Non Vanishing of Theta Functions and Sets of Small Multiplicative Energy 2

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In this case, there exists no algebraic reason forcing the L- function to vanish at s = 1/2. Therefore it is certainly expected that L(1/2) = 0 as ﬁrstly 6 conjectured by Chowla [Cho65] for quadratic characters. In the last century, the notion of family of L- functions has been important both as a heuristic guide to understand or guess many important statistical properties of L- functions. One of the main analytic tools is the study of moments and various authors have obtained results on the mean value of these L-series at their central point s = 1/2. The asymptotic of the ﬁrst two moments (as well as the fourth moment) is known:

L(1/2,χ) 2 p log p | | ∼ χ Xmod p (see [Ram79, Remark 3], [Bal81, Th. 3], or [HB81] for a more precise asymptotic expansion). This imply directly that there is a reasonable proportion of characters such that the L- function does not vanish at the central point.

Using the method of mollifiers, it was first proved by Balasubramanian and Murty [BM92] that there exists a positive proportion of characters such that the L- function does not vanish at s = 1/2. Their result was improved and greatly simplified by Iwaniec and Sarnak [IS99] enabling them to derive similar results for families of automorphic L-functions [IS00]. Since then, a lot of technical improvements and generalizations have been carried out, see for instance [Bui12, KN16, Sou00].

As initiated in previous works [LM13b, LM13a, MS16, Mun17], we would like to obtain similar results for moments of values at x = 1 of the theta functions θ(x,χ) associated with such Dirichlet L-functions. It was conjectured in [Lou07] that θ(1,χ) = 0 for every primitive character (see [CZ13] 6 for a case of vanishing in the non-primitive case). Using the computation of the first two moments of these theta functions at the central point x = 1, Louboutin and the author [LM13b] obtained that θ(1,χ) = 0 for at least 6 p/ log p even characters modulo p (for odd characters, it was already proven by Louboutin in [Lou99]). As in the case of L- functions, we would like to obtain a positive proportion of such characters. However, one backdraw in this situation is that the theta function does not have a representation as an Euler product which suggested the construction of mollifiers for Dirichlet L- functions. Thus, we need to proceed somehow differently in order to construct the mollifiers. Our goal in this note is to provide an argument which does not produce a positive proportion but improve the result coming from the evaluation of the second and fourth moments. Precisely, we prove

Theorem 1 Let x > 0. For all suﬃciently large prime p, there exists at least p/√log p even characters χ such that θ(x,χ) = 0. ≫ 6 Non vanishing of theta functions and sets of small multiplicative energy 3

Theorem 1 follows from a classical application of the Cauchy-Schwarz inequality:

M (p)2 1 M (p) (1) 1 ≤   2 χ mod p,χ(−1)=1  θ(x,χX)6=0    where we consider some molliﬁed moments

2 M1(p) := M(χ)θ(x,χ) and M2(p) := M(χ)θ(x,χ) . χ mod p χ mod p | | χ(X−1)=1 χ(X−1)=1 The molliﬁer M(χ) will be chosen as a suitable Dirichlet polynomial builded on a multiplicative subset of integers. The main idea is of combinatorial nature and relies on minimizing some GCD sums or equivalently the multiplicative energy of some well-chosen set.

In Section 2, we review some previous work as well as introduce the mol- liﬁers. In Section 3 and 4, we discuss the reduction of the problem to GCD sums which help us to prove Theorem 1 in Section 5. Section 6 concerns the case of odd characters and in Section 7 we address some open questions regarding the combinatorial problem which may lead to improve our main theorem.

2 Deﬁnitions and previous results

Let us ﬁrst restrict ourselves to the case of even Dirichlet characters, we refer the reader to Section 6 for the case of odd Dirichlet characters. Let X+ be the subgroup of order (p 1)/2 of the even Dirichlet characters mod p − p. We set πn2x/p + θ(x,χ)= χ(n)e− , χ X . ∈ p n 1 X≥ For χ = 1 we have 6 W θ(x,χ)= χ θ(1/x, χ¯) (x> 0) √x for some explicit complex number Wχ of absolute value equal to one. (e.g. see [Dav00, Chapter 9]). In particular, if θ(1,χ) = 0, then we can eﬃciently 6 compute numerical approximations to Wχ = θ(1,χ)/θ(1,χ) (see [Lou07] for an application). In order to prove that θ(1,χ) = 0 for many of the χ X+, 6 ∈ p one may proceed as usual and study the behavior of the moments of these theta functions at the central point x = 1 of their functional equations:

S+ (p) := θ(1,χ) 2k, k> 0. 2k | | χ X+ X∈ p Non vanishing of theta functions and sets of small multiplicative energy 4

Using the computation of the second and fourth moment, it was proved in [LM13b] that θ(1,χ) = 0 for at least p/ log p of the χ X+. Lower 6 ≫ ∈ p bounds of good expected order for the moments were obtained in [MS16] as well as nearly optimal upper bounds conditionally on GRH in [Mun17]. This can be related to recent results of [HNRl15], where the authors obtain the asymptotic behaviour of moments of Steinhaus random multiplicative function (a multiplicative random variable whose values at prime integers are uniformly distributed on the complex unit circle). This can reasonably be viewed as a random model for θ(1,χ). Indeed, the rapidly decaying factor 2 e πn /q is mostly equivalent to restrict the sum over integers n n (q) for − ≤ 0 some n (q) √q and the averaging behavior of χ(n) with n q1/2 is 0 ≈ ≪ essentially similar to that of a Steinhaus random multiplicative function. As noticed by Harper, Nikeghbali and Radziwill in [HNRl15], an asymptotic formula for the ﬁrst absolute moment θ(1,χ) would probably implies | | χ X+ X∈ p the existence of a positive proportion of characters such that θ(1,χ) = 0. 6 Though, quite surprisingly, Harper proved recently both in the random and deterministic case that the ﬁrst moment exhibits unexpectedly more than square-root cancellation [Hara, Harb]

1 √N χ(n) . p 1 ≪ min log log N, (log log log q)1/4 χ n N − X X≤

This shows that this approach would not in any case provide the existence of a positive proportion of “good” characters. In this note, we adapt another approach in order to improve on our previous results. Let M be a parameter which will be fixed later. For any even character χ, let us define M(χ)= cmχ¯(m) (2) m M X≤ where cm denotes the indicator function of some multiplicative set of integers , meaning that m,n implies mn . We consider the first and second A ∈A ∈A mollified moments M (p) := M(χ)θ(x,χ) and M (p) := M(χ)θ(x,χ) 2. (3) 1 2 | | χ X+ χ X+ X∈ p X∈ p While it seems plausible that we could obtain in some cases precise asymptotical formulas, we only give bounds in order to simplify the presentation. The main technical result is to suitable choose a set giving simultaneously A a good lower bound for M1(p) and a good upper bound for M2(p). Precisely, for a suitable choice of , we have asymptotically A p3/2 p2 M1(p) and M2(p) . (4) ≫ √log p ≪ √log p Non vanishing of theta functions and sets of small multiplicative energy 5

As already noticed, Theorem 1 follows from Cauchy-Schwarz inequality (1) combined with (4). In the next sections, we will explain how to construct a good set which veriﬁes (4) and address a related combinatorial problem which may lead to an improvement of our result.

3 Reduction of the problem to GCD sums

3.1 Lower bound on the ﬁrst molliﬁed moment Let us recall the classical orthogonality relations for the subgroup of Dirichlet + even characters Xp : (p 1)/2 if m n mod p and gcd(m,p) = 1, χ(m)¯χ(n)= − ≡± 0 otherwise. χ X+ X∈ p n Due to the fast decay of the exponential term in θ(x,χ), the main contribution to M1(p) comes from the terms less than √p, leading us to choose M = √p. It follows that

πn2x/p M1(p) = χ¯(m)cm χ(n)e− χ X+ m √p n 1 X∈ p X≤ X≥ p 1 πm2x/p = − c e− 2 m m √p X≤ p c (5) ≫ m m √p X≤ 2 where we used the fact that e πm x/p 1 for integers less than √p. − ≫ The problem boils down to choose a subset of integers of suﬃciently A high density in order to maximize M1(p) with the condition that it minimizes the second molliﬁer. We remark from the above inequality (5) that

M (p) p [1, M] (6) 1 ≫ |A ∩ | where [1, M] denotes the number of elements less than M in . By |A ∩ | A an abuse of notation, we will use in the following. |A| 3.2 Upper bound for the second molliﬁed moment and multiplicative energy The evaluation of the second moment is a bit more intricate.

2 2 π(n1+n2)x 1 p M2(p) = χ¯(m1)χ(m2)cm1 cm2 χ(n1− n2)e− χ X+ m1,m2 √p n1,n2 1 X∈ p X≤ X≥ Non vanishing of theta functions and sets of small multiplicative energy 6

p 1 π(n2+n2)x/p = − c c e− 1 2 2 m1 m2 m1,m2 √p m1n1≡±m2n2 (mod p) X≤ (n1Xn2,p)=1 πnx/p c c E (n)e− (7) ≪ m1 m2 m1,m2 m1,m2 √p n 1 X≤ X≥ where we used partial summation and

Em1,m2 (n)= 1. 2 2 n1X+n2≤n m1n1=±m2n2 mod p In [LM13b], in order to compute an asymptotic formula for the fourth moment of theta functions, the authors showed that the main contribution comes from the solutions m1n1 = m2n2 and obtained a precise asymptotic formula for the related counting function 3 m n = m n ,m2 + n2 + m2 + n2 x x log x. |{ 1 1 2 2 1 1 2 2 ≤ }| ∼ 8 If we want to improve on this result, we have to minimize the eﬀect of this logarithmic term. Let us recall a related bound which includes all the solutions modulo p. Taking initial intervals in [ACZ96, Lemma 1], we have

Lemma 2 Suppose m m and x p, then we have the following bound 1 ≤ 2 ≤ x(m ,m ) xm n ,n x,m n = m n mod p 1+ 1 2 1+ 2 | { 1 2 ≤ 1 1 ± 2 2 }|≪ m (m ,m )p 2 1 2 where as usual (m1,m2) denotes the greatest common divisor of m1 and m2. We deduce immediately the bound

n √nm2 (m1,m2) Em1,m2 (n) 1+ + + √n . (8) ≪ p p(m1,m2) m2 Truncating the series in (7) up to p2, inserting the bound (8) in (7) and using a comparison with an integral, the first three terms in the right hand 2 side of (8) contribute at most p m √p cm . The last term is more prob- ≤ lematic and we can summarize this in the inequality P 2 3/2 (m1,m2) M2(p) p cm + p cm1 cm2 . (9) ≪   m2 m √p m1 m2 √p X≤ ≤X≤ We recall that we assumed that the weights are all equal to 1 or 0 and are totally multiplicative meaning that our mollifier is contructed as the Non vanishing of theta functions and sets of small multiplicative energy 7 counting function of some multiplicative set . The first sum on the right A hand side of the inequality (9) gives the harmless contribution to M2(p)

2 p c = p 2.  m |A| m √p X≤   Hence, using Cauchy-Schwarz inequality with (6) and (9), we deduce the following lower bound

2 2 3/2 (m1,m2) 1 p / p cm1 cm2 . (10) + ≫ |A|  m2  χ∈X m1 m2 √p Xp ≤X≤ θ(x,χ)6=0   We need to construct a set in order to maximize this ratio. A 4 Good sets minimizing GCD sum

Let us consider the general setting of a subset [1,N] of integers. We B ⊂ are interested to minimize the quantity

(m ,m ) S( ) := 1 2 . B m2 m1,m2∈B mX1≤m2

More precisely, in view of (10), we want to maximize in terms of N the quantity

2 R( ) := N |B| . B S( ) B Restricting the sum to the couples (m ,m ) where m m we have 1 2 1 | 2 m1 1 S( ) = σ 1(m) . m d − B ≥ m ,m ∈B 2 ≥ ≥ |B| 1 2 m2 d m2 m mX1|m2 X∈B X| X∈B

In particular, we have the trivial bound

R( ) N . (11) B ≤ |B| Assume that is a set such that m and m m implies m , then B 2 ∈B 1 | 2 1 ∈B the divisor sum over d will be complete. Therefore, we need to construct a set such that, on average, every element has few small divisors. Precisely, we are looking for a set of reasonable density such that the GCD sum S( ) B is not too large. Non vanishing of theta functions and sets of small multiplicative energy 8

Let us discuss two extremal cases. Assume to be the set consisting B of all the primes less than N, we have S( ) N/ log N. The main con- B ≪ tribution comes from the diagonal terms p = q, in the other case p and q are coprimes and the sum is small. Thus, by the prime number theorem, R( ) N 2/ log N. Even though we are able to have good control on the B ≈ GCD sum, the set of primes has a too small density which prevents us to save any logarithm in R( ). Another extreme case is to take all the integers B up to N, a short computation shows that the GCD sum S( ) N log N B ≈ and again R( ) N 2/ log N. This is mainly equivalent as considering the B ≈ fourth moment of theta functions like in [LM13b]. We thus seek for an intermediate case of an high density set and relatively well controlled GCD sum S( ). B 4.1 Integers free of small prime factors As we remarked, taking the prime numbers is a good choice in order to have an optimal small GCD sum. Moreover, we can view the primes N ≤ as the set obtained after sieving out the ﬁrst N 1/2 primes. Our idea is to increase the size of this set sieving out small primes but at the same time control the GCD sum. Practically we will consider the set of numbers free of prime factors smaller than y for some parameter y which will be ﬁxed later. Denote by P −(n) the smalllest prime factor of an integer n. Let us write

Φ(x,y) := n x : P −(n) >y (x y 2). ≤ ≥ ≥ As for instance proved in [Ten95, Chapter 15], Brun’s sieve implies the asymptotic formula, valid uniformly for x y 2, ≥ ≥

x 1 Φ(x,y)= 1+ O (2 y x1/10 log2 x) (12) ζ(1,y) (log y)2 ≤ ≤ where 1 ζ(1,y)= (1 1/p)− − p y Y≤ is the partial zeta function. The following simple observation enlights the fact that if we want to construct a set such that two distinct integers have bounded gcd’s, we can remove the small prime factors.

Lemma 3 A pair of integers m n x such that P (mn) > y veriﬁes ≤ ≤ − either m n or (m,n) < x/y. | Proof. Suppose m ∤ n, then (m,n) is a proper divisor of m. Thus it has to divide m/p for some prime p>y which conludes the proof. • Non vanishing of theta functions and sets of small multiplicative energy 9

4.2 Sieve results Let = p prime such that p y . Then we have the classical following P { ≤ } result.

Lemma 4 We deﬁne = 1 n N : (n,p) = 1 for all p , where A { ≤ ≤ ∈ P} N is an arbitrary integer such that y N. There exists an absolute constant ≤ c such that

1 #A cN 1 . (13) ≤ − p p Y∈P Proof. This is an application of Brun’s sieve which follows from [HR74, Theorem 2.2, Equation (5.2)] or can also be deduced from Selberg’s sieve [HR74, Theorem 3.5]. • Sieving with logarithmic weights is very elementary.

Lemma 5 Assume as before that N is an arbitrary integer such that y N. ≤ Then we have 1 1 log N 1 . n ≪ − p n p X∈A Y∈P Proof. This is immediate

1 1 1 1 − 1 − 1 1 = 1 1 n ≤ − p − p − p n≤N y

y Y≤ Y≤ Y≤ 1 eγ log N 1 ∼ − p p y Y≤ by Mertens’ theorem. •

4.3 Estimate of R( ) for the set of integers free of small prime factors B We want to give an upper bound for R( ) in this intermediate setting where B is the set of integers free of small prime factors less than some parameter B y. By Lemma 3, we know that either m m or (m ,m ) < N/y. 1 | 2 1 2 4.3.1 Bound in the case m m 1 | 2 We have m 1 1 = . m2 d m1,m2≤N m2 N d m2 mX1|m2 X≤ X| Non vanishing of theta functions and sets of small multiplicative energy 10

We separate the contribution from d = 1 to the one coming from proper divisors of m2. We notice that proper divisors of m2 are greater than y. Thus,

(m ,m ) 1 1 1 2 = δ = . m2 m2 m2 m1,m2 δ≤N/y m1,m2∈B δ≤N/y m1,m2≤N/δ X∈B Xδ∈B (m1X,m2)=δ Xδ∈B m1,m2∈BX;(m1,m2)=1 We can forget the coprimality condition, so we have two independent sums over sifted sets. We seek for an upper bound sieve for the double sum over m and m . Noticing that m y if m and N/δ y, we have using 1 2 2 ≥ ∈ B ≥ twice Lemma 4 (ﬁrst for the sum over m1 and then for the sum over m2)

1 (1 1/p) m2 ≪ − δ≤N/y m1≤m2≤N/δ δ≤N/y m2≤N/δ p y Xδ∈B m1,m2∈BX;(m1,m2)=1 Xδ∈B mX2∈B Y≤ N/δ (1 1/p)2 . ≪ − δ≤N/y p y Xδ∈B Y≤ We conclude using Lemma 5 to handle the sum over δ (assuming N/y y) ≥ N/δ (1 1/p)2 N log N (1 1/p)3 . − ≪ − δ≤N/y p y p y Xδ∈B Y≤ Y≤ Combining all the previous inequalities, and assuming that the parameter y veriﬁes exp(logǫ N) y N ǫ, we have ≤ ≤ 3 S( ) max ,N log N |B| B ≪ |B| N ( ) Non vanishing of theta functions and sets of small multiplicative energy 11 where we used the asymptotic from (12)

N (1 1/p) . |B| ∼ − p y Y≤ This implies

N 3 R( ) min N , . B ≫ |B| log N |B| We can rewrite this in a more pleasant way. Setting = αN, we have |B| N 2 R( ) min N 2α, (14) B ≫ α log N 1 leading to the optimal choice α = √log N .

5 Proof of Theorem 1

In our context, we have N = √p and we can choose in (2) as the set A of numbers having all their prime factors greater than exp(√log p). Using the asymptotical formula (12) and Mertens’ theorem, we have √p . |A| ∼ √log p Thus inserting in (14), p R( ) N 2α = . A ≫ √log p This leads to the following lower bound on the proportion of non vanishing

2 p M1(p) /M2(p) , ≫ √log p which concludes the proof by (1).

6 The case of odd characters

If χ mod p is odd, then we set

πn2x/p θ(x,χ)= nχ(n)e− n 1 X≥ and we have the relations (p 1)/2 if b a (mod p) and gcd(a, p) = 1, − ≡ χ(a)¯χ(b)= (p 1)/2 if b a (mod p) and gcd(a, p) = 1,  − − ≡− χ X− 0 otherwise, X∈ p  where X is the set of the (p 1)/2 odd Dirichlet characters mod p 3. p− − ≥ Using a similar method and partial summation, we can show that θ(x,χ) = 6 0 for at least p/√log p of the (p 1)/2 characters χ X . ≫ − ∈ p− Non vanishing of theta functions and sets of small multiplicative energy 12

7 Combinatorial open questions and consequences

To summarize, we constructed a set [1,N] of density α = 1 such B ⊂ √log N that S( ) or in another words the multiplicative energy (as deﬁned B ≪ |B| in [Gow98, Tao08, TV06]) veriﬁes

E ( ,N) := ab = cd, 1 b, d N, a, c N . × B | { ≤ ≤ ∈B}≪ |B| This is of course optimal in terms of the size of the sets up to the constant. We address the following problem

Question 6 What is the maximal 0 <α< 1 (in terms of N) such that there exists a set of density α verifying E ( ,N) N ? B × B ≪ |B| 1 Our previous discussion shows that we can take α = √log N . We can 1 ask the maybe easier question of constructing a set of density (log N)β with β < 1/2 having this property of minimizing the multiplicative energy. In order to get any improvement of our main result, the actual reﬁned question would be suﬃcient

Question 7 Can we construct a set of density 1 such that B (log N)α E ( ,N) N (log N)β with α + β < 1/2. × B ≪ |B| Let us address the following related problem which might not have applications in the vanishing of theta functions but that we found interesting in its own right. Solymosi obtained in [Sol09] a beautiful upper bound E ( , ) + 2 log on the multiplicative energy of a set of reals. It × B B ≪ |B B| |B| implies particularly that it is good to choose a set with very small sumset in order to minimize the energy. We want to address the question whether we can get rid of this logarithmic term for suﬃciently high-density sets of integers.

Question 8 What is the maximal 0 <α< 1 (in terms of N) such that there exists a set of density α such that E ( , ) 2? B × B B ≪ |B| Acknowledgements

The author would like to thank St´ephane Louboutin and Igor Shparlinski for their comments on a ﬁrst version of the draft as well as the latter one for pointing him out that similar ideas where used in [KSY]. The author is supported by the Austrian Science Fund (FWF) project Y-901. Non vanishing of theta functions and sets of small multiplicative energy 13

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