Second moment of Dirichlet L -functions, character sums over subgroups and upper bounds on relative class numbers
aa r X i v : . [ m a t h . N T ] J u l Second moment of Dirichlet L -functions,character sums over subgroups,and upper bounds on relative class numbers. St´ephane R. LOUBOUTINAix Marseille Universit´e, CNRS, Centrale Marseille, I2M,Marseille, [email protected] MUNSCH5010 Institut f¨ur Analysis und Zahlentheorie8010 Graz, Steyrergasse 30, Graz, [email protected] 7, 2020
Abstract
We prove an asymptotic formula for the mean-square average of L - functions associatedto subgroups of characters of sufficiently large size. Our proof relies on the study of certaincharacter sums A ( p, d ) recently introduced by E. Elma. We obtain an asymptotic formula for A ( p, d ) which holds true for any divisor d of p − d . This anwers a question raised in Elma’s paper. Our proof relies both on estimates on thefrequency of large character sums and techniques from the theory of uniform distribution. Asan application we deduce the following bound h − p,d ≤ (cid:16) (1+ o (1)) p (cid:17) m/ on the relative classnumbers of the imaginary number fields of conductor p ≡ d and degree m = ( p − /d . Throughout the paper d ≥ p will be an odd prime satisfying p ≡ d ). We also write log j for the j-th iterated logarithm. We let X p denote the multiplica-tive cyclic group of order p − p and let X ∗ p denote the setwith p − p . We set m = ( p − /d , aneven integer, and χ p,m will denote any one of the φ ( m ) odd Dirichlet characters in X p of order m .Let h − p,d be the relative class number of the imaginary subfield K p,d of the cyclotomic field Q ( ζ p ) of even degree ( K p,d : Q ) = m and odd relative degree ( Q ( ζ p ) : K p,d ) = d (e.g. see [Was, L -functions, cyclotomic field, relative class number,character sums, multiplicative subgroups, exponential sums, Dedekind sums, discrepancy. econd moment of L -functions, character sums and relative class numbers d = 1, we have K p, = Q ( ζ p ) and it has long been known that h − p, = h − Q ( ζ p ) ≤ p (cid:16) p (cid:17) ( p − / = 2 p (cid:16) p (cid:17) m/ , (1)see [Met] and [Wal]. In [Lou93] it is explained how to improve upon this bound by taking valuesgreater than 24 for this denominator, in fact values as close to 4 π as desired. See also [Gra] formore subtle results.Denote by w p,d the number of complex roots of unity contained in K p,d , we have w p, = 2 p and w p,d = 2 for d >
1. The following bound holds: h − p,d = w p,d m/ Y j =1 √ p π L (1 , χ j − p,m ) ≤ w p,d (cid:18) pM ( p, m )4 π (cid:19) m/ , (2)where M ( p, m ) denotes the following mean square of L (1 , χ ): M ( p, m ) := 2 m m/ X j =1 | L (1 , χ j − p,m ) | . (3)Therefore explicit formulas (or asymptotic formulas) for M ( p, m ) allow to give precise upperbounds of type h − p,d ≤ C · C m/ . For d = 1, H. Walum deduced (1) in [Wal] by proving that M ( p, p −
1) = π (cid:18) − p (cid:19) (cid:18) − p (cid:19) . (4)For d = 3 ,
5, some explicit formulas for M ( p, m ) have been obtained in certain cases by thefirst author (see Section 2.2) allowing him to give upper bounds on h − p, and h − p, . In contrast,for a given even m , as p runs over the prime integers p ≡ m mod 2 m nothing better than h − p, ( p − /m ≪ m ( p log p ) m/ is known, by using (2) and the bound | L (1 , χ ) | ≪ log p . To beginwith, the following simple argument gives a trivial bound on M ( p, m ). Since in M ( p, m ) weconsider only m/ p − / M ( p, p − M ( p, m ) ≤ m p − M ( p, p −
1) = dM ( p, p −
1) = d π (cid:16) − p (cid:17) (cid:16) − p (cid:17) . (5)By (2) it implies h − p,d ≤ (cid:18) dp (cid:19) p − d = 2 (cid:18) dp (cid:19) m/ . (6)The aim of the paper is to give an asymptotic formula for M ( p, m ) when m is of reasonable sizewith respect to p and an upper bound when m is small. As a consequence we obtain a significantimprovement upon the trivial bound (6). econd moment of L -functions, character sums and relative class numbers Theorem 1 As p tends to infinity and d ≥ runs over the odd divisors of p − such that d ≤ (1 / − ε ) log p log p , we have the asymptotic formula M ( p, m ) = M ( p, ( p − /d ) = π (cid:16) O ( d (log p ) p − d − ) (cid:17) , (7) which implies the upper bound h − p,d ≤ (cid:18) (1 + o (1)) p (cid:19) m/ . (8) If the previous bound does not apply but d is such that log d = o (log p/ log p ) , we have for someabsolute constant C > M ( p, m ) = M ( p, ( p − /d ) ≤ C (log d ) (9) which implies, in this range of d , the upper bound h − p,d ≤ ( Cp (log d ) ) m/ . (10) Remarks 2
It should be emphasized that the error term in (7) is almost optimal, in view ofProposition 7. Furthermore (8) is in accordance with the known asymptotics (see [Lou96b,Theorem 4]) log h − p,d ∼ m + o (1)4 log p. For very large d , the bound M ( p, m ) ≪ log p remainsthe best known. This is not surprising if we look at the very extreme case d = ( p − / and p ≡ . In that situation, χ is the quadratic character given by the Legendre symbol χ ( n ) = (cid:16) np (cid:17) and M ( p, m ) = | L (1 , χ ) | . Under GRH, Littlewood [Lit] proved that L (1 , χ ) ≪ log p but improving upon the bound L (1 , χ ) ≪ log p remains out of reach unconditionally. Onthe other way we cannot expect an uniform bound for M ( p, m ) better than (log p ) . Indeed,Chowla [Cho] proved unconditionally that there are infinitely many quadratic characters χ suchthat L (1 , χ ) ≫ log p . This supports the hypothesis that the bound (9) could be sharp. The paper is organized as follows. To begin, in Section 2, we recall (and give a simple proofof) a formula discovered by Elma relating M ( p, m ) to certain character sums A ( p, d ) definedbelow in (11). In Section 3, Proposition 7, we show that for a certain family of primes p and d we can compute exactly M ( p, m ) using properties of Dedekind sums. Finally, in Section 4, weprove an asymptotic formula for A ( p, d ) which directly implies Theorem 1 (see Section 5). Acrucial point of the analysis comes from the fact that the average in (3) is made over a familyof m/ p . The same difficulty carries intothe analysis of A ( p, d ) with a character sum averaged over a subgroup of F ∗ p of size d . On onehand when d is small (see Theorem 10) we write A ( p, d ) as an average of a function evaluatedat equidistributed points modulo 1 and use techniques from discrepancy theory. On the otherhand, when d is large (see Theorem 19) we rely on character sums techniques and incorporaterecent estimates on the frequency of large character sums [BGGK]. econd moment of L -functions, character sums and relative class numbers Let χ be an odd Dirichlet character of (even) order m dividing p − p ≥ d = ( p − /m (an odd integer) and A ( p, d ) = 1 p − p − X N =1 X ≤ n ,n ≤ Nχ ( n χ ( n (11)(the results depends only on p and d , not on the choice of χ ). M ( p, m ) E. Elma proved a nice connection between the mean square values M ( p, m )’s and these charactersums A ( p, d ). We give a simple and short proof of [Elma, Theorem 1.1]: Theorem 3
Let χ be a primitive Dirichlet modulo f > , its conductor. Set S ( k, χ ) = P kl =0 χ ( l ) and let L ( s, χ ) = P n ≥ χ ( n ) n − s be its associated Dirichlet L -series. Then f − X k =1 | S ( k, χ ) | = f Y p | f (cid:18) − p (cid:19) + a χ f π | L (1 , χ ) | , where a χ := ( if χ ( −
1) = +1 , if χ ( −
1) = − . Proof.
Our simple proof is based on an easy to remember idea: we apply Parseval’s formula R | F ( x ) | d x = P ∞ n = −∞ | c n ( F ) | to the function x ∈ [0 , F ( x ) := P ≤ l ≤ fx χ ( l ) extendedto x ∈ R by 1-periodicity. The reader would be able to reconstruct the argument using thissimple idea. Let us now give all the details. Since χ is primitive, the Gauss sums τ ( n, χ ) = P fk =1 χ ( k ) exp(2 πink/f ) and τ ( χ ) = τ (1 , χ ) satisfy τ ( n, χ ) = χ ( n ) τ ( χ ) and | τ ( χ ) | = f , e.g.see [Was, Lemmas 4.7 and 4.8]. (These properties are easy to check when f = p ≥ x F ( x ) = S ( k, χ ) is constant for x ∈ [ k/f, ( k + 1) /f ), we have Z | F ( x ) | d x = 1 f f − X k =0 | S ( k, χ ) | . and the Fourier coefficients of F are given by c n ( F ) = Z F ( x ) exp( − πinx )d x = f − X k =0 S ( k, χ ) Z ( k +1) /fk/f exp( − πinx )d x. (12)Hence, by [Was, Theorem 4.2] we have c ( f ) = 1 f f − X k =0 S ( k, χ ) = 1 f f − X k =0 k X l =0 χ ( l ) = 1 f f − X l =0 ( f − l ) χ ( l ) = − f f − X l =0 lχ ( l ) = L (0 , χ ) econd moment of L -functions, character sums and relative class numbers n = 0 we have c n ( F ) = f − X k =0 S ( k, χ ) exp (cid:16) − πin ( k +1) f (cid:17) − exp (cid:16) − πinkf (cid:17) − πin (by (12))= f − X k =1 ( S ( k, χ ) − S ( k − , χ )) exp (cid:16) − πinkf (cid:17) πin (notice that S (0 , χ ) = S ( f − , χ ) = 0)= τ ( − n, χ )2 πin = τ ( χ )2 πi × χ ( − n ) n = − χ ( − c − n ( F ) . Now, L (0 , χ ) = 0 if χ ( −
1) = +1 and | L (0 , χ ) | = fπ | L (1 , χ ) | if χ ( −
1) = −
1, e.g. see [Was,Chapter 4, page 30]. Therefore, Parseval’s formula gives1 f f − X k =0 | S ( k, χ ) | = a χ fπ | L (1 , χ ) | + 2 X n ≥ n,f )=1 f π × n = a χ fπ | L (1 , χ ) | + f Y p | f (cid:18) − p (cid:19) and the desired result follows. Notice that this proof is similar to the ones in [BC]. • Corollary 4
Let χ be an odd Dirichlet character of (even) order m dividing p − and primeconductor p ≥ . Set d = ( p − /m (an odd integer) and let A ( p, d ) be as in (11). Then M ( p, m ) := 2 m m − X j =0 j odd | L (1 , χ j ) | = π p − p (12 A ( p, d ) − (4 d + 1) p − d − . (13) In particular, (4 d + 1) p + d + 112 ≤ A ( p, d ) ≤ (5 d + 1) p + d + 112 . (14) Moreover, by [Lou96a], we have ≤ M ( p, m ) ≤ (log p + 2 + γ − log π ) / . (15) Proof.
By Theorem 3, for j odd we have | L (1 , χ j ) | = − π (cid:18) − p (cid:19) + π p p − X k =1 | S ( k, χ j ) | The χ j ’s are primitive modulo p for 1 ≤ j ≤ m −
1, whereas χ is the non-primitive trivialDirichlet character modulo p . Therefore, on the one hand we have m − X j =0 p − X k =1 | S ( k, χ j ) | = m − X j =0 p − X k =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k X l =1 χ j ( l ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = m − X j =0 p − X k =1 X ≤ l ,l ≤ k χ j ( l ) χ j ( l ) = m ( p − A ( p, d ) , by using the orthogonality relation m − X j =0 χ j ( n ) χ j ( n ) = ( m if χ ( n ) = χ ( n ) = 0,0 otherwise. econd moment of L -functions, character sums and relative class numbers m − X j =1 p − X k =1 | S ( k, χ j ) | = ( m − p − mp π M ( p, m ) . Since p − X k =1 | S ( k, χ ) | = p − X k =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k X l =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = p − X k =1 k = ( p − p (2 p − , it follows that m ( p − A ( p, d ) = m ( p − p − (4 p + 1)12 + mp π M ( p, m ) . The desired identity (13) follows.Now, noticing that M ( p, m ) ≥
0, the lower bound on A ( p, d ) in (14) follows from (13).Finally, by (5) we have M ( p, m ) ≤ dπ (cid:18) − p (cid:19) . Plugging this bound in (13) we obtain the upper bound on A ( p, d ) in (14). • By (2) and (13), upper bounds on A ( p, d ) would yield upper bounds on h − p,d . More precisely,for d > M ( p, m ) ≤ π / A ( p, d ) < (cid:0) (4 d + 2) p + d + 2 (cid:1) /
12 would yield h − p,d ≤ p/ m/ . Remarks 5
Corollary 4 gives p +412 ≤ A ( p, ≤ p +412 , whereas A ( p,
3) = p +412 , by (19) below.Hence it should be possible to improve upon the upper bound in (14) . M ( p, m ) and A ( p, d ) in specific cases There are only four cases listed below where an explicit formula for A ( p, d ) is known.1. By (4) and (13), for d = 1 we have A ( p,
1) = p d + 1) p . (16)2. For d = 3 we proved in [Lou16, Theorem 1] that M ( p, ( p − /
3) = π (cid:18) − p (cid:19) (for p ≡ h − p, ≤ (cid:16) p (cid:17) ( p − / = 2 (cid:16) p (cid:17) m/ . (18)By (13), this gives for d = 3 and p ≡ A ( p,
3) = 7 p + 26 = (2 d + 1) p o ( p ) . (19) econd moment of L -functions, character sums and relative class numbers
73. For d = 5 we proved in [Lou16, Theorem 5] that M ( p, ( p − /
5) = π (cid:18) a ( a + 1) − p (cid:19) (for p > p = a − a − ) . (20)and the corresponding bound on the relative class number h − p, ≤ (cid:16) p (cid:17) ( p − / = 2 (cid:16) p (cid:17) m/ . (21)By (20) and (13), this implies A ( p,
5) = 11 p + 36 + a ( a + 1) p p −
1) = (2 d + 1) p o ( p ) . (22)4. For d = ( p − / < p ≡ χ is the quadratic charactergiven by the Legendre symbol χ ( n ) = (cid:16) np (cid:17) , L (1 , χ ) = πh Q ( √− p ) / √ p and (13) gives A ( p, ( p − /
2) = 4 p − p + 124 + ph Q ( √− p ) p − . Remarks 6
In fact, d = 1 is the only case for which we could come up with a direct proof of theformula for A ( p, d ) . Indeed, we have χ p,p − ( n ) = χ p,p − ( n ) if and only if n ≡ n (mod p ) .Hence, A ( p,
1) = 1 p − p − X N =1 N = p . It would be nice to have similar independent and direct proofs of (19) and (22). M ( p, m ) for primes p = ( a d − / ( a − ≡ d ) We gave an explicit formula for A ( p, A ( p, p of the form p = ( a − / ( a − a − ) / ( a −
2) or ( a − ) / ( a −
3) we could not guess any formula for M ( p, ( p − / A ( p, a < A ( p, d ). Instead we give an exact formula for M ( p, m ) and then use (13) to deduce an expression for A ( p, d ). econd moment of L -functions, character sums and relative class numbers Proposition 7
Set Q l ( X ) = ( X l − − l ( X − / ( X − ∈ Z [ X ] , l ≥ . Hence, Q ( X ) = 0 , Q ( X ) = 1 and Q l ( X ) = X l − + 2 X l − + · · · + ( l − X + ( l − for l ≥ . Let d ≥ be a primeinteger. For a prime integer of the form p = ( a d − / ( a − for some a = − , , , we have M ( p, ( p − /d ) = π a ( a + 1) Q d − ( a ) − p ! , (23) A ( p, d ) = (2 d + 1) p + d +12 pp − · a ( a + 1) Q d − ( a )6 = 2 d + 16 p + O (cid:16) p − d − (cid:17) , (24) and h − p,d ≤ p/ m/ for p = ( a d − / ( a − with a ≤ − . Proof.
We keep the notation of [Lou16], use the properties of Dedekind sums s ( c, d ) = 14 d d − X n =1 cot (cid:16) πnd (cid:17) cot (cid:16) πncd (cid:17) ( c ∈ Z , d ∈ Z \ {− , , } )recalled in [Lou16] and set l = ( d − /
2. To deal in one stroke with the two cases a ≤ − a ≥ d to be negative. Letting ǫ ( d ) ∈ {± } denote the sign of 0 = d ∈ Z , the reciprocity and complementary laws for thesegeneralized Dedekind sums are s ( c, d ) + s ( d, c ) = c + d − ǫ ( c ) ǫ ( d ) cd + 112 cd and s (1 , d ) = d − ǫ ( d ) d + 212 d . Set p k = ( a k − / ( a −
1) and ǫ = ǫ ( a ). Then ǫ ( p ) = 1, ǫ ( a k ) = ǫ k and ǫ ( p k ) = ǫ k +1 . We have M ( p, ( p − /d ) = π (cid:18) Np (cid:19) , where N = 24 l X k =1 s ( a k , p ) ! − p . Now, p ≡ p k (mod a k ) and a k ≡ p k ) for 1 ≤ k ≤ d . Hence s ( a k , p ) = a k + p − ǫ k a k p + 112 a k p − s ( p, a k ) = a k + p − ǫ k a k p + 112 a k p − s ( p k , a k )and s ( p k , a k ) = p k + a k − ǫp k a k + 112 p k a k − s ( a k , p k ) = p k + a k − ǫp k a k + 112 p k a k − s (1 , p k ) , by the reciprocity law for Dedekind sums. Since s (1 , p k ) = p k − ǫ k +1 p k + 212 p k , by the complementary law for Dedekind sums, we obtain s ( a k , p k ) = a k + p − ǫ k a k p + 112 a k p − (cid:18) p k + a k − ǫp k a k + 112 p k a k − p k − ǫ k +1 p k + 212 p k (cid:19) econd moment of L -functions, character sums and relative class numbers s ( a k , p k ) = a k + p − a k p + 112 a k p + ( a − p k + 1 − a k a k . Notice that the more natural congruence p ≡ p k − (mod a k ) and a k ≡ a (mod p k − ) would leadto slightly more complicated computations. An easy but boring computation using P lk =1 b k = b ( b l − / ( b −
1) then finally yields N = 2 a ( a + 1) Q l ( a ) −
1, as desired. • Remarks 8
It is widely believed since a long time that there are infinitely many primes ofthe form p = ( a d − / ( a − , as firstly investigated in the special case of Mersenne primes p − ( a = 2 ). More precise results about the number of such primes less than x are expected.This is sometimes called Lenstra-Pomerance-Wagstaff conjecture (see the survey [Pom] for moreinformation and references on this topic). Let us remark that A ( p, d ) dp/ pM ( p, m )2 π d ( p −
1) + 1 + 14 d + 14 p + 14 dp , by (13). Hence, as d/ log p tend to infinity, we have by (15) A ( p, d ) ∼ dp d . Our goal in this section is to prove that (25) holds true without any restrictionon the size of the parameter d . Moreover when d is constant, we obtain a refined asymptoticformula A ( p, d ) ∼ d +16 which is in accordance with the exact formulas from Section 2.2. Theorem 9 As p tends to infinity and d ≥ runs over the odd divisors of p − , we have A ( p, d ) = dp o ( dp ) . We will split the discussion into two cases depending on whether d goes or not to infinity.Theorem 9 follows from Theorems 10 ( d small) and 19 ( d large) proved below. In the formercase, we obtain the more precise asymptotic expansion A ( p, d ) ∼ d +16 . By (13), this allows usto deduce an asymptotic formula for M ( p, m ). In the latter case, Theorem 9 is not sufficient toinfer an asymptotic formula for M ( p, m ) and only implies an upper bound. d ’s Our goal in this section is to prove the following theorem which gives Theorem 9 for small d ’s: econd moment of L -functions, character sums and relative class numbers Theorem 10
Let d range over the odd integers. Set γ ( d ) = max k | d φ ( k ) . Hence γ ( d ) ≤ d − ,with equality whenever d is an odd prime. Let p range over the prime integers such that p ≡ d ) . Then we have the following asymptotic formula A ( p, d ) = 2 d + 16 p + O (cid:16) d (log p ) p − /γ ( d ) (cid:17) where the implicit constant in the error term is absolute.In particular if d ≤ c log p log p with c < / , we have A ( p, d ) ∼ d + 16 p. Remarks 11
Let us point out that (24) shows that the power of p in the error term of Theorem10 is optimal. For any fixed integer s , we consider the s -dimensional cube I s = [0 , s equipped with its s -dimensional Lebesgue measure λ s . We denote by B the set of rectangular boxes of the form s Y i =1 [ α i , β i ) = { x ∈ I s , α i ≤ x i < β i } where 0 ≤ α i < β i ≤ . If S is a finite subset of I s , we define the discrepancy D ( S ) by D ( S ) = sup B ∈B (cid:12)(cid:12)(cid:12)(cid:12) B ∩ S ) S − λ s ( B ) (cid:12)(cid:12)(cid:12)(cid:12) . The discrepancy measures in a quantitative way the deviation of a pointset S from equidistribu-tion. In particular a sequence of sets S n is uniformly distributed if and only if D ( S n ) −−−→ n →∞ Koksma-Hlawka inequality : Theorem 12 [DT, Theorem . ] Let f ( x ) a function of bounded variation on I s in the senseof Hardy and Krause and x , . . . , x N a finite sequence of points in I s . Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N X i =1 f ( x i ) − Z I s f ( u ) dλ s ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ V ( f ) D ( S ) where V ( f ) is the Hardy-Krause variation of f (see also [KN, Chapter ]). In order to estimate the discrepancy, we recall the inequality of
Erd˝os-Tur´an-Koksma : Theorem 13 [DT, Theorem . ]. Let S = { x , . . . , x N } be a set of points in I s and H apositive integer. Then we have D ( S ) ≤ (cid:18) (cid:19) s H + 1 X < k h k ∞ ≤ H r ( h ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N X n =1 e ( h h , x N i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (26) econd moment of L -functions, character sums and relative class numbers where e ( z ) = exp(2 πiz ) , r ( h ) = s Y i =1 max { , | h i |} for h = ( h , . . . , h s ) ∈ Z s and h , i denotes thestandard inner product in R s . In the rest of the paper the reults of the previous section will only be used for s = 2.We introduce some tools from the theory of pseudo-random generators and optimal coefficientsin a very basic situation. We refer for more information to the survey of Korobov [Kor], thework of Niederreiter [Nied77, Nied78] or the book of Konyagin and Shparlinski [KS, Chapter 12]and keep their notations. For any prime p and integer 1 ≤ λ ≤ p − σ ( λ, p ) := X < k h k ∞ ≤ p − δ p ( h + h λ ) r ( h )where δ p ( a ) = 1 if a = 0 mod p and δ p ( a ) = 0 otherwise.For any λ , we define ρ ( λ, p ) = min h =0 r ( h )where the min is taken over all non trivial solutions h = ( h , h ) of the congruence h + h λ = 0 mod p. These two quantities are relatively close to each other:
Lemma 14 [Nied77, Theorem . ]. There exists C > such that, for any prime p ≥ , and λ ∈ { , . . . , p − } we have ρ ( λ, p ) ≤ σ ( λ, p ) ≤ C (log p ) ρ ( λ, p ) . (27)In some cases which are of interest for our problem, we can control from below ρ ( λ, p ): Lemma 15
Let λ be an element order k ≥ in the multiplicative group F ∗ p . Then ρ ( λ, p ) ≥ p /φ ( k ) / √ , where φ denotes as usual the Euler’s totient function. Proof.
Let Φ k ( X ) = X ≤ l ≤ φ ( k ) a l X l = Y ≤ l ≤ k gcd( k,l )=1 ( X − ζ lk )denote the k -th cyclotomic polynomial. Set k Φ k ( X ) k = X ≤ l ≤ φ ( k ) | a l | / = (cid:18) π Z π | Φ k ( e it ) | dt (cid:19) / ≤ φ ( k ) . econd moment of L -functions, character sums and relative class numbers k ( λ ) = 0 mod p . For h = ( h , h ) = 0 we define P ( X ) = h + h X. Assumethat P ( λ ) = 0 mod p , then p divides the resultant R = Res( P, Φ k ). The polynomial Φ k beingirreducible of degree ≥
2, we deduce that R = 0. It follows that | R | ≥ p . Since R is thedeterminant of the Sylvester matrix of P ( X ) and Φ k ( X ), by Hadamard’s inequality we have | R | ≤ k P ( X ) k deg Φ k ( X )2 k Φ k ( X ) k deg P ( X )2 ≤ (cid:0) h + h (cid:1) φ ( k ) / φ ( k ) ≤ (max( | h | , | h | )) φ ( k ) φ ( k ) / . Hence we have r ( h ) ≥ max( | h | , | h | ) ≥ | R | /φ ( k ) / √ ≥ p /φ ( k ) / √ . All together we obtain the lower bound ρ ( λ, p ) ≫ p /φ ( k ) / √ . • Set H = ker( χ ), the subgroup of F ∗ p of order d . We interpret the condition χ ( n ) = χ ( n ) as n n − ∈ H. We write H as a disjoint unionH = [ k | d H k , where H k := { θ ∈ H , ord ( θ ) = k } . Proposition 16
For any pair ( x, y ) of I we define f d ( x, y ) = xd − x, y ) . We have the following relation A ( p, d ) = 1 p − X ≤ n ,n ≤ p − χ ( n χ ( n min( n , n ) = p ( p − X k | dk =1 X θ ∈ H k X x mod p f d (cid:18) xp , xθp (cid:19) . Proof.
Changing the order of summation in (11) and making the change of variables ( n , n ) ( p − n , p − n ), we do have( p − A ( p, d ) = X ≤ n ,n ≤ p − χ ( n χ ( n ( p − max( n , n )) = X ≤ n ,n ≤ p − χ ( n χ ( n min( n , n ) . Now we have X ≤ n ,n ≤ p − χ ( n χ ( n min( n , n ) = X x mod p x + X θ ∈ H θ =1 min( x, θx ) . We remark that if θ = 1, we havemin( x, θx ) = pf d (cid:18) xp , xθp (cid:19) − xd − . Using the decomposition H = S k | d H k and summing over H, the proposition follows. • Remarks 17
The reader might wonder why we did not express directly the sum A ( p, d ) using themore natural function on I d given by g ( x , . . . , x d ) = P di =1 min( x , x i ) evaluated at the points (cid:16) xp , xλp , . . . , xλ d − p (cid:17) , where λ generates H. This comes from the fact that these points are notequidistributed in I d because they lie in the hyperplane of equation x + · · · + x d = 0 . econd moment of L -functions, character sums and relative class numbers We introduce the set of points in I : S θ = (cid:26)(cid:18) xp , xθp (cid:19) , x mod p (cid:27) for any θ ∈ H \{ } . By Theorem 12 we have for any θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p X x mod p f d (cid:18) xp , xθp (cid:19) − Z I f d ( u, v ) dudv (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ V ( f d ) D ( S θ ) . It is easy to compute the integral and obtain Z I f d ( u, v ) dudv = 12( d −
1) + 13 . Applying Proposition 16 and simplifying, we obtain the equation A ( p, d ) = 2 d + 16 p + O ( ET ) (28)where the error term is ET := pV ( f d ) X k | dk =1 X θ ∈ H k D ( S θ ) . (29)The readers can easily convince themselves that V ( f d ) ≪ d (for instance lookat the variation of f d over the rectangle R := [ x , x ] × [ x , y ], namely v R ( f ) := f d ( x , y ) − f d ( x , y ) − f d ( x , y ) + f d ( x , y ). The Vitali variation can then be obtained by summing v R ( f )over a partition of I and taking the supremum over all possible partitions) . Hence to finishthe proof, we need to bound the sum of discrepancies. Applying Theorem 26 with H = p − D ( S θ ) ≤ (cid:18) (cid:19) p + X < k h k ∞ ≤ p − r ( h ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p p X x =1 e (cid:18) h x + h xθp (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Using the orthogonality relations X b mod p e ( bn/p ) = (cid:26) p, if n ≡ p ),0 , if n p ),we can bound the sum over h by σ ( θ, p ) := X < k h k ∞ ≤ p − δ p ( h + h θ ) r ( h )using the notations of subsection 4.1.2. For θ ∈ H k , we apply consecutively Lemma 14 andLemma 15 to obtain σ ( θ, p ) ≤ C (log p ) /p /φ ( k ) The Hardy-Krause variation is then obtained as a sum of the Vitali variations of f d , f d ( x,
1) and f d (1 , y ). econd moment of L -functions, character sums and relative class numbers C . Hence recalling that γ ( d ) = max k | d φ ( k ) and summing over k , wearrive at ET = pV ( f d ) X k | dk =1 X θ ∈ H k D ( S θ ) ≪ d (log p ) p − /γ ( d ) . This concludes the proof of Theorem 10, in view of Equation (28). d ’s For a given non-principal Dirichlet character χ mod p , where p is a prime, let M ( χ ) := max ≤ x ≤ p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X n ≤ x χ ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) and its renormalization m ( χ ) = M ( χ ) e γ √ p/π . The P´olya–Vinogradov Theorem states that m ( χ ) ≪ log p (30)for all non-principal characters χ mod p . Apart from some improvements on the implicit con-stant, this remains the state-of-the-art for the general non-principal character. However, formost of the characters M ( χ ) is much smaller and we can study how often M ( χ ) is large. Thebest result in this direction was obtained in [BGGK]: Theorem 18
Let η = e − γ log 2 . If ≤ τ ≤ log p − M for some M ≥ , then Φ p ( τ ) := 1 p − { χ mod p : m ( χ ) > τ } ≤ exp (cid:26) − e τ − − η τ (1 + O ((log τ ) /τ )) (cid:27) . We are now in a position to prove Theorem 9 for large d ’s. Theorem 19
Assume that d → + ∞ and log d = o (log p/ log p ) . Then we have the followingasymptotic formula A ( p, d ) = dp/ O (cid:0) p (log d ) (cid:1) . For any d (in particular for larger d ’s), the following holds A ( p, d ) = dp/ O (cid:0) p (log p ) (cid:1) . Remarks 20
The condition log d = o (log p/ log p ) could be made more explicit by specifyingconstants in the proof below. Notice also that whereas Theorem 9 follows from Theorems 10 and19, it does not follow from Theorem 10 and (25) . econd moment of L -functions, character sums and relative class numbers Proof.
The second part of the Theorem follows directly from (13) and the inequality | L (1 , χ ) ≪ log p . This could also be proved following our argument below and using only P´olya–Vinogradovinequality. Let us now focus on the case log d = o (log p/ log p ). The condition χ ( n ) = χ ( n ) isequivalent to n n − lying in the kernel of χ , which is a subgroup of order d of the multiplicativecyclic group F ∗ p . We apply the orthogonality of characters in the subgroup < χ > of order m generated by χ ∈ X p and rewrite the sum A ( p, d ) defined in (11) as A ( p, d ) = 1( p − p − X N =1 m X Ψ ∈X p Ψ m = χ X ≤ n ,n ≤ N Ψ( n n − ) . Separating the contribution of the trivial character from the others, this leads us to the equation A ( p, d ) = d ( p − p − X N =1 N + 1( p − p − X N =1 m X Ψ ∈X∗ p Ψ m = χ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ≤ n ≤ N Ψ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . We have trivially d ( p − p − X N =1 N = dp dp p −
1) = dp/ O ( d ) . Therefore we are left to bound the contribution of non-trivial characters and A ( p, d ) = dp/ O (cid:18) d R ( p − (cid:19) (31)where R := p − X N =1 X Ψ ∈X∗ p Ψ m = χ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ≤ n ≤ N Ψ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Let us set the parameter τ = min { C (log d ) , log p − M } where M is the constant appearingin Theorem 18 and C is some large constant which will be specified later. We introduce thefollowing set of characters X τp, = (cid:8) Ψ ∈ X ∗ p : m (Ψ) ≤ τ (cid:9) and further define for every integer 1 ≤ j ≤ J X τp,j := (cid:8) Ψ ∈ X ∗ p : 2 j − τ < m (Ψ) ≤ j τ (cid:9) where J is chosen in order to allow an application of Theorem 18. Precisely, we choose J suchthat τ J ≤ log p − M < τ J +1 . We now split the characters appearing in the summation in R as follows X ∗ p = J [ j =0 X τp,j [ (cid:8) Ψ ∈ X ∗ p : m (Ψ) > J τ (cid:9) . econd moment of L -functions, character sums and relative class numbers τ = log p − M then J = 0 and we only split the summation depending on whether m (Ψ) ≤ log p − M or not. Remark that here are at most m characters Ψ ∈ X τp, appearing inthe sum. Hence, it follows from Theorem 18 and the inequality (30) that R ≪ p − X N =1 mpτ + p J X j =1 τ j Φ p ( τ j − ) + p (log p ) Φ p ( τ J ) ≪ p − X N =1 mpτ + p J X j =1 τ j exp ( − e τ j − τ j ) + p (log p ) exp ( − e τ J τ J ) . (32)The summation over j in the right hand side of (32) is clearly dominated by its first term. Thuswe obtain after summing over N and recalling our choice of J : R ≪ p d τ + p τ exp (cid:26) − c e τ τ (cid:27) + p (log p ) e − c log p/ log log p (33)for some absolute constants c , c >
0. We insert (33) in (31) and choose C large enough inthe definition of τ to ensure that the second and third term in the right hand side of (33)have negligible contribution. This is indeed possible due to the restriction on the size of d andconcludes the proof. • The first part of Theorem 1 follows from Theorem 10 and (13). The second part follows fromTheorem 19 and (13).
We solved Elma’s question about the asymptotic behavior of the character sums A ( p, d ) regard-less of the size of d . As already noticed above, for d large, this is not precise enough to deduce anasymptotic formula for the mean-square value M ( p, m ). To conclude, let us say that the upperbound (9) could be obtained by working directly with L (1 , χ ) following our method of proofof Theorem 19. This requires results about the distribution of L (1 , χ ) as the ones obtained byGranville and Soundararajan [GS1, GS2] instead of Theorem 18. Funding
This work was supported (for M. M) by the Austrian Science Fund (FWF) [P-33043].
Acknowledgements
The second author would like to thank Igor Shparlinski for sketching a refinement of our argu-ment in the proof of Theorem 19 leading to a better result. econd moment of L -functions, character sums and relative class numbers References [BC] P. T. Bateman and S. Chowla. Averages of character sums.
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