Download REPRESENTATION OF EVEN NUMBERS VIA THE SUM OF TWO

Russian Mathematics (Iz. VUZ)
Vol. 44, No. 8, pp.1{13, 2000
Izvestiya VUZ. Matematika
UDC 511.216
REPRESENTATION OF EVEN NUMBERS VIA THE SUM OF TWO
ODD PRIME NUMBERS FROM ARITHMETIC PROGRESSION
I.A. Allakov
1. Introduction
p Let X and P be suciently large real numbers, while N a natural number with the condition
N < X N p, p1 , p2 prime numbers D = p , where p > 2, a positive integer MD (X ) the set
of even natural numbers n X , which (perhaps) cannot be represented in the form
n = p1 + p2 pi li (mod D) (li D) = 1 i = 1 2
(1.1)
ED (X ) = card MD (X ) R(n) the number of representations of n in the form (1.1) c, cj (j = 1 2 : : : )
some positive constants. Next, stands for Vinogradov's symbol, '(a) is the Euler function.
In 1] an asymptotic formula for R(n) was obtained, which is valid for all even n X , except
for at most ED (X ) X ln;A X (where A > 0 is an arbitrary constant) values of n. pLater in 2]
and 3], for ED (X ) in the case where D = 1, it was proved that E1 (X ) < X exp(;c ln X ) and
E1 (X ) < X 1; , respectively, 0 < < 1. In 4] the estimate from below for R(n) was found as
D = 1 for all n X except at most X 1; values of n.
In this article, by combining the technique suggested in 1], 2], we shall prove the following
Theorem 1. For D = p and D lnA X , the estimates are valid
p
ED (X ) X';1 (D) exp(;c1 ln X )
and, for n 2= MD (X ), n X ,
lnApn exp ; c2 pln n
n
R(n) 1
;
4
'(D) ln2 n
exp(c2 ln n)
where the constants c1 and c2 do not depend on A.
Let s be a complex variable, q (n) the Dirichlet character modulo q, and L(s q ) the Dirichlet
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L-function dened for Re s > 1 via the equality L(s q ) = P q (n)n;s . If a real (exclusive)
n=1
zero with the condition > 1 ; c3 ln;1 q of the Dirichlet L-function for all q P does not exist
(in this case we can set E = 0), then, by using the technique of this article, one can obtain the
asymptotic formula for R(n). If such an exclusive zero exists (we then put E = 1), we also arrive
at an asymptotic formula, but, in this formula, along with the usual principal term, another term
corresponding to the exclusive zero will take part (actually, its order will be the same as that of
the principal term). In other words, the following theorem is valid.
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