Extreme values of derivatives of zeta and L‐functions

Abstract It is proved that as T→∞, uniformly for all positive integers ℓ⩽(log3T)/(log4T), we have maxT⩽t⩽2Tζ(ℓ)1+it⩾Yℓ+o1log2Tℓ+1, where Yℓ=∫0∞uℓρ(u)du. Here, ρ(u) is the Dickman function. We have Yℓ>eγ/(ℓ+1) and logYℓ=(1+o(1))ℓlogℓ when ℓ→∞, which significantly improves previous results in [17, 40]. Similar results are established for Dirichlet L‐functions. On the other hand, when assuming the Riemann hypothesis and the generalized Riemann hypothesis, we establish upper bounds for |ζ(ℓ)(1+it)| and |L(ℓ)(1,χ)|. Furthermore, when assuming the Granville–Soundararajan conjecture is true, we establish the following asymptotic formulas: maxχ≠χ0χ(modq)L(ℓ)(1,χ)∼Yℓlog2qℓ+1,asq→∞, where q is prime and ℓ∈N is given.


INTRODUCTION
This paper establishes the following results for extreme values of derivatives of the Riemann zeta function on the 1-line. Throughout the paper, we define Y ℓ = ∞ 0 u ℓ ρ(u)d u and ρ(u) denotes the Dickman function.
Our result on the Riemann zeta function can be generalized to L-functions. In the following theorem, we consider the case of Dirichlet L-function L(s, χ) associated with non-principal characters χ(mod q). Theorem 2. Let q be prime, then uniformly for all positive integers ℓ log 3 q/ log 4 q, we have Remark 3. 1) The above result does not hold for general moduli q. For instance, assume q = p X p · m , with m ∈ N and X = 1 2 log q. This assumption will force χ(k) = 0 if ∃p X , p|k and thus will make L (ℓ) (1, χ) small. 2) However, if q is not divisible by small primes (for instance, consider the case that any prime factor of q is larger than q 1 10 ), then the above theorem will still hold. For simplicity, we state the result for prime moduli.
Theorem 4. Fix ℓ ∈ N. Let χ be any non-principal character (mod q), and assume GRH for L(s, χ). Then The key ingredient to prove Theorem 4 is the following theorem of Granville and Soundararajan [18,Theorem 2].
To state their result, we need some definitions. Let f be an arithmetic function. Define the functions Ψ(x, y) and Ψ(x, y; f ) as Theorem (Granville-Soundararajan). Let χ be any non-principal character (mod q), and assume the Riemann Hypothesis for L(s, χ). If 1 x q and y log 2 q log 2 x(log 2 q) 12 , then n x Further n x and so the following estimate holds n x when log x/ log 2 q → ∞ as q → ∞.
When g (n) = n −i t , ∀n ∈ N, we write Ψ(x, y; t ) in place of Ψ(x, y; g ). Then we have the following result analogous to the Granville-Soundararajan Theorem.
Theorem 5. Assume RH and let T be sufficiently large. If 2 x T , T + y + 3 t T 1000 and y log 2 T log 2 x(log 2 T ) 12 , then n x 1 and so the following estimate holds when log x/ log 2 T → ∞ as T → ∞.
In [18], Granville and Soundararajan also made the following conjecture.
Conjecture (Granville-Soundararajan). There exists a constant A > 0 such that for any nonprincipal character χ (mod q), and for any 1 x q we have, uniformly, n x A consequence of the Granville-Soundararajan Conjecture is that the constant appearing in Theorem 2 is sharp.
Combining with the lower bound, we immediately get the following asymptotic formulas.
We have the following analogous conjecture, which can imply that the constant appearing in Theorem 1 is sharp.

Conjecture 1.
There exists a constant A > 0 such that for any 1 x T , 2T t 5T , we have, Littlewood's result has been improved in the past century by Levinson [30], by Granville-Soundararajan [21], and by Aistleitner-Mahatab-Munsch [2], who established that max T t T |ζ(1 + i t )| e γ (log 2 T +log 3 T +C ), for some constant C which can be effectively computed. Littlewood also established conditional results for the upper bound of |ζ(1+i t )|. When assuming the truth of the Riemann hypothesis (RH), he proved that [38,Thm 14.9]). Furthermore, he conjectured that the maximum of |ζ(1 + i t )| on the interval [1, T ] should satisfy the asymptotic formula In [21], Granville and Soundararajan made the stronger con- is also effectively computable.
In the past five years, the problem of obtaining extreme values of derivatives of the Riemann zeta function have been studied. In [28], Kalmynin obtained Ω-results for the Riemann zeta function and its derivatives In [40], we established Ω-results for ζ (ℓ) (σ + i t ) when ℓ ∈ N and σ ∈ [1/2, 1) are given.
These results are comparable with the best currently known lower bounds for maximum of ζ(σ + i t ) . When σ = 1, we obtain lower bounds different from the case of ζ(1 + i t ) . Namely, uniformly for all positive integers ℓ (log T )/(log 2 T ), when T is sufficiently large. On the other hand, in [41] we proved that on RH, where the implied constants are effectively computable. Refining methods of [40], Z. Dong and B. Wei [17] proved that This is due to the following identity and the fact that ρ(u) is always positive.
By Theorem 3, assuming RH, we have which improve corresponding results in [41].
where h is the class number of Q( d ), and ω denotes the number of roots of unity in Q( d ).
Let χ be any non-principal character (mod q). Assuming GRH, Littlewood [31] proved that In another direction, Chowla [10] showed that there exist arbitrarily large q and non- As in the proof of Theorem 6, the Granville-Soundararajan Conjecture implies L(1, χ) e γ + o(1) log 2 q for nonprincipal characters χ. We mention three best results known for L (1, χ). For further information and results about extreme values of L-functions, we refer to the survey [37] and [3,14,19,20,21,29,36]. In [21], Granville and Soundararajan established that for sufficiently large prime q and any given A 10 there are at least q 1−1/A characters χ(mod q) for which for some absolute constant C . In [3], Aistleitner, Mahatab, Munsch and Peyrot proved that when fix ǫ > 0, then for all sufficiently large prime q, we have max In [29], when assuming GRH, Lamzouri-X. Li-Soundararajan obtained the following upper bound for primitive character χ modulo q In The study of character sums is another central problem in number theory. In many cases, one would like to know when the following character sum is o(x), n x where χ is a non-principal Dirichlet character χ(mod q).
In [33], Montgomery and Vaughan show that the above character sums can be conditionally approximated by character sums over integers with small prime factors. More precisely, they prove that if χ(mod q) is non-principal and GRH holds then n x when log 4 q y x q. One of main results in [33] states that on GRH, for any non-principal character χ modulo q and any x. On GRH, Granville and Soundararajan [22] find an implicit constant in (4) for primitive character χ modulo q. The upper bound (4) can be used to improve the error term in the approximation formula (16) for L (ℓ) (1, χ). In We will use Soundararajan's resonance methods [36] to prove Theorem 1 and Theorem 2.
The key ingredient is the following Proposition 1.

Proposition 1. As T → ∞, uniformly for all positive numbers
where the supremum is taken over all functions r : N → C satisfying that the denominator is not equal to zero, when the parameter T is given.
Notations: in this paper, γ denotes the Euler constant. We write log j for the j -th iterated logarithm, so for example log 2 T = log log T, log 3 T = log log log T . P + (n) denotes the largest prime factor of n. p denotes a prime number and p n denotes the n-th prime.

PRELIMINARY RESULTS
Recall that the function Ψ(x, y) = # n x P + (n) y counts the number of integers n not exceed x with prime factors at most y. The Dickman function ρ(u) is a continuous function defined by the initial condition ρ(u) = 1 for 0 u 1 and satisfies the following differential equation From the definition, the Dickman function ρ(u) is a positive decreasing function. In 1930, Dickman [15] proved that for fixed u > 0, lim x→∞ Ψ(x, x 1 u )/x exists and equals to ρ(u). We will use the following strong form of this asymptotic formula and an asymptotic formula for ρ(u). In the following lemma, (6) is due to Hildebrand [25]. The upper bound of (7) is due to de Bruijn [13], while the lower bound of (7) is due to Hildebrand [25]. And the asymptotic formula (8) for ρ(u) was obtained by de Bruijn [12].
The following lemma is on the Laplace transform of the Dickman function, which is useful for us to compute Y ℓ , as mentioned in Remark 1.

Lemma 2 (Lemma 2.6 [26], Thm 7.10 [34]). For any real or complex number s we have
We have the following conditional approximation formula for log ζ(σ+i t ), which is adapted from Lemma 1 of [21].
We have the following unconditional approximation formula for ζ (ℓ) (σ + i t ). The constant 6.28 can be replaced by any positive number smaller than 2π (see [23,Lemma 2] ).

PROOF OF PROPOSITION 1
Proof. Let T be large. Let w = π(y). Define y, b and P(y,b) as follows Define K : = k ∈ N k exp log 2 T · log 3 T , P + (k) y and its two subsets K 1 and K 2 to be , where k has the prime factorization as k = p , where k has the prime factorization as k = p Clearly, K 1 is a subset of M . Let k be any given element of K 1 , then the inner sum in (10) tends to 1, as T → ∞. More precisely, we have 1 |M | n∈M To see this, assume that k = p and (11) follows from the condition w i =1 α i (log 2 T ) 3 log 3 T . Now consider upper bounds for the sum of reciprocals of elements of K 2 . By Rankin's trick and dropping conditions for α i , we have where in the last inequality we use the Mertens' theorem. By the definition of K 2 , when k ∈ K 2 , we have logk (log 2 T ) · (log 3 T ). Thus we find that In order to compute a lower bound for the outer sum in (10), we first compute the sum over the set K , then by (12) we restrict the sum to over its subset K 1 , which is also a subset of M .
Let R = exp log 2 T · log 3 T . And we keep in mind that ℓ (log 3 T )/(log 4 T ) in the following computations.
We split the sum into two parts as follows The first sum is By partial summation, the second sum is By (6), we have Applying (14) into (13), and using (5) and (8), we obtain We immediately get Together with (12), we have Since K 1 is a subset of M , we find that By (10), (11) and (15), we are done.

PROOF OF THEOREM 1
Proof. By [40, page 496], we have By Proposition 1, we finish the proof of Theorem 1.

PROOF OF THEOREM 2
Proof. First note that we have the following approximation formula by partial summation and Pólya-Vinogradov inequality ( [34,Thm 9.18]) In order to use Soundararajan's resonance method [36] to produce extreme values, we define V 2 (q) and V 1 (q) as follows (also see [14, page 129]) where L (ℓ) (1, χ; N ) and the resonator R χ are defined by We chose T = q By Cauchy's inequality, we have Thus we can bound L (ℓ) (1, χ Above upper bound together with the orthogonality of characters gives that Combining (18) with (17), we have max By (19), (16) and Proposition 1, we obtain (1).

PROOF OF THEOREM 3
Proof. Let x 1 = exp (log 2 T ) 2 , x 2 = T , and y j = log 2 T log 2 x j (log 2 T ) 12 for j = 1, 2. Note that we have log y 1 ∼ 2 log 2 T, as T → ∞. By taking σ = 1 and ǫ = (log 2 T ) −1 in (9), we have We split the sum in the above approximation formula into two parts as follows For the first sum, we have For the second sum, by partial summation, we have By (3) of Theorem 5 and Lemma 1, we have Clearly, we have (log y 1 ) ℓ y 1 By (2) of Theorem 5, (5) and Lemma 1, we have Again, by (2), (5) and Lemma 1, we have As a result, we obtain Since t ∈ [2T, 5T ], we are done.

PROOF OF THEOREM 5
The proof is almost the same as the proof of the  Note that log ζ(s; y) = ∞ n=2 Λ y (n)(log n) −1 n −s , where the generalized von Mangoldt function Λ y (·) is defined as Λ y (n) = log p if n = p k and p > y, otherwise Λ y (n) = 0.
So we have log ζ(s + i t ; y) j / j ! = ∞ m=1 a j (m, y)m −s−i t , where |a j (m, y)| 1 for all m, j and y. All other steps are the same as the proof of the Granville-Soundararajan Theorem.

PROOF OF THEOREM 4
Proof. Let x 1 = exp (log 2 q) 2 , x 2 = q 3 4 , and y j = log 2 q log 2 x j (log 2 q) 12 for j = 1, 2. We will use the approximation formula (16) for L (ℓ) (1, χ) and other steps are the same as the proof of Theorem 3.

PROOF OF THEOREM 6
Proof. Let x 1 = exp (log 2 q) 2 , x 2 = q 3 4 , and y j = (log q + log 2 x j )(log 2 q) A for j = 1, 2. We again use (16) and other steps are the same as the proof of Theorem 3. Note that now we have log y 1 ∼ log 2 q, as q → ∞. Thus in the end, we obtain Y ℓ log 2 q ℓ+1 instead of Y ℓ 2 log 2 q ℓ+1 in Theorem 4 .

PROOF OF THEOREM 7
Proof. For the upper bound, let x 1 = exp (log 2 T ) 2 , x 2 = T , and y j = (log T + log 2 x j )(log 2 T ) A for j = 1, 2. And other steps are the same as the proof of Theorem 3. Combining with the lower bound, we are done.

A MIXED CONJECTURE
Combining the Granville-Soundararajan Conjecture and Conjecture 1, we pose the following mixed conjecture.