The universality of quadratic L-series for prime discriminants

Transcription

1 ACTA ARITHMETICA ) The universality of quadratic L-series for prime discriminants by Hidehiko Mishou Nagoya) and Hirofumi Nagoshi Niigata). Introduction and statement of results. For an odd prime p let λ p denote the real Dirichlet character modulo p given by the Legendre symbol p). Let Ls, χ) be the Dirichlet L-function associated with a character χ. The value distribution of Ls, λ p ) for a complex number s with Res > / as p varies over the odd primes is investigated e.g. in [El]. The main purpose of the present paper is to study the functional distribution of Ls, λ p ) on D, as p varies over the primes in an arithmetic progression; here and henceforth D denotes the strip {s C / < Re s < }. More precisely, we shall establish the so-called universality theorem for Ls, λ p ) in the p-aspect. The universality theorem was first discovered by Voronin [Vo], [KV]) for the Riemann zeta-function ζs) in the t-aspect; he showed the following. Theorem [Vo]). Let 0 < r < /4 and hs) be a continuous function on the disk s r which is holomorphic and has no zeros in s < r. Then for any ε > 0 there exists a real number t such that max ζs + 3/4 + it) hs) < ε. s r The universality theorem for a Dirichlet L-function Ls, χ) in the t- aspect was obtained by Bagchi [B], [B], Gonek [Go] and Voronin see [KV, Chapter VII, Section 3]) independently; indeed, the joint universality theorem was shown. Furthermore, Bagchi [B], Eminyan [Em] and Gonek [Go] independently showed an analogous result for Dirichlet L-functions in another aspect. In fact, they established the universality theorem for the family of Ls, χ) s as χ varies over the set of characters modulo q with q large. We denote by R, R +, Z and N the set of all real numbers, positive real numbers, integers and positive integers, respectively. For a discriminant d, let χ d denote the real Dirichlet character modulo d defined by the Kronecker 000 Mathematics Subject Classification: Primary M06, 4A30; Secondary R4. [43]

2 44 H. Mishou and H. Nagoshi symbol ) d. Letting γ stand for the plus sign or the minus sign, we define K { {d > 0 d is a square-free integer, d mod 8, d } if γ is +, D γ := {d < 0 d is a square-free integer, d mod 8} if γ is, and D γ X := {d Dγ d X} for X R +. The authors [MN] have recently obtained the following universality theorem, which is an analogue of Bagchi, Eminyan and Gonek s result above for the family {Ls, χ d ) d D γ } of L-functions associated with real characters χ d : Let Ω, hs) and K be as in Theorem. below. Then for any ε > 0 we have.) lim inf X #D γ X #{d D γ X max Ls, χ d) hs) < ε} > 0. In the present paper we investigate the universality theorem for Ls, λ p ) in the prime p-aspect, as mentioned above. Noting that Ls, λ p ) is equal to Ls, χ q ) with a certain integer q see 3.3)), we will deal with Ls, χ q ) instead of Ls, λ p ). Throughout let γ {+, } and let m and a = aγ) be any fixed positive integers such that gcd =, 8 m, a mod 4 if γ is + and a 3 mod 4 if γ is. We define { {p p is a prime, p a mod m} if γ is +, P γ := { p p is a prime, p a mod m} if γ is, and P γ X := {q Pγ q X} for X > 0. By the prime number theorem for arithmetic progressions,.) #P γ X ϕm) X log X as X, where ϕm) denotes the Euler totient function. Every integer q in P γ is a prime discriminant for its definition, see e.g. [Ay, p. 30], [Da, p. 4]). In the following, the letter p will stand for a prime number and q for a prime discriminant. Theorem.. Let γ {+, }. Let m, a N be as above. Let Ω be a simply connected region in D which is symmetric with respect to the real axis. Suppose that hs) is a holomorphic function on Ω which has no zeros on Ω and is R + -valued on the set Ω R. Let K be a compact set in Ω, and ε > 0. Then there exist infinitely many q P γ such that max Ls, χ q ) hs) < ε. More precisely, we have.3) lim inf X #P γ X #{q Pγ X max Ls, χ q) hs) < ε} > 0.

3 Universality of quadratic L-series 45 It should be noted that the results.) and.3) do not directly imply each other, because the density of the set P γ in D γ is 0 in the sense that #P γ X /#Dγ X 0 as X see [MN, Lemma 4.] and.)). In the same way as in the present paper, we can generalize.) to the result in which d varies over the fundamental discriminants in the arithmetic progression {km + a k Z}, where m, a N are as in Theorem.. Theorem. yields the following corollaries, for example. First we get a denseness result on values of Ls, χ q ) s for fixed s D and variable q P γ. This is analogous to Bohr Courant s result [BC] on values of the Riemann zeta-function ζs). Corollary.. ) Let any s 0 D with Im s 0 0 be fixed. Then the set {Ls 0, χ q ) q P γ } is dense in C. More precisely, for any z 0 C and ε > 0 we have.4) lim inf X #P γ X #{q Pγ X Ls 0, χ q ) z 0 < ε} > 0. ) Let / < σ 0 < be fixed. Then the set {Lσ 0, χ q ) q P γ } is dense in R +. More precisely, for any x 0 R + and ε > 0 we have lim inf X #P γ X #{q Pγ X Lσ 0, χ q ) x 0 < ε} > 0. Next we have a non-vanishing result for Ls, χ q ) s on D, and the following stronger result. Corollary.3. Let α, β be any positive real numbers with α < β. Let K be a compact set in D. Then lim inf X #P γ X #{q Pγ X α < Ls, χ q) < β uniformly for s K} > 0. Noting that Ls, χ q ) is R-valued on the real segment /, ), we can obtain a result concerning the horizontal distribution of zeros of the derivatives L r) s, χ q ) on /, ) in the q-aspect. Corollary.4. Let α, β R with / < α < β < and r, N N. Then there exist infinitely many q P γ such that for every integer r with r r the rth derivative L r) s, χ q ) has at least N zeros on the interval [α, β] R. More precisely, lim inf X #P γ X #{q Pγ X Lr) s, χ q ) has at least N zeros on [α, β] for every r =,..., r } > 0.

4 46 H. Mishou and H. Nagoshi We shall also study the denseness result on values of Ls, χ q ) for a fixed complex number s with Res = and variable q P γ. Theorem.5. Let t R {0} be fixed. Then the set {L+it, χ q ) q P γ } is dense in C. More precisely, for any z 0 C and ε > 0 we have lim inf X #P γ X #{q Pγ X L + it, χ q) z 0 < ε} > 0. In [MN] the authors showed an analogue of Theorem.5 for L, λ p ) and deduced from it a quantitative result for a problem of Ayoub Chowla Walum on certain character sums.. Denseness lemma. The purpose of this section is to show Proposition.3 below. For s C we write s = σ +it with σ, t R. The next lemma is proved in [MN, Proposition.4]. Lemma.. Let Ω be a simply connected region in D symmetric with respect to the real axis, as in Theorem.. Let U be a bounded, simply connected region in Ω which is symmetric with respect to the real axis and which satisfies U Ω, where U denotes the closure of U. Suppose that gs) is a holomorphic function on Ω which is R-valued on the interval Ω R. Let y > 0 be fixed. Then for any ε > 0 there exist ν R + and c p {, }, for each prime p with y p ν, such that gs) c p dσ dt < ε. U p s y p ν The next lemma is a generalization of [Ti, p. 303, Lemma] and was obtained in [MN, Lemma.5]. Lemma.. Let U be a bounded region in C. Let K be a compact subset of C such that K U. Let B > 0. Suppose that fs) is a holomorphic function on U satisfying Ì U fs) dσ dt B. Then max fs) bu, K)B /, where bu, K) is a certain positive constant depending only on U and K. Proposition.3. Let Ω be a simply connected region in D symmetric with respect to the real axis. Suppose that gs) is a holomorphic function on Ω which is R-valued on Ω R. Let K be a compact set in Ω and ν R + with ν > m+. Let a p {, } for each prime p with p m. Then for any ε > 0 there exist ν > ν and a p {, }, for each prime p with p ν and p m, such that max gs) log a ) p p s < ε, p ν

5 where log p ν Universality of quadratic L-series 47 a ) p p s = log a ) p p s p ν = p ν n= a n p np ns. Proof. Take a bounded, simply connected region U in Ω which is symmetric with respect to the real axis and which satisfies K U and U Ω. Set σ := min{re s s U} > /. Let ε > 0 be arbitrary. Fix a real number y satisfying y > ν and y σ /σ ) < ε. Then we have.) p y n= np nσ p y n= n y, n N p nσ = p y p σ p σ n σ y σ σ < ε. Set a p = for each prime p with p < y and p m. From Lemma. it follows that there exist ν y and c p {, }, for each prime p with y p ν, such that gs) a n ) p np ns c p p s dσ dt < ε. p<y n= y p ν U This and Lemma. yield.) max gs) a n p np ns p<y n= c p p s y p ν U,K ε. For each prime p with y p ν we set a p = c p. Then we obtain, by.) and.), max gs) log a ) p p s p ν = max gs) a n p np ns c p p s c n p np ns p<y n= y p ν y p ν n= max gs) a n p np ns c p p s + max c n p np ns p<y U,K ε + p y n= which completes the proof. n= np nσ ε, y p ν y p ν n=

6 48 H. Mishou and H. Nagoshi 3. Approximation by finite Euler products. As usual, let πx) denote the number of primes not exceeding X R +. For large X R +, let R X denote the set {s = σ + it C / + log log log X) / σ 5/4, t < X 3 σ ) } and put 3.) h X := log log X). The next lemma is obtained in [El, Lemma 8]. Lemma 3.. For all large X and uniformly for s R X we have Ls, λ r) λ ) rp) p s πx)h σ X log h X) 3 σ ) 4. 3 r X r : prime Recall that for an odd prime r and a positive integer n we have the relation see e.g. [Ay, p. 90, Lemma.]) ) r ) n if r mod 4, n K 3.) = ) r r if r 3 mod 4, n and hence { Ls, χr ) if r mod 4, 3.3) Ls, λ r ) = Ls, χ r ) if r 3 mod 4. K Proposition 3.. Let ε > 0 and K be a compact set in the region / < Re s < 5/4. Define A γ X = Aγ X m, a, ε, K) by A γ X {q := P γ X max Ls, χ q) χ ) qp) p s < ε}. Then if X is sufficiently large. #A γ X #P γ X > ε Proof. Take an open rectangle U of the form {s C σ < Re s < σ, Ims < A} satisfying / < σ < min{re s s K} max{re s s K} < σ < 5/4 and max{ Ims s K} < A. Then K U. For large X R + we define Ãγ X to be the set 3.4) {q P γx Ls, χ q) U χ ) qp) p s dσ dt< ε } bu, K),

7 Universality of quadratic L-series 49 where bu, K) is the constant in Lemma.. By Lemma., 3.5) Ã γ X Aγ X. From Lemma 3., 3.), 3.3), the prime number theorem, and.), we infer that for all large X, 3.6) Ls, χ q) χ ) qp) p s dσ dt q P γ X m,a) U 3 r X r : prime U Ls, λ r) U πx)h σ X log h X ) 3 σ ) 4 λ ) rp) p s dσ dt ϕm)#p γ X h σ X log h X ) 3 σ ) 4. Since h σ X log h X ) 3 0 as X, it follows from 3.6) that there exists a large number X 0 = X 0 ε, U, K, m) such that for all X > X 0, 3.7) Ls, χ q) χ ) qp) p s dσ dt q P γ X m,a) U < ε 3 bu, K) #Pγ X. Now assume that there exists a real number X > X 0 such that #P γ X Ãγ X ) ε#pγ X. For this X we have, by 3.4), Ls, χ q) χ ) qp) p s dσ dt q P γ X m,a) U q P γ X m,a) Ãγ X m,a) U Ls, χ q) ε#p γ X ε bu, K) = ε 3 bu, K) #Pγ X. χ ) qp) p s dσ dt However, this contradicts 3.7). Hence for any X > X 0 we have #P γ X Ãγ X ) < ε#pγ X, that is, #Ãγ X /#Pγ X > ε. This and 3.5) complete the proof. 4. Results on characters χ q for prime discriminants q. The aim of this section is to obtain Proposition 4.3. As before, the letter γ denotes the plus sign or the minus sign. For X R + we define I X to be the interval

8 50 H. Mishou and H. Nagoshi [0, X] if γ is +, and [ X, 0] if γ is. We define { if γ is +, 4.) δ = δγ) = if γ is. Lemma 4.. Fix a number ν R + such that πν) > πm). Let a p {, } for each prime p satisfying p ν and p m. Define P γ X,ν = P γ X,ν m, a, {a p}) to be the set {q P γ X χ qp) = a p for every prime p with p ν and p m}, and put C ν m) := p ν, p m. Then #P γ X,ν lim X #P γ X = C νm). Proof. In general, for n N and b Z, we denote by [b] n the set of all integers x such that x b mod n, that is, the residue class mod n which b belongs to. Let p be an odd prime. Let Q p be the set of all residue classes [b] p mod p such that b is a quadratic residue mod p, other than the residue class [0] p, and let Q p be the set of all residue classes [c] p mod p such that c is a quadratic non-residue mod p. It is well known that 4.) #Q p = #Q p = p. In view of the definitions of Kronecker s symbol and Legendre s symbol, a discriminant q satisfies χ q p) = a p if and only if q belongs to one of residue classes in Q p if a p = and in Q p if a p =. From this, 4.) and the Chinese remainder theorem, it follows, for an integer r such that δr is a prime number, that r satisfies r δa mod m i.e. r P γ ) and χ r p) = a p for every prime p with p ν and p m if and only if r belongs to one of exactly p ν, p m p )/ distinct residue classes mod Q, where Q = Qm, ν) := m p ν, p m and δ is as in 4.). Let R γ = R γ m, a, ν) denote the set of those residue classes mod Q, so that 4.3) #R γ = p ν, p m p. p

9 Thus Universality of quadratic L-series 5 4.4) P γ X,ν = {r I X δr is a prime, r δa mod m, We note that if [c] Q R γ then χ r p) = a p for every prime p with p ν and p m} = [c] Q R γ {r I X δr is a prime, r c mod Q}. 4.5) gcdc, Q) =, since [0] p / Q p and [0] p / Q p for all primes p with p ν and p m, and gcda, m) =. From 4.4) we have #P γ X,ν = [c] Q R γ p X p δc mod Q r I X, δr : prime r c mod Q = [c] Q R γ p X p δc mod Q By the prime number theorem for arithmetic progressions and 4.5), 4.6) X as X. ϕq) log X Note that the right-hand side of 4.6) is independent of [c] Q R γ. Therefore for fixed ν we have #P γ #Rγ X 4.7) X,ν ϕq) log X ) X = ϕm) log X = C νm) X ϕm) log X p ν, p m as X, using 4.3) and the fact 4.8) ϕq) = ϕm) ϕp) = ϕm) p ν, p m Thus 4.7) and.) give us #P γ X,ν #P = #P γ X γ X,ν C ν m) X ϕm) log X as X. This completes the proof. C ν m) X ϕm) log X X ϕm) log X p ν, p m X ϕm) log X p ).. #P γ X C νm) Lemma 4.. Fix ν R + such that πν) > πm). Let a p {, } for each prime p with p ν and p m. Let P γ X,ν and C νm) be as in Lemma 4., h X = log log X) be as in 3.), and σ > /. Then for all

10 5 H. Mishou and H. Nagoshi large X and uniformly for s C with Re s σ we have 4.9) χ q p) p s ν σ σ C νm) #P γ X. ν< q P γ X,ν m,a) Proof. Let Q and R γ be as in the proof of Lemma 4.. From 4.4) it follows that χ q p) p s = χ r p) 4.0) p s ν< ν< q P γ X,ν m,a) For [c] Q R γ we have 4.) χ δup) p s u X, u:prime ν< = u X, u :prime = ν< p s + p,p :primes, p p ν<p,p h X = S + S, say. [c] Q R γ r I X, δr :prime r c mod Q = [c] Q R γ ν< χ δup) p s + u X,u :prime p s ps χ δu p) u X, u:prime u X,u :prime p,p : primes, p p ν<p,p h X χ δu p )χ δu p ) χ δup) p s. ν< ) χ δu p )χ δu p ) p s ps Using the prime number theorem for arithmetic progressions, we deduce that for all s C with Res σ S 4.) p σ ν< u X, u:prime n σ n>ν, n N ν σ σ ϕq) ) ϕq) X log X. X log X

11 Universality of quadratic L-series 53 Next we shall consider the sum S. Fix two distinct primes p, p satisfying ν < p h X and ν < p h X. Then by the definition of the Kronecker symbol and the orthogonality relation for Dirichlet characters, we have ) ) δu δu 4.3) χ δu p )χ δu p ) = u X, u:prime = u X, u:prime = δ ϕq) p ) δu δu p p ) ) δ p λ mod Q u X,u :prime ) ϕq) λδc) λ mod Q p p λu)λδc) u X, u:prime ) u u p p ) λu), where λ mod Q means the sum over all the Dirichlet characters λ mod Q. Since p, p and Q are relatively prime in pairs, we find from the Chinese remainder theorem that for any character λ mod Q the product ) ) p p λ ) is a non-principal Dirichlet character mod p p Q. From this, the Siegel Walfisz theorem see [Da, p. 3, 3)]) and partial summation, it follows, for fixed ν, that for all large X and all pairs of distinct primes p, p ) satisfying ν < p h X and ν < p h X, we have ) ) u u λu) Xe b log 4.4) X, u X, u:prime p p where b is an absolute positive constant. From this and 4.3) we infer 4.5) S χ δu p )χ δu p ) = p,p : primes, p p ν<p,p h X p,p : primes, p p ν<p,p h X p σ X = o log X ). p σ pσ p σ pσ u X, u:prime OXe b log X ) ) OXe b log X ) h XXe b log X Consequently, for fixed ν we find, from 4.), 4.) and 4.5), that for all large X and uniformly for s C with Res σ, χ δup) 4.6) p s ν σ X σ ϕq) log X. ν< u X, u:prime

12 54 H. Mishou and H. Nagoshi Note that the right-hand side of 4.6) is independent of [c] Q R γ. Combining 4.6), 4.0), 4.3), 4.8) and.), we conclude that χ q p) p s #R γ ν σ X σ ϕq) log X ν< q P γ X,ν m,a) #R γ ν σ ϕm) σ ϕq) #Pγ ν σ X σ C νm)#p γ X. This completes the proof. To obtain 4.4) we have used the Siegel Walfisz theorem. We remark that actually, instead of the Siegel Walfisz theorem, a weaker result e.g. [Da, p. 3]) is sufficient since p p Q ν h X = log log X)4. Proposition 4.3. Let σ > / and K be a compact subset of C such that K {s C Re s > σ }. Let ε > 0. Then there exists a large real number ν 0 = ν 0 σ, K, ε, m) depending only on σ, K, ε and m, and satisfying πν 0 ) > πm) and the following. Fix any real number ν > ν 0. Let a p {, } for each prime p satisfying p ν and p m. Let P γ X,ν, C νm) and h X be as in Lemma 4. for large X. Define B γ X,ν = Bγ X,ν m, a, ε, σ, K, {a p }) by B γ X,ν {q := P γ X,ν max χ q p) p s }. < ε ν< Then for all sufficiently large X we have #B γ X,ν #P γ X > C νm). Proof. Set σ = + sup{re s s K} and A = + sup{ Ims s K}. Let U be the open rectangle {s C σ < Re s < σ, Ims < A} in C, and then U K. Take a large real number ν 0 = ν 0 σ, K, ε, m) satisfying πν 0 ) > πm) and 4.7) dσ dt U )c ν σ 0 σ < ε 4bU, K), where c is the absolute constant implied by the symbol in 4.9), and bu, K) is the constant in Lemma.. Note that ν 0 depends only on σ, K, ε and m. In the following we fix any ν > ν 0. For large X we define 4.8) Bγ X,ν := {q P γx,ν χ q p) p s ν< U dσ dt < ε } bu, K).

13 By Lemma., Universality of quadratic L-series ) Bγ X,ν Bγ X,ν. By Lemma 4. and 4.7), we have, for all large X, 4.0) χ q p) p s dσ dt q P γ X,ν m,a) U ν< dσ dt )c ν σ σ C νm) #P γ X < U ε 4bU, K) C νm) #P γ X. Now we assume that there exists a large number X such that #P γ X,ν B γ X,ν ) 4 C νm) #P γ X. Then for this X we have, using 4.8), χ q p) p s dσ dt q P γ X,ν m,a) U ν< χ q p) p s dσ dt q P γ X,ν m,a) B γ X,ν m,a) U ν< 4 C νm) #P γ X ε bu, K). However, this contradicts 4.0). Hence for all large X we have so 4.) #P γ X,ν B γ X,ν ) < 4 C νm) #P γ X, # B γ γ X,ν #PX,ν > #P γ X 4 C νm). #P γ X Further, Lemma 4. implies that 4.) #P γ X #P γ X,ν #P γ X > 3 4 C νm) if X is large enough. Combining 4.9), 4.) and 4.), we conclude that if X is large enough then #B γ γ X,ν # B X,ν #P γ X > 3 4 C νm) 4 C νm) = C νm).

14 56 H. Mishou and H. Nagoshi 5. Proofs of Theorem. and its corollaries Proof of Theorem.. Let ε > 0 be an arbitrary small number. Take a real number σ > / such that K {s C Re s > σ }. Fix a large positive number ν satisfying ν > ν 0 σ, K, ε, m) and ν σ /σ ) < ε, where ν 0 σ, K, ε, m) is the constant in Proposition 4.3. We set a to be if a or 7 mod 8, and if a 3 or 5 mod 8. Further, we set a p = ) δa p for each odd prime p with p m, where δ is as in 4.). As is shown in [MN], there exists a holomorphic function gs) on Ω such that gx) R for any x Ω R and 5.) hs) = e gs). Now Proposition.3 implies that there exist ν > ν and a p {, }, for each prime p with p ν and p m, such that max gs) log a ) p 5.) p s < ε. p ν For those a p s, where p ν and p m, we apply Proposition 4.3. Then for the above number ν and all large X, we have 5.3) #B γ X,ν #P γ X > C νm). Since 8 m, we have q δa mod 8 and q δa mod p for q P γ and a prime p with p m. This and the definition of Kronecker s symbol yield χ q ) = a and χ q p) = q p) = δa ) p = ap for q P γ and an odd prime p with p m. Hence, from the definition of B γ X,ν we find that for every q B γ X,ν and all large X, 5.4) max log a ) p p s log χ ) qp) p s p ν = max χ q p) p s + χ q p) n np ns ν< ν< n= max χ q p) p s + max χ q p) n np ns ν< n= since np ns ν< n= ε + Oε) ε, ν< p σ ν σ ν< σ < ν σ σ < ε.

15 Universality of quadratic L-series 57 From 5.4) and 5.) we deduce, for every q B γ X,ν, max gs) log χ ) qp) p s ε and therefore 5.5) max χ qp) p s ) hs) = max hs) χ q p)/p s ) ) hs) max hs) max elog p h χ q p)/p s ) gs) X K, hs) ε, using 5.) and the fact that e z z if z is small. Let ε be a small positive number such that { ε < min ε, C } νm). According to Proposition 3., if we put 5.6) A γ X { := q P γ X max Ls, χ q) then for all large X, 5.7) 5.8) #A γ X #P γ X > ε. χ ) } qp) p s < ε, By 5.6) and 5.5), every q A γ X Bγ X,ν satisfies max Ls, χ q) hs) K, hs) ε. Furthermore, from 5.3) and 5.7) it follows that for the above number ν and all large X, 5.9) #A γ X Bγ X,ν ) #A γ X + #Bγ X,ν #Pγ X Cν m) ε )#P γ X. Since C ν m)/ ε > 0, 5.8) and 5.9) yield.3). This completes the proof.

16 58 H. Mishou and H. Nagoshi From Theorem. we can prove Corollaries..4 by the same arguments as in the proofs of Corollaries..4 in [MN], respectively. 6. On the line Re s =. In this section we prove Theorem.5. The next lemma is proved in [MN]. Lemma 6.. Let t R + and y R + be fixed. Then for any z 0 C and ε > 0, there exist ν y and c p {, }, for each prime p with y p ν, such that z 0 c p < ε. p +it y p ν Proposition 6.. Let t R + be fixed. Let z C and ν R + with ν > m +. Let a p {, } for each prime p with p m. Then for any ε > 0 there exist ν > ν and a p {, }, for each prime p with p ν and p m, such that z log a ) p p +it < ε, p ν where log p ν a ) p p +it = log a ) p p +it p ν = a n p np n+it). p ν n= Proof. The proof is similar to that of Proposition.3. Let ε > 0 be arbitrary. Take a large number y > ν such that /y < ε. Then 6.) p y n= np n p y p y < ε. Set a p = for each prime p with p < y and p m. From Lemma 6. it follows that there exist ν y and c p {, }, for each prime p with y p ν, such that 6.) z p<y n= a n p np n+it) ) c p p +it y p ν < ε. For each prime p with y p ν we set a p = c p. Then we obtain, by 6.) and 6.),

17 z log p ν = z p<y n= z p<y a p p +it) < ε + p y n= n= which completes the proof. Universality of quadratic L-series 59 a n p np n+it) y p ν a n p np n+it) np n ε, c p p +it y p ν c p p +it c n p np n+it) y p ν n= + c n p np n+it) y p ν n= Proof of Theorem.5. The proof is similar to that of Theorem. in Section 5. Since L+it, χ q ) = L it, χ q ), it suffices to verify the assertion in the case t > 0. Moreover, it suffices to consider the case z 0 C {0}, since the set C {0} is dense in C. Fix z 0 C {0} and t > 0. Take a complex number z such that z 0 = e z. Let ε > 0 be an arbitrary small number. Take σ R with / < σ <, and set K = { + it}. Take ν R + so large that /ν < ε and ν > ν 0 σ, K, ε, m), where ν 0 σ, K, ε, m) is the constant in Proposition 4.3. We set a to be if a or 7 mod 8, and if a 3 or 5 mod 8. Further, we set a p = ) δa p for each odd prime p with p m. According to Proposition 6., there exist ν > ν and a p {, }, for each prime p with p ν and p m, such that 6.3) z log a ) p p +it < ε. p ν For those a p s, where p ν and p m, we apply Proposition 4.3. Then for the above number ν and all large X, we have 6.4) #B γ X,ν #P γ X > C νm). Noting χ q ) = a and χ q p) = a p for q P γ and an odd prime p with p m, we have, for every q B γ X,ν and all large X, 6.5) log a ) p p +it log χ ) qp) p +it p ν = χ q p) p +it + χ q p) n np n+it) ε + Oε) ε, ν< n= ν<

18 60 H. Mishou and H. Nagoshi since ν< n= np n ν< p ν < ν < ε. By 6.3) and 6.5), every q B γ X,ν satisfies z log χ ) qp) p +it ε and hence 6.6) χ qp) p +it ) z 0 = p h z X χ q p)/p +it ) 0 ) z 0 = z 0 e log χ q p)/p +it ) z z0 ε. Let ε be a small positive number such that ε < min{ε, C ν m)/}. Proposition 3. implies that if we put 6.7) A γ X then 6.8) := { q P γ X L + it, χ q ) #A γ X #P γ X > ε χ ) } qp) p +it < ε for all large X. Hence by 6.6) and 6.7) we conclude that every q A γ X Bγ X,ν satisfies 6.9) L + it, χ q ) z 0 z0 ε. Furthermore, from 6.4) and 6.8) we see that for the above number ν and all X sufficiently large, 6.0) #A γ X Bγ X,ν ) Cν m) ε )#P γ X. Since C ν m)/ ε > 0, 6.9) and 6.0) complete the proof. Acknowledgments. The authors would like to thank the referee for kind comments. References [Ay] R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 974.

19 Universality of quadratic L-series 6 [B] B. Bagchi, The statistical behaviour and universality properties of the Riemann zeta-function and other allied Dirichlet series, Ph.D. thesis, Indian Statistical Institute, Calcutta, 98. [B], A joint universality theorem for Dirichlet L-functions, Math. Z. 8 98), [BC] H. Bohr und R. Courant, Neue Anwendungen der Theorie der Diophantischen Approximationen auf die Riemannsche Zetafunktion, J. Reine Angew. Math ), [Da] H. Davenport, Multiplicative Number Theory, 3rd ed., Springer, 000. [El] P. D. T. A. Elliott, On the distribution of the values of quadratic L-series in the half-plane σ >, Invent. Math. 973), [Em] K. M. Eminyan, χ-universality of the Dirichlet L-function, Mat. Zametki ), 3 37 in Russian); English transl.: Math. Notes ), [Go] S. M. Gonek, Analytic properties of zeta and L-functions, Ph.D. thesis, University [KV] of Michigan, 979. A. A. Karatsuba and S. M. Voronin, The Riemann Zeta-Function, de Gruyter, 99. [MN] H. Mishou and H. Nagoshi, Functional distribution of Ls,χ d ) with real characters and denseness of quadratic class numbers, Trans. Amer. Math. Soc., to appear. [MN],, Character sums and class numbers of quadratic fields with prime discriminants, submitted. [Ti] [Vo] E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, nd ed., Oxford Univ. Press, 986. S. M. Voronin, Theorem on the universality of the Riemann zeta-function, Izv. Akad. Nauk SSSR Ser. Mat ), in Russian); English transl.: Math. USSR Izv ), Graduate School of Mathematics Nagoya University Chikusa-ku, Nagoya , Japan m9808a@math.nagoya-u.ac.jp Department of Mathematics Faculty of Science Niigata University Niigata 950-8, Japan nagoshih@ybb.ne.jp Received on and in revised form on )