Abstract
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Reports of the Academy of Sciences of the USSR
- Volume 132, No. 6
MATHEMATICS
A. F. LAVRIK
ON THE DISTRIBUTION OF PRIME (k)-TWINS
(Presented by Academician I. M. Vinogradov on 25 II 1960)
§ 1. The laws of distribution of pairs (p,\ p+2k) of prime (k)-twins, for a given integer (k \ge 1), are very difficult and at the present time have been little studied. By the method of trigonometric sums of I. M. Vinogradov ((^1)), for the number of pairs of (k)-twins belonging to an interval of given length, it is possible to prove asymptotic relations for almost all integers (k \ge 1). In the present communication new theorems on (k)-twins are given, obtained by combining the aforementioned result of I. M. Vinogradov’s method with results of A. Selberg’s sieve method, namely, theorems concerning the difference between consecutive pairs of prime (k)-twins.
Theorem 1. The assertion that the inequality
[
\left|p_{ki}-p_{k(i-1)}\right|>
\frac{\ln^2 p_{ki}}{f(p_{ki})}
]
holds for almost all neighboring pairs ([p_{ki}; p_{ki}+2k]), ([p_{k(i-1)}; p_{k(i-1)}+2k]) of prime (k)-twins, where (f(t)\to\infty) is any function such that (\ln^2 t / f(t)) increases monotonically, is true for almost all integers (k \ge 1).
A somewhat more precise result, explicitly connected with the length of the interval, is supplied by Theorem 2.
Theorem 2. For almost all neighboring pairs ([p_{ki}; p_{ki}+2k]), ([p_{k(i-1)}; p_{k(i-1)}+2k]) of prime (k)-twins from the interval ((0,x)), for each (2 \le 2k \le x/\ln x), excluding no more than (cx/(\ln x)^M) of them, where (M>1) is arbitrary fixed and (c) depends only on (M), the inequality
[
\left|p_{ki}-p_{k(i-1)}\right|>
\frac{\ln^2 p_{ki}}{\alpha_k\,\omega(p_{ki})},\qquad
\alpha_k=\prod_{\substack{p\mid k\ p>5}}\frac{p-2}{p-4},
]
is valid, where (\omega(t)\to\infty) is any function such that (\ln^2 t/\omega(t)) increases monotonically, and (p) runs through the prime divisors of (k).
The following assertion, which is an analogue of a theorem of P. Erdős ((^2)), gives a characterization of irregularity in the distribution of pairs of prime numbers (p,\ p+2k).
Theorem 3. Let (x>x_0); (E(x)=\ln^2 x/\omega(x)); (0<D(x)\le E(x)), where (\omega(x)\to\infty) as (x\to\infty). Then for almost every integer (2 \le 2k \le x/\ln x) there exist
[
N=\left[\alpha_k\frac{\ln^2 x}{D(x)}\right],\qquad
\alpha_k=\beta\prod_{\substack{p\mid k\ p>3}}\frac{p-4}{p-2},
]
consecutive pairs
[
[p_{ki};p_{ki}+2k],\ldots,[p_{k(i+N)};p_{k(i+N)}+2k]
]
of prime (k)-twins, lying in the interval ((0,x)), such that
[
p_{k(i+s)}-p_{k(i+s-1)}>D(x)\qquad (s=1,2,\ldots,N),
]
where (\beta>0) is an absolute constant.
The obtained order of growth of the quantity (N) is apparently close to the final one.
§ 2. The derivation of the formulated propositions is based on the following theorem A, obtained by the author by the method of I. M. Vinogradov.
Theorem A. For every integer (2 \leq 2k \leq x/\ln x), with the exception of no more than (cx/(\ln x)^M) of them, where (M>1) is arbitrary fixed and (c) depends only on (M), the number (\pi_k(x)) of pairs of prime (k)-twins from the interval ((0,x)) is expressed in the form
[
\pi_k(x)=2\prod_{p>2}\frac{p(p-2)}{(p-1)^2}\,\frac{x}{\ln^2 x}\prod_{\substack{p\mid k\ p>2}}\frac{p-1}{p-2}
+O\left(\frac{x}{\ln^3 x}\right).
]
Let us indicate the main points of the proof of the theorems.
Let (x) be sufficiently large; (F(x)=\ln^2 x/\omega(x)). Choose an arbitrary number (2k) under the condition (F(x)<2k\leq x/\ln x) and not exceptional in the sense of theorem A.
The set (\mathfrak M) of all pairs
[
[p_{k1},\,p_{k1}+2k],\ldots,[p_{ks};\,p_{ks}+2k]
]
of prime (k)-twins lying on the segment ((0,x)), we split into two subsets (\mathfrak M_1) and (\mathfrak M_2):
[
[p_{ki};\,p_{ki}+2k]\in\mathfrak M_1,\quad
\text{if } p_{ki}-p_{k(i-1)}>F(x);
]
the complement of (\mathfrak M_1) in (\mathfrak M) we denote by (\mathfrak M_2). We estimate from above the number of numbers from (\mathfrak M_2).
We have
[
\sum_{\mathfrak M_2}1=
\sum_{0<2n\leq F(x)}
\sum_{\substack{2n=p_{ki}-p_{k(i-1)}\ 2\leq i\leq s}}1.
]
Next we note that the inner sum is not smaller than the number of solutions of the system of equations
[
p_2=p_1-2k;\qquad p_3=p_1-2n;\qquad p_4=p_1-2(n+k)
]
in primes (p_i) ((i=1,\ldots,4)), belonging to the interval ((0,x)), so that, applying the result of the sieve of A. Selberg ((^3)), we shall have
[
\sum_{\mathfrak M_2}1\ll
\frac{x}{\ln^4 x}
\sum_{2<2n\leq F(x)}
\prod_p
\frac{p-\gamma(p)}
{p\left(1-\frac1p\right)^4},
\tag{1}
]
where (\gamma(p)) is the number of residue classes to which the numbers (0,2k,2n,2(n+k)) belong modulo the prime modulus (p). Putting now (\Delta=kn(k^2-n^2)), we find that for (p\mid\Delta), (1\leq\gamma(p)\leq3), and for (p\nmid\Delta), (\gamma(p)=4). Therefore for the product from (1) we have
[
\prod_p\ll
\prod_{\substack{p\mid k\ p>2}}\frac{p-1}{p-2}
\prod_{\substack{p\mid k\ p>3}}\frac{p-2}{p-4}
\prod_{\substack{p\mid n(k^2-n^2)\ p\nmid k,\ p>3}}\frac{p-1}{p-4}.
\tag{2}
]
Introducing the notation:
[
\Pi_a=\prod_{p\mid a}\left(1+\frac{15}{p}\right)^4;\qquad
Q(x)=\sum_{2k<t\leq 2k+F(x)}\Pi_t;\qquad
V(x)=\sum_{2k-F(x)\leq t\leq 2k}\Pi_t
]
and (T_k) instead of the first two products in (2), we obtain for the sum on the right in relation (1) the estimate
[
\ll T_k\,(Q(x)V(x))^{1/4}
\left[
\sum_{2\leq n\leq F(x)}(\Pi_n)^{1/2}
\right]^{1/2}
\ll T_k\,(F(x)F(x))^{1/4}(F(x))^{1/2}.
\tag{3}
]
Combining the estimates (1)—(3), we find
[
\sum_{\mathfrak M_2} 1 \ll
\frac{x}{\ln^2 x \cdot \omega(x)}
\prod_{\substack{p/k\ p>3}}\frac{p-1}{p-4}.
]
Taking now into account the relation
[
\sum_{\mathfrak M_1} 1+\sum_{\mathfrak M_2} 1=\pi_k(x),
]
by virtue of Theorem A we obtain Theorem 2, and from this latter, owing to the fact that (\prod \frac{p-2}{p-4}) is on average a constant, Theorem 1 follows.
§ 3. For the derivation of Theorem 3 the same means suffice. The idea of the derivation is as follows. Instead of the function (F(x)), introduced in § 2, we take the function (D(x))
[
0<D(x)\ll \frac{\ln^2 x}{\omega(x)}.
]
In this case it turns out that the set (\mathfrak M_1) is nonempty, while for the number (T(x;D(x))) of numbers of the set (\mathfrak M_2) the estimate
[
T(x;D(x))\ll
\frac{xD(x)}{\ln^4 x}
\prod_{\substack{p/k\ p>3}}\frac{p-1}{p-4}
]
is valid.
Further, to each sequence of prime numbers (p_{k(s+1)}, \ldots, p_{k(s+r)}) belonging to (\mathfrak M_1) there corresponds a number (p_{ks}\in \mathfrak M_2). Denoting by (N) the maximal length of such a sequence from (\mathfrak M_1), we shall have
[
\pi_k(x)<2NT(x;D(x)).
]
Hence, applying Theorem A, we obtain
[
N>\beta\,\frac{\ln^2 x}{\alpha_kD(x)},\qquad
\alpha_k=\prod_{\substack{p/k\ p>3}}\frac{p-2}{p-4},
]
where (\beta>0) is an absolute constant, which proves the theorem.
Received
24 II 1960
CITED LITERATURE
¹ I. M. Vinogradov, Selected Works, Publishing House of the Academy of Sciences of the USSR, 1952. ² P. Erdős, Acta math. Szeged, 13, 57 (1949). ³ N. I. Klimov, UMN, 13, no. 3 (81), 146 (1958).