fact ( ) ⩾ 1 for every andˆ(log ) ⩾ ∕2 if ⩽ , one gets ) .
( 1 Hence, Make slightly larger in the beginning then one can get ( ). □

PROOF OF PROPOSITION 2
The idea of the proof is basically the same as in the proof of Proposition 1. The new ingredient is Gál's identity. In this section, in order to avoid confusion about notations, we use the notation ( ⊗ ) for the ordered pair of and . Proof.
Following Lewko-Radziwiłł in [19], we use Gál's identity for the GCD sum and then split the product into two parts: By Mertens' theorem, the first product is asymptotically equal to ( log ) 2 ∼ ( log log ) 2 as → ∞. The second product converges as → ∞ to Next, let = ⋅ (2 + 1) −1 and define the sets  (1) ,  (2) as follows: Then define  to be the union of the above two sets and  to be the complement of  in  × : Now we split the GCD sum into two parts: By symmetry, we have By the definition of  (1) and Gál's identity, we have YANG Again, we have ∼ , the first product converges to 6∕ 2 , and the third product is asymptotically equal to ( log ) 2 ∼ ( log log ) 2 as → ∞.
For the second product, it can be bounded as And by Mertens' theorem and the prime number theorem, we have As a result, we obtain that Hence by (38) ) log By our choice of = ⋅ (2 + 1) −1 , we are done. □
We are also interested in extreme values of | ( ) ( + )| in the left half strip. It is unlike the situation of the zeta function, where the values on the left half strip can be easily determined by the right half strip via the functional equation. Thus it is worth to study Γ ( ) ( ) when < 1 2 , even for this reason.

J O U R N A L I N F O R M AT I O N
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