A dichotomy for extreme values of zeta and Dirichlet L$L$ ‐functions

We exhibit large values of the Dedekind zeta function of a cyclotomic field on the critical line. This implies a dichotomy whereby one either has improved lower bounds for the maximum of the Riemann zeta function, or large values of Dirichlet L$L$ ‐functions on the level of the Bondarenko–Seip bound.


introduction
Extreme values play a key role in the value distribution theory of L-functions.In [13], Soundararajan introduced a versatile method for obtaining large values of L-functions in a variety of families.Applied to the Riemann zeta function, this gave max improving on previous results of Balasubramanian-Ramachandra [3].By combining a modification of Soundararajan's version of the resonance method along with the burgeoning connections with GCD sums and insights of Aistleitner [1], the first and fifth authors showed [4] that log T log log log T log log T .
The constant 1/ √ 2 was subsequently improved to 1 in [5] with the current best being √ 2 due to de la Bretèche-Tenenbaum [9].
These techniques have since been applied in a variety of settings [2,6,7,8,15].A severe constraint, however, is that the L-functions under consideration must have positive coefficients.This excludes many L-functions of interest and so, for instance, it is not known whether Dirichlet L-functions exhibit such large values.Currently, Soundararajan's resonance method gives the best known bounds a cyclotomic field K = Q(ω q ) with ω q a qth root of unity for q > 2. Of special interest is the factorisation where the product is over all non-principal Dirichlet characters χ modulo q and χ ′ is the character which induces χ if χ is not primitive and χ ′ = χ otherwise.We will establish the following bound which thus yields an assertion about the interplay between extreme values of the functions L(s, χ).
Theorem 1.Let K = Q(ω q ) and A be an arbitrary positive number.If T is sufficiently large, then uniformly for q ≪ (log 2 T ) A , (2) max Thanks to the extra factor φ(q) in the exponent on the right-hand side of (2), we obtain the following consequence of Theorem 1 and the factorisation (1).where one could take any c < 1/ √ 2 for Galois extensions and any c < 1/( √ 2[K : Q]) in general.However, with these bounds one cannot deduce our dichotomy.
Our improvement of the constant in (3) is afforded by the following observation.Note that by (1) we may write ζ Q(ωq) (s) = n 1 a(n)n −s where for primes p ∤ q we have a(p) = φ(q) if p ≡ 1 (mod q), 0 otherwise.
For the purposes of this discussion we may ignore primes p|q since they are only finite in number.On applying the resonance method we obtain a lower bound for the maximum which is roughly of the form where r are the resonator coefficients.The fact that φ(q) p≡1(mod q), p x 1 means that this lower bound is essentially p (1 + r(p)/p 1/2 ), i.e. we are in the same situation as for the Riemann zeta function and nothing seems to have been gained.However, the fact that our resonator only needs to be supported on a smaller set of primes, p ≡ 1( mod q), allows us to take it larger whilst still matching the other constraints of the argument.Precisely, we can take it larger by a factor of φ(q).In order to balance this, we need to take larger primes than usual.
It is likely that the methods of de la Bretèche-Tenenbaum [9] can improve the exponent on the right hand side of (2) by a factor of √ 2. However, this would not affect our dichotomy in any essential way and so in the interests of keeping our exposition simpler we have not pursued this line of inquiry.
We close this introduction by mentioning the possibility of extending our result to other Dedekind zeta functions.To this end, let L/Q be a Galois extension.Then the The density of the primes that split completely in L is 1/[L : Q] by Chebotarev's density theorem.This should be thought of as the analogue of 1/φ(q) in the Siegel-Walfisz theorem in our case.It seems plausible then that one should be able to establish a counterpart to our main theorem, with constant [L : Q] instead of φ(q) by modifying the resonator coefficients slightly.When L/Q is Galois, ζ L (s) factors into a product of Artin L-function according to the decomposition of the regular representation of Gal(L/Q) into irreducible representations.We could then get a similar dichotomy as in the cyclotomic case.In particular, when Gal(L/Q) is abelian, Dirichlet L-functions would be replaced by Hecke L-functions.For the non-abelian case this is more subtle in general.
This paper contains two additional sections.We prepare for the proof of Theorem 1 in the next section by developing the required novel extremal GCD-type sums.The actual proof of Theorem 1 is carried out in Section 3.
We will in what follows use the notations log 2 x := log log x and log 3 x := log log log x, and we will use the convention that for f an integrable function on R.

Extremal GCD-type sums
We will now construct the extremal versions of the sums that appear in the resonance method.We follow the scheme of [4, Sec.2] closely, the main differences being that we need to account for the coefficients of the Dedekind zeta function and that we are picking primes that are congruent to 1 modulo q.
Let 0 < γ < 1 be a parameter to be chosen later and P q the set of all primes p such that p ≡ 1 (mod q) and eφ(q) log N log 2 N < p φ(q) log N exp ((log 2 N) γ ) log 2 N.
Here N is a large integer to be chosen later as N = ⌊T η ⌋.For any c q 1, we define the multiplicative function f (n) to be supported on the set of square-free numbers such that Eventually we will take c q = φ(q), but we keep it general for the time being.Let P k,q be the set of all primes p such that p ≡ 1 (mod q) and φ(q)e k log N log 2 N < p φ(q)e k+1 log N log 2 N for k = 1, . . ., ⌊(log 2 N) γ ⌋.Fix a satisfying 1 < a < 1/γ.Let M k,q be the set of those integers having at least a log N k 2 log 3 N prime divisors in P k,q .Also let M ′ k,q be the set of integers from M k,q that have prime divisors only in P k,q .Set Lemma 1.We have |M q | N uniformly for q ≪ (log 2 N) A .
Proof.Note that By the Siegel-Walfisz theorem, we have 1))e k+1 log N provided q log A (φ(q)e k+1 log N log 2 N) which is satisfied for q ≪ (log 2 N) A .The remainder of the proof now follows directly that of Lemma 2 of [4].
Since our resonator only interacts with p ≡ 1 mod q we need not compute a(p) on the ramified primes p|q.We now consider the quantity Lemma 2. Suppose that c q φ(q) and q ≪ (log 2 N) A .Then Proof.Note that if c q φ(q), then f (p) = o(1) for all p ∈ P q .Thus By the Siegel-Walfisz theorem along with partial summation, we have Lemma 3. Suppose that c q φ(q) and q ≪ (log 2 N) A .Then Proof.We begin with By the Siegel-Walfisz theorem, for q ≪ (log 2 N) A , we have with the latter bound following since k (log 2 N) γ and φ(q) ≪ (log 2 N) A .Since every number in M ′ k,q has at least a log N k 2 log 3 N prime divisors and f (n) is a multiplicative function, for any b > 1, we have Observe that by the Siegel-Walfisz theorem, Combining all these estimates, we find that Since c q φ(q) and a > 1, on taking b sufficiently close to 1 the exponent is negative giving the result.

Proof of Theorem 1
We follow the setup from [4,Sec. 3].Let J q be the set of integers j such that Also let m j be the minimum of [(1 + T −1 ) j , (1 + T −1 ) j+1 ] ∩ M q for all j in J q .Set M ′ q := {m j : j ∈ J q } , and for every We take our resonator to be Set N = [T η ] for some 0 < η 1, and Φ(t) := e −t 2 /2 so that Lemma 4. Suppose that 1/2 σ < 1 and let K(x + iy) be an analytic function in the horizontal strip σ − 2 y 0 satisfying Then for all real t, we have Proof.This follows exactly as in Lemma 1 of [5].
Proof.The first part follows similarly to Lemma 5 of [5].For the second part we have We wish to lower bound this using positivity.From the properties of K n (v) given in Lemma 5, in particular the derivative bound (5), we see that if . By (6) we have c n ∼ 1 and hence on restricting the sum to log k 1 3n ǫ log T , say, the above is by positivity.We now lower bound by summing over those integers m, n ∈ M ′ q such that |km/n−1| 3/T .For such terms we have Φ(T log(km/n)) ≫ 1 and also, similarly to equation (21) of [5], we find This gives the lower bound It remains to remove the restriction on the divisor d since then the result will follow from Lemmas 2 and 3. Following Lemma 3 of [4], we note that since f for p ∈ P q and there are ≪ log T / log 3 T prime factors of any given n in M q , the above is Proof of Theorem 1.Note that (7) Thus it remains to extend the integrals on the left to R.
max t∈[T,2T ] |L(1/2 + it, χ)| exp (1 + o(1)) log T log log T for non-principal Dirichlet characters χ.A plentiful source of L-functions with positive coefficients are provided by the Dedekind zeta functions.In this paper, we consider the Dedekind zeta function ζ K (s) attached to Research supported in part by Grant 275113 of the Research Council of Norway.The work of Darbar is funded by that grant through the Alain Bensoussan Fellowship Programme of the European Research Consortium for Informatics and Mathematics.