Proceedings of the Steklov Institute of Mathematics, Vol. 218, 1997, pp. 129–145. Translated from Trudy Matematicheskogo Instituta imeni V. A. Steklova, Vol. 218, 1997, pp. 134–150. Original Russian Text Copyright \begin{matrix}c \\ ◯ \end{matrix} 1997 by Gritsenko. English Translation Copyright \begin{matrix}c \\ ◯ \end{matrix} by ���� �����/Interperiodica Publishing (Russia).

On the Zeros of Linear Combinations of the Analogs of Riemann’s Function

S. A. Gritsenko

Received February, 1997

Selberg defined in [1] the class S of Dirichlet series

F (s) = ∑_{n=1}^{∞} a(n)n^{−s}, Re s > 1,

satisfying the following conditions:

(1) there exists a nonnegative integer m such that the function (s − 1)^{m}F (s) is entire,

(2) the Dirichlet coefficients a(n) satisfy the inequalities a(n) ≪_{ε} n^{ε} for any positive ε, and a(1) = 1,

(3) there exists a function γ_{F} (s) of the form

γ_{F} (s) = ε_{1}Q^{s} ∏_{ν=1}^{k} Γ(λ_{ν}s + μ_{ν}),

where |ε_{1}| = 1, k = k_{F} is a positive integer, Q > 0, λ_{ν} > 0, Re μ_{ν} ≥ 0, the function being such that the identity Φ(s) = \overline{Φ}(1 − \overline{s}) is valid for the function Φ(s) = γ_{F} (s)F (s),

(4) for Re s > 1 the function F (s) is decomposed in the Euler product

F (s) = ∏_{p}(1 + a(p)p^{−s} + a(p^{2})p^{−2s} + ⋯),

and p runs through the prime numbers

log F (s) = ∑_{n=1}^{∞} b(n)n^{−s},

where b(n) = 0 if n is not equal to the positive power of the prime number, with b(n) ≪ n^{θ} for a certain θ ≤ 1/2.

The class S includes the Dirichlet series which are defined by a functional equation of the Riemann type and a decomposition in the Euler product, in particular, in Riemann’s zeta function, in Dirichlet’s L-functions etc.

In the article [2], for any function F (s) from S the degree of F (s) is defined as

d_{F} = 2 ∑_{ν=1}^{k} λ_{ν}.

For primitive functions from this class, i.e., functions which are not representable as

F_{1}(s)F_{2}(s), F_{1}(s), F_{2}(s) ∈ S,

Selberg stated in [1] a number of conjectures, in particular, the following conjectures. 1. As x → ∞, the asymptotic formula

∑_{p≤x} |a(p)|^{2}p^{−1} = log log x + O(1)

holds true.

2. As x → ∞, the estimate

∑_{p≤x} a(p)\overline{a}′(p)p^{−1} = O(1),

where p runs through the prime numbers and a(n) and a′(n), which are Dirichlet coefficients of the primitive functions F (s) and F′(s), F (s) and F′(s), are distinct, holds true.

Let F (s) be a function from the class S. We say that the zeros of F (s) which coincide with the poles of the function γ_{F} (s) are trivial and the other zeros are nontrivial.

It is known (see [1, p. 368]) that the nontrivial zeros of F (s) lie in the strip 1 − A ≤ Re s ≤ A, A = A_{F} > 0.

In addition, (see [1, p. 368]), since F (s) satisfies a functional equation of the Riemann type, it follows that

N(T) ∼ cT log T, c = c_{F} > 0,

where N(T) is the number of nontrivial zeros of F (s) in the rectangle 1 − A ≤ Re s ≤ A, 0 < Im s ≤ T.

We define the following three functions:

ρ(s) = \overline{γ}_{F} (1 − \overline{s})γ_{F}^{−1}(s) = \overline{ε}_{1}^{2}Q^{1−2s} ∏_{ν=1}^{k} Γ(λ_{ν}(1 − s) + \overline{μ}_{ν})Γ^{−1}(λ_{ν}s + μ_{ν}),

exp(iθ(t)) = (ρ(\frac{1}{2} + it))^{−1/2}, Z(t) = exp(iθ(t))F (\frac{1}{2} + it).

It follows from the functional equation that Z(t) assumes real values for real t. The function Z(t) is an analog of Riemann’s function

Ξ(t) = −\frac{1}{2}(T^{2} + 1)π^{−1/4−it/2}Γ(\frac{1}{4} + \frac{1}{2}it)ζ(\frac{1}{2} + it)

(see [3, pp. 249–250, 253] and [4, pp. 218–219]).

Let us define f(t), which is a linear combination of the analogs of Riemann’s function:

f(t) = Φ_{1}(\frac{1}{2} + it) = ∑_{j=1}^{N} a_{j}(ρ_{j}(\frac{1}{2} + it))^{−1/2}F_{j}(\frac{1}{2} + it) = ∑_{j=1}^{N} a_{j}Z_{j}(t).  (1)

In what follows we assume that T ≥ T_{0} > 0, where T_{0} is a sufficiently large constant which depends only on F_{1}(s), …, F_{N}(s).

ON THE ZEROS OF LINEAR COMBINATIONS OF THE ANALOGS OF RIEMANN’S FUNCTION 131

Let N_{0}(T) be the number of zeros of the function Φ_{1}(s) on the interval (1/2, 1/2 + iT] or, what is the same, the number of zeros of the function f(t) on the interval (0, T].

The following inequality is “expected” for the quantity N_{0}(T): N_{0}(T) ≥ c_{4}T log T, where c_{4} > 0 is a constant.

Note that if F_{1}(s), …, F_{N}(s) are functions of degree 1 from the class S, then the application of the Hardy–Littlewood–Landau method to the function f(t) yields the inequality N_{0}(T) ≥ c_{5}T (c_{5} is an absolute positive constant).

For the first time a principally more exact estimate of the quantity N_{0}(T), which corresponds to the Davenport–Heilbronn function, was obtained by Voronin, who proved in 1980, in the article [5], that

N_{0}(T) ≥ c_{6}T exp(\frac{1}{20}\sqrt{log log log log T} ),

(c_{6} is an absolute positive constant). The last inequality means that the straight line Re s = 1/2 is a “special set” for the zeros of the Davenport–Heilbronn function. The Davenport–Heilbronn function is defined in the half-plane Re s > 0 by the Dirichlet series

∑_{n=1}^{∞} r(n)n^{−s},  where r(n) = \{ \begin{matrix}1 & if n ≡ 1 (mod 5) \\ κ & if n ≡ 2 (mod 5) \\ −κ & if n ≡ 3 (mod 5) \\ −1 & if n ≡ 4 (mod 5), \end{matrix}  κ = \frac{\sqrt{10 − 2\sqrt{5}} − 2}{\sqrt{5} − 1}.

It is intensively studied as the simplest Dirichlet series for which the analog of Riemann’s conjecture is not satisfied.

In 1989 Karatsuba worked out a new method for estimating from below the number of zeros of certain Dirichlet series on segments of a critical line and used this method to prove the following statement for the Davenport–Heilbronn functions [6]:

N_{0}(T) ≥ T (log T)^{0.5}(log T)^{−ε},  (2)

where ε > 0 is an arbitrarily small fixed number, T ≥ T_{0}(ε) > 0.

Then, in his article [7], Karatsuba used his method to estimate the number of zeros on the critical line of the function Φ_{1}(s) for the case where F_{1}(s), …, F_{N}(s) are Dirichlet’s L-functions which correspond to the irreducible characters (in our terminology they refer to functions from the class S of degree 1) and got the following estimates for the quantity N_{0}(T):

N_{0}(T) ≥ T (log T)^{2β}(log T)^{−ε},  (3)

if Dirichlet characters χ_{1}, …, χ_{N} are of the same parity,

N_{0}(T) ≥ T (log T)^{β} (log T)^{−ε}  (4)

otherwise, here β = (φ(K))^{−1}, K is the least common multiple of the positive integers k_{1}, …, k_{N} which are moduli of the characters χ_{1}, …, χ_{N}, φ(K) is Euler’s function, ε is an arbitrarily small fixed number, T ≥ T_{0}(ε) > 0.

Later, in the work [8] the factor (log T)^{−ε} on the right-hand sides of inequalities (2)–(4) was replaced by the slower decreasing factor exp(−c_{4}\sqrt{log log T} ), where c_{4} is an absolute positive constant.

At present, estimates (2)–(4) are the strongest among the known estimates.

Suppose now that the functions F_{1}(s), …, F_{N}(s) in relation (1) belong to the Selberg class and are of degree 2. The problem of estimating the quantity N_{0}(T) from below was considered in the works [9–11], where the following theorems were proved.

Theorem 1. Let N be an arbitrary positive integer and a_{1}, …, a_{N} be arbitrary real numbers, T ≥ T_{0} > 0. Suppose that f(t) is the function defined by relation (1) and F_{1}(s), …, F_{N}(s) are functions of degree 2 from S which satisfy the following conditions.

1. As x → ∞, the asymptotic formulas

∑_{p≤x} |a_{j}(p)|^{2} log p = x + O(x(log x)^{−1}),

hold true, where a_{j}(n) are Dirichlet coefficients of the functions F_{j}(s) and p runs through the prime numbers.

2. As x → ∞, the estimates

∑_{p≤x} a_{j}(p)\overline{a}_{i}(p)p^{−1} = O(1)

are valid, where a_{j}(n) and a_{i}(n) are Dirichlet coefficients of any functions F_{j}(s), F_{i}(s), 1 ≤ j ≠ i ≤ N.

3. For any j, 1 ≤ j ≤ N, any prime numbers p, and any real t there exist positive constants c_{F_{j}} such that

|1 + a_{j}(p)p^{−1/2−it} + a_{j}(p^{2})p^{−1−2it} + ⋯ | > c_{F_{j}}.

4. Let F (s) be any one of the functions F_{1}(s), …, F_{N}(s). Then, as x → ∞, the asymptotic formula

∑_{n≤x} |a(n)|^{2}n^{−1} = A_{F} log x + B_{F} + O((log x)^{−4})

holds true, where a(n) are Dirichlet coefficients of the function F (s), A_{F} > 0, B_{F} are constants dependent only on F (s).

Then the inequality

N_{0}(T) > T exp(\sqrt{log log log T} )

holds true.

Theorem 2. Suppose that F_{1}(s), …, F_{N}(s) are Hecke’s L-functions corresponding to Hecke’s complex characters and f(t) is the function defined by relation (1). Then the inequality

N_{0}(T) > T exp(\sqrt{log log log T} )

holds true.

In this article we continue to study the distributions of zeros of the function f(t). We formulate our main results.

Theorem 3. Suppose that N is an arbitrary positive integer, a_{1}, …, a_{N} are arbitrary real numbers, T ≥ T_{0} > 0. Let f(t) be the function defined by relation (1) and F_{1}(s), …, F_{N}(s) be functions of degree 2 from S which satisfy the following conditions.

1. As x → ∞, the asymptotic formulas

∑_{p≤x} |a_{j}(p)|^{2} log p = x + O(x(log x)^{−1})

hold true, where a_{j}(n) are Dirichlet coefficients of the functions F_{j}(s) and p runs through prime numbers.

2. As x → ∞, the estimates

∑_{p≤x} a_{j}(p)\overline{a}_{i}(p)p^{−1} = O(1)

hold true, where a_{j}(n) and a_{i}(n) are Dirichlet coefficients of any functions F_{j}(s), F_{i}(s), 1 ≤ j ≠ i ≤ N.

3. For any j, 1 ≤ j ≤ N, any prime numbers p, and any real t there exist positive constants c_{F_{j}} such that

|1 + a_{j}(p)p^{−1/2−it} + a_{j}(p^{2})p^{−1−2it} + ⋯ | > c_{F_{j}}.

4. Let F (s) be any one of the functions F_{1}(s), …, F_{N}(s). Then, as x → ∞, the asymptotic formula

∑_{n≤x} |a(n)|^{2}n^{−1} = A_{F} log x + B_{F} + O((log x)^{−4})

holds true, where a(n) are Dirichlet coefficients of the function F (s), A_{F} > 0, B_{F} are constant dependent only on F (s).

Then there exists a positive constant η, dependent only on F_{1}(s), …, F_{N}(s), such that the inequality

N_{0}(T) > T (log log T)^{η}

holds true.

Theorem 4. Let F_{1}(s), …, F_{N}(s) be Hecke’s L-functions corresponding to Hecke’s complex characters and let f(t) be the function defined by relation (1). Then the inequality

N_{0}(T) > T (log log T)^{η}

holds true, where η is the same constant as in Theorem 3.

It follows from our estimates that the straight line Re s = 1/2 is a special set for the zeros of the function f(t). Obviously, in addition, they are less exact than Karatsuba’s estimates (2)–(4).

Theorem 3 can be proved by Voronin’s method (see [5]) which he worked out in 1980 in order to estimate the number of zeros of the Davenport–Heilbronn function which lie on a segment of the critical line.

We begin with a brief description of the proof of Theorem 3. It is based on the idea, which belongs to Hardy, Littlewood, and Landau, of comparing integrals

I(t) = |∫_{0}^{h} ∫_{0}^{h} f(t + u_{1} + u_{2}) du_{1} du_{2}|  and  J(t) = ∫_{0}^{h} ∫_{0}^{h} |f(t + u_{1} + u_{2})| du_{1} du_{2},

where h is a small parameter whose exact value will be chosen later and f(t + u_{1} + u_{2}) is a linear combination of the analogs of Riemann’s function defined by (1).

First the interval [T, 2T] is divided into two subsets

S = {t ∈ [T, 2T] | I(t) = J(t)}, E = [t, 2T] \ S.

The integral

J = ∫_{S} J(t) dt

is estimated from above and from below. By virtue of the definition of the set S, we have

J = ∫_{S} J(t) dt ≤ ∫_{T}^{2T}I(t) dt = I

and the integral I is estimated, with the standard use of the Hardy–Littlewood–Landau method, as O(T h^{3/2}).

Let us estimate the integral J from below. With the use of elementary arguments, we derive the inequality

J ≥ h^{2}J_{1} − ∑_{j=1}^{N} ∑_{k=1}^{N} |a_{j}a_{k}|I_{jk},

where

J_{1} = min_{0≤u≤2h} ∫_{S} \frac{|∑_{j=1}^{N} a_{j} exp(iθ_{j}(t + u))P_{j}(1/2 + it + iu)|^{2}}{1 + ∑_{j=1}^{N} |a_{j}| × |P_{j}(1/2 + it + iu)|} dt,

\begin{aligned}I_{jk} & = ∫_{T}^{2T}|∫_{0}^{h} ∫_{0}^{h} \frac{P_{j}(1/2 + it + iu)\overline{P}_{k}(1/2 + it + iu)}{1 + ∑_{j=1}^{N} |a_{j}| × |P_{j}(1/2 + it + iu)|} \\ & × exp(i(θ_{j}(t + u_{1} + u_{2}) − θ_{k}(t + u_{1} + u_{2})))(Q_{j}(\frac{1}{2} + it + iu) − 1) du_{1} du_{2}| dt, \end{aligned}

P_{j}(s) = ∏_{p≤y}(1 + a_{j}(p)p^{−s} + a_{j}(p^{2})p^{−2s} + ⋯),  Q_{j}(s) = F_{j}(s)P_{j}^{−1}(s),  y = (log T)^{ε}

(ε > 0 is an arbitrarily small number which is smaller than 0.001).

The interals I_{jk} are estimated from above. The more difficult part of the solution of this problem is the estimation of the sum V of the form

V = ∑_{\begin{matrix}n≤U \\ (n,P)=1 \end{matrix}} |a_{j}(n)|^{2}, where P = ∏_{p≤y} p.

For any U, y < U, we must prove that V = O(U(log y)^{−1}). To this end, we must use the small sieve method since for “insufficiently large” values of U, as compared to y, the method of generating functions does not work.

For the sum ∑_{j=1}^{N} ∑_{k=1}^{N} |a_{j}a_{k}|I_{jk} we get the estimate

∑_{j=1}^{N} ∑_{k=1}^{N} |a_{j}a_{k}|I_{jk} = O(T h^{3/2}),

from which, by virtue of J = O(T h^{3/2}) and

J ≥ h^{2}J_{1} − ∑_{j=1}^{N} ∑_{k=1}^{N} |a_{j}a_{k}|I_{jk},

it follows that h^{2}J_{1} = O(T h^{3/2}).

In order to estimate J_{1} from below, we show that there exists a set E_{1} of a large measure mes(E_{1}) > c_{0}T (log log T)^{−2η} such that for any t from E_{1} the value of the function

\frac{|∑_{j=1}^{N} a_{j} exp(iθ_{j}(t))P_{j}(1/2 + it)|^{2}}{1 + ∑_{j=1}^{N} |a_{j}| × |P_{j}(1/2 + it)|}

is “large”, to be more precise,

\frac{|∑_{j=1}^{N} a_{j} exp(iθ_{j}(t))P_{j}(1/2 + it)|^{2}}{1 + ∑_{j=1}^{N} |a_{j}| × |P_{j}(1/2 + it)|} > (log log T)^{4η},

whence it follows that

J_{1} ≫ (log log T)^{4η} mes(S ∪ (E_{1} − u)).

If we had the inequality

mes(S ∪ (E_{1} − u)) > \frac{1}{2} mes(E_{1}) > T (log log T)^{−2η},

then it would follow that

J_{1} > \frac{1}{2}T (log log T)^{2η},

and this contradicts the estimate h^{2}J_{1} = O(T h^{3/2}) for h = (log log t)^{−3.5η}. Hence

mes(S ∪ (E_{1} − u)) ≤ \frac{1}{2} mes(E_{1}),

and, consequently, the inequality

mes(E) > \frac{1}{4}T (log log t)^{−2η}

holds true, from which it is easy to obtain the statement of the theorem.

Note that in problems on the estimation of the number of zeros of Dirichlet series on a critical line one usually tries to get the inequality

mes(E) > c_{0}T,  where c_{0} > 0 is a constant.  (5)

With the use of our approach, we can prove the validity of estimate (5) only for

h ≥ exp(−c_{1}\sqrt{log log y}),

where c_{1} > 0 is a constant.

It turns out that the result is somewhat better if we replace estimate (5) by an estimate, which is less exact but which acts for smaller values of the parameter h.

The proof of Theorem 1 is given in the article [9], and in order to prove Theorem 3, it suffices to use the same scheme only with the application of the following lemmas at appropriate places.

Lemma 1. Suppose that N is an arbitrary positive integer, F_{1}(s), …, F_{N}(s) are pairwise distinct primitive functions of degree 2 from Selberg’s class S which satisfy the conditions of Theorem 1 and a_{j}(n) are their Dirichlets coefficients, j = 1, …, N.

Let a and c be nonnegative real numbers with a satisfying the inequalities 0 < a ≤ ξ\sqrt{log log y}, where ξ = min(0.01, ξ_{1}) and ξ_{1} is such a small positive number that

ξ_{1} ∑_{p≤y} ∑_{α=3}^{∞} |a_{j}(p^{α})p^{−0.5α} ≤ 0.01

for any 1 ≤ j ≤ N. Let b assume one of the three values, 0, a, 2a.

Let

g_{1}(t) = (log log y)^{−1/2} Re(∑_{p≤y} a_{1}(p)p^{−1/2−it}) − c,

g_{j}(t) = −(log log y)^{−1/2} Re(∑_{p≤y} a_{j}(p)p^{−1/2−it}) − c, j = 2, …, N,

y = (log T)^{ε} (ε is an arbitrarily small number, smaller than 0.001).

Let n be any positive integer such that 1 ≤ n ≤ N.

Then the inequalities

0.9T exp(\frac{a^{2}}{4} (N − 1) + \frac{b^{2}}{4} − ca(N − 1) − 2b) ≤ I ≤ 1.1T exp(\frac{a^{2}}{4} (N − 1) + \frac{b^{2}}{4} − ca(N − 1) − 2b)

are valid for I,

I = ∫_{T}^{2T}exp(a ∑_{\begin{matrix}j=1 \\ j≠n \end{matrix}}^{N} g_{j}(t) + bg_{n}(t)) dt.

Proof. We introduce the notation

β_{j}(p) = \frac{δ(j)}{\sqrt{log log y}} Re(\frac{a_{j}(p)}{\sqrt{p}}p^{−it}), where δ(j) = \{ \begin{matrix}1 & if j = 1 \\ −1 & if j ≠ 1. \end{matrix}

Then

exp(a ∑_{\begin{matrix}j=1 \\ j≠n \end{matrix}}^{N} g_{j}(t) + bg_{n}(t)) = exp(−2a(N − 1) − 2b) ∏_{p≤y} ∏_{\begin{matrix}j=1 \\ j≠n \end{matrix}}^{N} exp(aβ_{j}(p)) exp(bβ_{n}(p)).

ON THE ZEROS OF LINEAR COMBINATIONS OF THE ANALOGS OF RIEMANN’S FUNCTION 137

According to the hypothesis,

|aβ_{j}(p)| ≤ 0.5, |bβ_{n}(p)| ≤ 0.5.

For any real α, |α| ≤ 0.5, the inequality

(1 + α + \frac{α^{2}}{2} )(1 − |α|^{3}) ≤ exp(α) ≤ (1 + α + \frac{α^{2}}{2} )(1 + |α|^{3})

holds true, and therefore

Pr_{1} × ∏_{p≤y} ∏_{\begin{matrix}j=1 \\ j≠n \end{matrix}}^{N}(1 + aβ_{j}(p) + \frac{a^{2}}{2} β_{j}^{2}(p))(1 + bβ_{n}(p) + \frac{b^{2}}{2} β_{n}^{2}(p))

≤ ∏_{p≤y} ∏_{\begin{matrix}j=1 \\ j≠n \end{matrix}}^{N} exp(aβ_{j}(p)) exp(bβ_{n}(p))

≤ Pr_{2} × ∏_{p≤y} ∏_{\begin{matrix}j=1 \\ j≠n \end{matrix}}^{N}(1 + aβ_{j}(p) + \frac{a^{2}}{2} β_{j}^{2}(p))(1 + bβ_{n}(p) + \frac{b^{2}}{2} β_{n}^{2}(p)),

where

Pr_{1} = ∏_{p≤y} ∏_{\begin{matrix}j=1 \\ j≠n \end{matrix}}^{N}(1 − \frac{a^{3}|a_{j}(p)|^{3}}{(p log log y)^{3/2}} )(1 − \frac{b^{3}|a_{n}(p)|^{3}}{(p log log y)^{3/2}} ),
Pr_{2} = ∏_{p≤y} ∏_{\begin{matrix}j=1 \\ j≠n \end{matrix}}^{N}(1 + \frac{a^{3}|a_{j}(p)|^{3}}{(p log log y)^{3/2}} )(1 + \frac{b^{3}|a_{n}(p)|^{3}}{(p log log y)^{3/2}} ).

Since Pr_{1} ≥ 0.99, Pr_{2} ≤ 1.01 by the hypothesis of Lemma 1, it follows that in order to prove Lemma 1, it suffices to derive for

I′ = ∫_{T}^{2T} ∏_{p≤y} ∏_{\begin{matrix}j=1 \\ j≠n \end{matrix}}^{N}(1 + aβ_{j}(p) + \frac{a^{2}}{2} β_{j}^{2}(p))(1 + bβ_{n}(p) + \frac{b^{2}}{2} β_{n}^{2}(p)) dt

an asymptotic formula of the form

I′ = T exp(\frac{a^{2}}{4} (N − 1) + \frac{b^{2}}{4} )(1 + O((log log y)^{−1})).

PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS  Vol. 218  1997

138 GRITSENKO

By definition we have

1 + aβ_{j}(p) + \frac{a^{2}}{2} β_{j}^{2}(p) = 1 + \frac{a}{2}δ(j)(log log y)^{−1/2} \frac{a_{j}(p)}{\sqrt{p}}p^{−it} + \frac{a}{2}δ(j)(log log y)^{−1/2} \frac{\overline{a}_{j}(p)}{\sqrt{p}}p^{it}

+ \frac{a^{2}}{8} (log log y)^{−1} \frac{a_{j}^{2}(p)}{p}p^{−2it} + \frac{a^{2}}{8} (log log y)^{−1} \frac{\overline{a}_{j}^{2}(p)}{p}p^{2it}

+ \frac{a^{2}}{4} (log log y)^{−1} \frac{|a_{j}(p)|^{2}}{p} , j = 1, …, N, j ≠ n,

1 + bβ_{n}(p) + \frac{b^{2}}{2} β_{n}^{2}(p) = 1 + \frac{b}{2}δ(n)(log log y)^{−1/2} \frac{a_{n}(p)}{\sqrt{p}}p^{−it} + \frac{b}{2}δ(n)(log log y)^{−1/2} \frac{\overline{a}_{n}(p)}{\sqrt{p}}p^{it}

+ \frac{b^{2}}{8} (log log y)^{−1} \frac{a_{n}^{2}(p)}{p}p^{−2it} + \frac{b^{2}}{8} (log log y)^{−1} \frac{\overline{a}_{n}^{2}(p)}{p}p^{2it}

+ \frac{b^{2}}{4} (log log y)^{−1} \frac{|a_{n}(p)|^{2}}{p} .

(6)

We take the product Pr_{3},

Pr_{3} = ∏_{p≤y} ∏_{\begin{matrix}j=1 \\ j≠n \end{matrix}}^{N}(1 + aβ_{j}(p) + \frac{a^{2}}{2} β_{j}^{2}(p))(1 + bβ_{n}(p) + \frac{b^{2}}{2} β_{n}^{2}(p)) dt.

By virtue of relations (6), it can be represented as the sum of 6^{Nπ(y)} terms. We combine the terms independent of t (nonoscillating); their sum is

Pr_{4} = ∏_{p≤y}(1 + \frac{a^{2}}{4} (log log y)^{−1} ∑_{\begin{matrix}j=1 \\ j≠n \end{matrix}}^{N} \frac{|a_{j}(p)|^{2}}{p} + \frac{b^{2}}{4} (log log y)^{−1} \frac{|a_{n}(p)|^{2}}{p}

+ \frac{a^{2}}{4} (log log y)^{−1} ∑_{\begin{matrix}j=1 \\ j≠n \end{matrix}}^{N} ∑_{\begin{matrix}m=1 \\ m≠n \end{matrix}}^{N} δ(j)δ(m)p^{−1}(a_{j}(p)\overline{a}_{m}(p) + \overline{a}_{j}(p)a_{m}(p))

+ \frac{ab}{4}(log log y)^{−1} ∑_{\begin{matrix}j=1 \\ j≠n \end{matrix}}^{N} δ(j)δ(n)p^{−1}(a_{j}(p)\overline{a}_{n}(p) + \overline{a}_{j}(p)a_{n}(p)) + O((log log y)^{−3/2}p^{−4/3})).

The terms dependent on t (oscillating) (they are less in number than 6^{Nπ(y)}) have the form

d(\frac{l}{g} )^{±it},  (7)

where |d| ≪ 1, l and g are positive integers not exceeding

∏_{p≤y} p^{2} < exp(3y),

PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS  Vol. 218  1997

and therefore the integral over t of the terms of (7) can be estimated as O(exp(3y)), and the integral of the sum of the oscillating terms can be estimated as

O(6^{Nπ(y)} exp(3y)) = O(T^{ε})

for any positive ε. Consequently, I′ = Pr_{4} T + O(T^{ε}).

Finally, we get for Pr_{4} the asymptotic formula

Pr_{4} = exp(\frac{a^{2}}{4} (N − 1) + \frac{b^{2}}{4} ) × (1 + O((log log y)^{−1})).

We have proved Lemma 1.

Corollary 1. Suppose that the conditions of Lemma 1 are satisfied and, in addition,

c = \frac{ξ}{16N} \sqrt{log log y},  a = 4c(1 + 0.01ξ),  D = \frac{3}{4}a^{2} − ca + 0.01ξa.

Then the inequality

mes(E_{2}) > \frac{1}{2}T exp(\frac{a^{2}}{4} N − DN − caN)

holds true.

The proof follows immediately from Lemma 1, as well as from Lemma 7 and the corollary of Lemma 9 of the work [9] (it should only be pointed out that the function H_{1}(x) in Lemma 7 [9] must be defined as H_{1}(x) = exp(ax − D)).

Setting now η = 0.99ξ/(64N), we have mes(E_{1}) > T (log log T)^{−2η}, where

E_{1} = \{t ∈ [T, 2T] | \frac{|∑_{j=1}^{N} a_{j} exp(iθ_{j}(t + u))P_{j}(1/2 + it + iu)|^{2}}{1 + ∑_{j=1}^{N} |a_{j}| × |P_{j}(1/2 + it + iu)|} > (log log T)^{4η}\}.

We have shown how the integral J can be estimated from below. In order to prove Theorem 3, it is necessary to estimate the integrals I_{jk} from above. The estimation is carried out just as in the article [9]. However, since the definitions that we gave here do not completely coincide with those from [9], we must additionally use the following lemma.

Lemma 2. Suppose that N is an arbitrary positive integer, F_{1}(s), …, F_{N}(s) are Dirichlet series from the Selberg class, and a_{j}(n) are their Dirichlet coefficients, j = 1, …, N.

Suppose that there exist positive constants c_{F_{1}}, c_{F_{2}}, …, c_{F_{N}} such that for any real t and any prime number p the inequalities

|1 + ∑_{k=1}^{∞} \frac{a_{j}(p^{k})}{p^{k/2}}p^{−ikt}| > c_{F_{j}}, j = 1, 2, …, N,

hold true.

Let

P_{j}(\frac{1}{2} + it) = ∏_{p≤y}(1 + ∑_{k=1}^{∞} \frac{a_{j}(p^{k})}{p^{k/2}}p^{−ikt}) j = 1, 2, …, N,

y = (log T)^{ε}, ε is an arbitrary positive number smaller than 0.001.

Let

g_{1}(\frac{1}{2} + it) = (1 + ∑_{j=1}^{N} |a_{j}| × |P_{j}(\frac{1}{2} + it)|)^{−1},

where a_{1}, …, a_{N} are arbitrary constants.

Then there exists a number sequence x_{m,n} such that for any A > 0 the relation

g_{1}(\frac{1}{2} + it) = ∑_{\begin{matrix}m,n∈M \\ (m,n)=1 \\ m,n≤T^{1/200} \end{matrix}} x_{m,n}(\frac{m}{n})^{it} + O((log T)^{−A})

is valid, where M is a set consisting of 1 and those positive integers all of whose prime divisors do not exceed y.

Proof. 1. Let 2, 3, …, p, …, q be all prime numbers not exceeding y and let θ_{2}, θ_{3}, …, θ_{p}, …, θ_{q} be real variables indexed by prime numbers.

Let us consider the function Ψ(θ_{2}, θ_{3}, …, θ_{p}, …, θ_{q}) of the form

Ψ(θ_{2}, θ_{3}, …, θ_{p}, …, θ_{q}) = (1 + Σ(θ_{2}, θ_{3}, …, θ_{p}, …, θ_{q}))^{−1},

where

Σ(θ_{2}, θ_{3}, …, θ_{p}, …, θ_{q}) = ∑_{j=1}^{N} |a_{j}| ∏_{p≤y}|1 + ∑_{k=1}^{∞} \frac{a_{j}(p^{k})}{p^{k/2}} exp(2πikθ_{p})|

= ∑_{j=1}^{N} |a_{j}| × | ∑_{k_{2}=0}^{∞} ∑_{k_{3}=0}^{∞} ⋯ ∑_{k_{p}=0}^{∞} ⋯ ∑_{k_{q}=0}^{∞} \frac{a_{j}(2^{k_{2}}3^{k_{3}} ⋯ p^{k_{p}} ⋯ q^{k_{q}})}{\sqrt{2^{k_{2}}3^{k_{3}} ⋯ p^{k_{p}} ⋯ q^{k_{q}}}}

× exp(2πi(k_{2}θ_{2} + k_{3}θ_{3} + ⋯ + k_{p}θ_{p} + ⋯ + k_{q}θ_{q}))|.

It follows from the conditions of the lemma that

Ψ(θ_{2}, θ_{3}, …, θ_{p}, …, θ_{q}) = g_{1}(\frac{1}{2} + it)

for θ_{p} = −(t log p)/(2π), 2 ≤ p ≤ y. Moreover, by the hypothesis we have

|1 + ∑_{k=1}^{∞} \frac{a_{j}(p^{k})}{p^{k/2}} exp(2πikθ_{p})| > c_{F_{j}},

and therefore the function

|1 + ∑_{k=1}^{∞} \frac{a_{j}(p^{k})}{p^{k/2}} exp(2πikθ_{p})|

is infinitely differentiable with respect to θ_{p}.

Consequently, the function Ψ(θ_{2}, θ_{3}, …, θ_{p}, …, θ_{q}) is infinitely differentiable with respect to each variable and periodic with respect to each variable with period 1. By virtue of the well-known theorem from analysis, it can be expanded in the Fourier series which can be termwise differentiated with respect to each variable any number of times.

The sum

∑_{\begin{matrix}m,n∈M \\ (m,n)=1 \\ m,n≤T^{1/200} \end{matrix}} x_{m,n}(\frac{m}{n})^{it}

that appears in the formulation of the lemma is a partial sum of this Fourier series at the point

(−\frac{t log 2}{2π}, −\frac{t log 3}{2π}, …, −\frac{t log p}{2π}, …, −\frac{t log q}{2π} ).

2. Since F_{1}(s), …, F_{N}(s) are functions from the Selberg class, there exists a constant d > 1 such that the inequality |a_{j}(n)| ≤ dn^{0.01} holds true for any j, 1 ≤ j ≤ N, and any n; if, in addition, d^{100} < n, then |a_{j}(n)| ≤ n^{0.02}.

Let ξ = max(d^{100}, e^{25π}), and 2, 3, …, p_{1} be prime numbers not exceeding ξ.

Using the boundedness of all partial derivatives of the function Ψ(θ_{2}, θ_{3}, …, θ_{p}, …, θ_{q}), whose orders are bounded, we obtain the relation

Ψ(θ_{2}, θ_{3}, …, θ_{p}, …, θ_{q}) = ∑_{|k_{2}|≤(log T)^{0.01}} ⋯ ∑_{|k_{p_{1}}|≤(log T)^{0.01}} exp(2πi(k_{2}θ_{2} + ⋯ + k_{p_{1}}θ_{p_{1}}))

× ∫_{0}^{1} ⋯ ∫_{0}^{1} Ψ(θ_{2}, θ_{3}, …, θ_{p}, …, θ_{q}) exp(−2πi(k_{2}θ_{2} + ⋯ + k_{p_{1}}θ_{p_{1}})) dθ_{2} ⋯ dθ_{p_{1}} + O((log T)^{−A}).

3. Let p_{2}, …, p, …, q be prime numbers lying between ξ and y. Let

Φ(θ_{p_{2}}, …, θ_{p}, …, θ_{q}) = ∫_{0}^{1} ⋯ ∫_{0}^{1} Ψ(θ_{2}, θ_{3}, …, θ_{p}, …, θ_{q}) exp(−2πi(k_{2}θ_{2} + ⋯ + k_{p_{1}}θ_{p_{1}})) dθ_{2} ⋯ dθ_{p_{1}}.

We expand Φ(θ_{p_{2}}, …, θ_{p}, …, θ_{q}) in the Fourier series

Φ(θ_{p_{2}}, …, θ_{p}, …, θ_{q}) = ∑_{\overline{k}} b\overline{k} exp((\overline{k}, \overline{θ})),

where

\overline{k} = (k_{p_{2}}, …, k_{p}, …, k_{q}), \overline{θ} = (θ_{k_{p_{2}}} , …, θ_{k_{p}}, …, θ_{k_{q}}),

b\overline{k} = ∫_{0}^{1} ⋯ ∫_{0}^{1} Φ(θ_{p_{2}}, …, θ_{p}, …, θ_{q}) exp(−2πi(\overline{k}, \overline{θ})) dθ_{p_{2}} ⋯ dθ_{p} ⋯ dθ_{q}.

Then, by virtue of the relation obtained at the end of the preceding section, we have

Ψ(θ_{2}, θ_{3}, …, θ_{p}, …, θ_{q}) = ∑_{|k_{2}|≤(log T)^{0.01}} ⋯ ∑_{|k_{p_{1}}|≤(log T)^{0.01}} exp(2πi(k_{2}θ_{2} + ⋯ + k_{p_{1}}θ_{p_{1}}))

× ∑_{k_{p_{2}}=−∞}^{∞} ⋯ ∑_{k_{p}=−∞}^{∞} ⋯ ∑_{k_{q}=−∞}^{∞} b\overline{k} exp(k_{p_{2}}θ_{k_{p_{2}}} + ⋯ + k_{p}θ_{k_{p}} + ⋯ + k_{q}θ_{k_{q}}) + O((log T)^{−A})).

142 GRITSENKO

Setting θ_{p} = −(t log p)/(2π), 2 ≤ p ≤ y, and introducing the notation

m = ∏_{\begin{matrix}p≤ξ \\ 0≤k_{p}≤(log T)^{0.01} \end{matrix}} p^{k_{p}} ∏_{\begin{matrix}ξ<p≤y \\ 0≤k_{p} \end{matrix}} p^{k_{p}},  n = ∏_{\begin{matrix}p≤ξ \\ 0<−k_{p}≤(log T)^{0.01} \end{matrix}} p^{k_{p}} ∏_{\begin{matrix}ξ<p≤y \\ 0<−k_{p} \end{matrix}} p^{k_{p}},

and x_{m,n}, which is the corresponding Fourier coefficient, we obtain

g_{1}(\frac{1}{2} + it) = Ψ(θ_{2}, θ_{3}, …, θ_{p}, …, θ_{q}) = ∑_{\begin{matrix}m,n∈M \\ (m,n)=1 \end{matrix}} x_{m,n}(\frac{m}{n})^{−it} + O((log T)^{−A}).

Let us estimate the sum

∑_{\begin{matrix}m,n∈M \\ (m,n)=1 \\ max(m,n)>T^{1/200} \end{matrix}} x_{m,n}(\frac{m}{n})^{−it}

trivially:

∑_{\begin{matrix}m,n∈M \\ (m,n)=1 \\ max(m,n)>T^{1/200} \end{matrix}} x_{m,n}(\frac{m}{n})^{−it} ≪ (log T)^{0.01ξ} ∑_{\overline{k} ; |k_{p_{2}}|+···+|k_{q}|>(1/400) log T (log y)^{−1}} |b\overline{k}|.

Indeed, let, for instance, m > T^{1/200}. Then the inequality

|k_{p_{2}}| + ⋯ + |k_{q}| ≤ \frac{1}{400} log T (log y)^{−1}

is not satisfied since otherwise it would yield

log m ≤ ξ(log T)^{0.01} + \frac{1}{400} log T < \frac{1}{200} log T.

4. Let us estimate the sum W,

W = ∑_{\overline{k} ; |k_{p_{2}}|+···+|k_{q}|>(1/400) log T (log y)^{−1}} |b\overline{k}|.

Let l be a positive integer whose value will be chosen later. Integrating by parts, we arrive at the inequality

|b\overline{k}| ≤ \frac{1}{(2π)^{2l}} ∫_{0}^{1}⋯∫_{0}^{1}| ∑_{ξ<p≤y} \frac{∂^{2l}}{∂θ_{p}^{2l}} Φ(θ_{p_{2}}, …, θ_{p}, …, θ_{q})| dθ_{p_{2}} ⋯ dθ_{p} ⋯ dθ_{q}×(k_{p_{2}}^{2l}+⋯+k_{p}^{2l}+⋯+k_{q}^{2l})^{−1},

from which it follows that

W ≤ \frac{1}{(2π)^{2l}} ∫_{0}^{1} ⋯ ∫_{0}^{1} ∑_{ξ<p≤y}|\frac{∂^{2l}}{∂θ_{p}^{2l}} Φ(θ_{p_{2}}, …, θ_{p}, …, θ_{q})| dθ_{p_{2}} ⋯ dθ_{p} ⋯ dθ_{q} × U,

where

U = ∑_{|k_{p_{2}}|+···+|k_{q}|>(1/400) log T (log y)^{−1}}(k_{p_{2}}^{2l} + ⋯ + k_{p}^{2l} + ⋯ + k_{q}^{2l})^{−1}.

In order to estimate W, it suffices now to estimate the quantities

∑_{ξ<p≤y}|\frac{∂^{2l}}{∂θ_{p}^{2l}} Φ(θ_{p_{2}}, …, θ_{p}, …, θ_{q})|

and U, the statement of the lemma will follow immediately from the inequality W ≤ (log T)^{−2A}, where A is an arbitrary positive number.

5. We begin with estimating

|\frac{∂^{2l}}{∂θ_{p}^{2l}} Φ(θ_{p_{2}}, …, θ_{p}, …, θ_{q})|.

We use the fact that by virtue of the choice of ξ, for any p > ξ and any j, 1 ≤ j ≤ N, the inequalities

|∑_{k=1}^{∞} \frac{a_{j}(p^{k})}{p^{k/2}} exp(2πikθ_{p})| ≤ ∑_{k=1}^{∞} p^{−0.48k} < \frac{1}{2}

hold true, from which it follows that

(1 + ∑_{k=1}^{∞} \frac{a_{j}(p^{k})}{p^{k/2}} exp(2πikθ_{p}))^{1/2} = 1 + ∑_{n=1}^{∞} \frac{α_{j}(p^{n})}{p^{n/2}} exp(2πinθ_{p}),

where

α_{j}(p^{n}) = ∑_{l=1}^{n} \binom{1/2}{l} ∑_{k_{1}=1}^{n} ⋯ ∑_{k_{l}=1}^{n}_{k_{1}+···+k_{l}=n} a_{j}(p^{k_{1}}) ⋯ a_{j}(p^{k_{l}})

with

|α_{j}(p^{n})| ≤ ∑_{l=1}^{n} \binom{n − 1}{l − 1}p^{0.02n} = (2p^{0.02})^{n}.

From the relations obtained we have

Ψ(θ_{2}, θ_{3}, …, θ_{p}, …, θ_{q}) = (1 + ∑_{j=1}^{N} |a_{j}| ∏_{\begin{matrix}p_{3}≤y \\ p_{3}≠p \end{matrix}}|1 + ∑_{k=1}^{∞} \frac{a_{j}(p_{3}^{k})}{p_{3}^{k/2}} exp(2πikθ_{p_{3}})|G_{1j}(θ_{p})G_{2j}(θ_{p}))^{−1},

where

G_{1j}(θ_{p}) = 1 + ∑_{k=1}^{∞} \frac{α_{j}(p^{k})}{p^{k/2}} exp(2πikθ_{p}), G_{2j}(θ_{p}) = 1 + ∑_{k=1}^{∞} \frac{\overline{α}_{j}(p^{k})}{p^{k/2}} exp(−2πikθ_{p}).

Let z_{p} be a complex number such that |z_{p}| ≤ 1 and θ_{p} be an arbitrary real number.

Using the inequalities ξ < p and |α_{j}(p^{n})| ≤ (2p^{0.02})^{n}, we get one more inequality:

|∑_{k=1}^{∞} \frac{α_{j}(p^{k})}{p^{k/2}} exp(2πik(θ_{p} + z_{p}))| < ∑_{k=1}^{∞} \frac{|α_{j}(p^{k})|}{p^{k/2}} exp(2πk) < ∑_{k=1}^{∞}(\frac{2p^{0.02}}{\sqrt{p}} )^{k} < ∑_{k=1}^{∞} p^{−0.3k} < \frac{1}{4}.

It follows from the last inequality that

Re(G_{1p}(θ_{p} + z_{p})G_{2p}(θ_{p} + z_{p}))

> 1 − |∑_{k=1}^{∞} \frac{α_{j}(p^{k})}{p^{k/2}} exp(2πik(θ_{p} + z_{p}))| − |∑_{k=1}^{∞} \frac{\overline{α}_{j}(p^{k})}{p^{k/2}} exp(−2πik(θ_{p} + z_{p}))|

− |∑_{k=1}^{∞} \frac{α_{j}(p^{k})}{p^{k/2}} exp(2πik(θ_{p} + z_{p}))| × |∑_{k=1}^{∞} \frac{\overline{α}_{j}(p^{k})}{p^{k/2}} exp(−2πik(θ_{p} + z_{p}))|

> 1 − \frac{1}{2} − \frac{1}{16} = \frac{7}{16} > 0.

We have proved that in the circle |z_{p}| ≤ 1 the function

Ψ(θ_{2}, θ_{3}, …, θ_{p} + z_{p}, …, θ_{q})

is analytic, with

|Ψ(θ_{2}, θ_{3}, …, θ_{p} + z_{p}, …, θ_{q})| ≤ 1.

(the last inequality follows immediately from

Re(1 + ∑_{j=1}^{N} |a_{j}| ∏_{\begin{matrix}p_{3}≤y \\ p_{3}≠p \end{matrix}}|1 + ∑_{k=1}^{∞} \frac{a_{j}(p_{3}^{k})}{p_{3}^{k/2}} exp(2πikθ_{p_{3}})|G_{1j}(θ_{p})G_{2j}(θ_{p})) ≥ 1.)

In order to estimate

|\frac{∂^{2l}}{∂θ_{p}^{2l}} Φ(θ_{2}, …, θ_{p}, …, θ_{q})|,

we apply the integral Cauchy formula. We have

\frac{∂^{2l}}{∂θ_{p}^{2l}} Φ(θ_{2}, …, θ_{p}, …, θ_{q}) = \frac{(2l)!}{2πi} ∫_{γ} \frac{Φ(θ_{2}, …, ζ_{p}, …, θ_{q})}{(ζ_{p} − θ_{p})^{2l+1}} dζ_{p},

where γ is a circle with center at the point θ_{p} of radius 1 which is traversed counterclockwise. From this we find that

|\frac{∂^{2l}}{∂θ_{p}^{2l}} Φ(θ_{2}, …, θ_{p}, …, θ_{q})| ≤ (2l)! .

6. We have thus obtained the estimate

W ≤ \frac{(2l)!}{(2π)^{2l}} π(y)U,

PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS  Vol. 218  1997

where

U = ∑_{|k_{p_{2}}|+···+|k_{q}|>(1/400) log T (log y)^{−1}}(k_{p_{2}}^{2l} + ⋯ + k_{p}^{2l} + ⋯ + k_{q}^{2l})^{−1}.

For the given natural N, under the condition max(|k_{p_{2}}|, …, |k_{q}|) = N, the number of inte-

ger points (k_{p_{2}}, …, k_{q}) ∈ ℝ^{π(y)−π(ξ)} does not exceed 2π(y)(2N + 1)^{π(y)−1}, and therefore, setting l = π(y), we obtain

U ≤ 2π(y)3^{π(y)} ∑_{N>(1/400) log T (log y)^{−1}} N^{−1−π(y)} ≤ 4π(y)(\frac{log T}{1200 log y} )^{−π(y)},

and, consequently,

W ≤ 4π(y)(\frac{log T}{4800π^{2}(y) log y} )^{−π(y)}.

Since y ≤ (log T)^{0.001} by the hypothesis, the inequalities

4800π^{2}(y) log y < \sqrt{log T} , W ≤ (log T)^{−2A}

are now valid for any A > 0. We have proved the lemma.

ACKNOWLEDGMENTS

The work was supported by the Russian Foundation for Fundamental Research (project no. 97-01-00721).

REFERENCES

1. Selberg, A., Old and New Conjectures and Results about a Class of Dirichlet Series, in: Proc. of the Amalfi Conf. on Analytic Number Theory, Univ. di Salerno, 1992, pp. 365–387.

2. Conrey, J. B. and Ghosh, A., On the Selberg Class of Dirichlet Series: Small Degrees, Duke Math. J., 1993, vol. 72, no. 3, pp. 673–695.

3. Titchmarsh, E., The Zeta-Function of Riemann, Oxford, 1953.

4. Riemann, B., Collected Works [Russ. transl.], Moscow, Leningrad, 1948.

5. Voronin, S. M., On the Zeros of Some Dirichlet Series Lying on a Critical Line, Izv. Akad. Nauk SSSR, Ser. Mat., 1980, vol. 44, no. 1, pp. 63–91.

6. Karatsuba, A. A., On the Zeros of the Davenport–Heilbronn Function Lying on a Critical Line, Izv. Akad. Nauk SSSR, Ser. Mat., 1990, vol. 54, no. 2, pp. 303–315.

7. Karatsuba, A. A., On the Zeros of a Special Kind of Functions Related to Dirichlet Series, Izv. Akad. Nauk SSSR, Ser. Mat., 1991, vol. 55, no. 3, pp. 483–514.

8. Karatsuba, A. A., A New Approach to the Problem of Zeros of Some Dirichlet Series, Tr. Mat. Inst. Akad. Nauk, 1994, vol. 207, pp. 180–196.

9. Gritsenko, S. A., On the Zeros of Linear Combinations of a Special Kind of Functions Related to Dirichlet Series from Selberg’s Class, Izv. Ross. Akad. Nauk, Ser. Mat., 1996, vol. 60, no. 4, pp. 3–42.

10. Gritsenko, S. A., On the Zeros of a Special Kind of Functions Related to Dirichlet Series from Selberg’s Class, Mat. Zametki, 1996, vol. 60, no. 4, pp. 601–605.

11. Gritsenko, S.A., On the Zeros of Linear Combinations of a Special Kind of Functions Related to Hecke L-Functions of Imaginary Quadratic Fields, Izv. Ross. Akad. Nauk, Ser. Mat., 1997, vol. 61, no. 1, pp. 45–68.

Translated by I. Aleksandrova