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Zerofree regions for Dirichlet Lfunctions and the least prime in an arithmetic progression
 Proc. Lond. Math. Soc
, 1992
"... The classical theorem of Dirichlet states that any arithmetic progression a(mod q) in which a and q are relatively prime contains infinitely many prime numbers. A natural question to ask is then, how big is the first such prime, P (a, q) say? In one direction we have trivially ..."
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The classical theorem of Dirichlet states that any arithmetic progression a(mod q) in which a and q are relatively prime contains infinitely many prime numbers. A natural question to ask is then, how big is the first such prime, P (a, q) say? In one direction we have trivially
Integers represented as a sum of primes and powers of two
 Asian J. Math
"... It was shown by Linnik [10] that there is an absolute constant K such that every sufficiently large even integer can be written as a sum of two primes and at most ..."
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It was shown by Linnik [10] that there is an absolute constant K such that every sufficiently large even integer can be written as a sum of two primes and at most
The quadratic WaringGoldbach problem
 J. Number Theory
"... It is conjectured that Lagrange’s theorem of four squares is true for prime variables, i.e. all positive integers n with n 4 ðmod 24Þ are the sum of four squares of primes. In this paper, the size for the exceptional set in the above conjecture is reduced to OðN38þeÞ: ..."
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It is conjectured that Lagrange’s theorem of four squares is true for prime variables, i.e. all positive integers n with n 4 ðmod 24Þ are the sum of four squares of primes. In this paper, the size for the exceptional set in the above conjecture is reduced to OðN38þeÞ:
On Goldbach’s conjecture in arithmetic progressions
 Stud. Sci. Math. Hungar
"... ABSTRACT. It is proved that for a given integer N and for all but (log N)B prime numbers k ≤ N5/48−ε the following is true: For any positive integers bi, i ∈ {1, 2, 3}, (bi, k) = 1 that satisfy N ≡ b1 + b2 + b3 (mod k), N can be written as N = p1+p2+p3, where the pi, i ∈ {1, 2, 3} are prime number ..."
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ABSTRACT. It is proved that for a given integer N and for all but (log N)B prime numbers k ≤ N5/48−ε the following is true: For any positive integers bi, i ∈ {1, 2, 3}, (bi, k) = 1 that satisfy N ≡ b1 + b2 + b3 (mod k), N can be written as N = p1+p2+p3, where the pi, i ∈ {1, 2, 3} are prime numbers that satisfy pi ≡ bi (mod k). 1. Introduction. Vinogradov [17] has proved that every sufficiently large odd positive integer can be written as the sum of three primes. This theorem has been generalized in many ways. In 1953, Ayoub [1] proved the following result: If k is a fixed positive integer, bi, i = 1, 2, 3, are integers with (bi, k) = 1 and J(N; k, b1, b2, b3) is the number of
A Numerical Bound For Baker's Constant  Some Explicit Estimates For Small Prime Solutions Of Linear Equations
"... INTRODUCTION Let a 1 ; a 2 ; a 3 be any nonzero integers such that gcd(a 1 ; a 2 ; a 3 ) = 1: (1.1) Let b be any integer satisfying b j a 1 + a 2 + a 3 (mod2) and gcd(b; a i ; a j ) = 1 for 1 i ! j 3: (1.2) Write A := maxf3; ja 1 j; ja 2 j; ja 3 jg. In [11], M.C. Liu and K. M. Tsang studied a pr ..."
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INTRODUCTION Let a 1 ; a 2 ; a 3 be any nonzero integers such that gcd(a 1 ; a 2 ; a 3 ) = 1: (1.1) Let b be any integer satisfying b j a 1 + a 2 + a 3 (mod2) and gcd(b; a i ; a j ) = 1 for 1 i ! j 3: (1.2) Write A := maxf3; ja 1 j; ja 2 j; ja 3 jg. In [11], M.C. Liu and K. M. Tsang studied a problem of A. Baker [1], namely, the solubility and the size of small prime solutions p 1 ; p 2 ; p 3 of the linear equation a 1 p 1<
A note on the least totient of a residue class
"... Let q be a large prime number, a be any integer, ε be a fixed small positive quantity. Friedlander and Shparlinksi [4] have shown that there exists a positive integer n ≪ q 5/2+ε such that φ(n) falls into the residue class a (mod q). Here, φ(n) denotes Euler’s function. In the present paper we impro ..."
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Let q be a large prime number, a be any integer, ε be a fixed small positive quantity. Friedlander and Shparlinksi [4] have shown that there exists a positive integer n ≪ q 5/2+ε such that φ(n) falls into the residue class a (mod q). Here, φ(n) denotes Euler’s function. In the present paper we improve this bound to n ≪ q 2+ε. 2000 Mathematics Subject Classification: 11L40 1
THE PARITY PROBLEM FOR REDUCIBLE CUBIC FORMS
, 2005
"... Let f ∈ Z[x, y] be a reducible homogeneous polynomial of degree 3. We show that f(x, y) has an even number of prime factors as often as an odd number of prime factors. ..."
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Let f ∈ Z[x, y] be a reducible homogeneous polynomial of degree 3. We show that f(x, y) has an even number of prime factors as often as an odd number of prime factors.
On Waring’s problem: some consequences of Golubeva’s method, submitted
 School of Mathematics, University of Bristol, University Walk
"... Abstract. We investigate sums of mixed powers involving two squares, two cubes, and various higher powers, concentrating on situations inaccessible to the HardyLittlewood method. 1. ..."
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Abstract. We investigate sums of mixed powers involving two squares, two cubes, and various higher powers, concentrating on situations inaccessible to the HardyLittlewood method. 1.
On a WaringGoldbach type problem for fourth powers
"... In this paper, we prove that every sufficiently large positive integer satisfying some necessary congruence conditions can be represented by the sum of a fourth power of integer and twelve fourth powers of prime numbers. ..."
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In this paper, we prove that every sufficiently large positive integer satisfying some necessary congruence conditions can be represented by the sum of a fourth power of integer and twelve fourth powers of prime numbers.