MEAN REPRESENTATION NUMBER OF INTEGERS AS THE SUM OF PRIMES

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MEAN REPRESENTATION NUMBER OF INTEGERS AS THE SUM OF PRIMES GAUTAMI BHOWMIK AND JAN-CHRISTOPH SCHLAGE-PUCHTA Abstract. Assuming the Riemann Hypothesis we obtain asymptotic esti- mates for the mean value of the number of representations of an integer as a sum of two primes. By proving a corresponding ?-term, we show that our result is essentially the best possible. 1. Introduction and Results When studying the Goldbach conjecture that every even integer larger than 2 is the sum of two primes it is natural to consider the corresponding problem for the von Mangoldt function ?. Instead of showing that an even integer n is the sum of two primes, one aims at showing that G(n) =∑k1+k2=n ?(k1)?(k2) is sufficiently large, more precisely, G(n) > C√n implies the Goldbach conjecture. It is known since long that this result is true for almost all n. It is easy to see that if f is an increasing function such that the Tchebychev function ?(x) = x +O(f(x)), then the mean value of G(n) satisfies the relation ∑ n≤x G(n) = x2/2 +O(xf(x)). If we consider the contribution of only one zero of the Riemann zeta function ?, an error term of size O(f(x)2) appears, which, under the current knowledge on zero free regions of ?, would not be significantly better than O(xf

• summation over primitive


• over


• th china-japan


• characters ? modulo


• japan acad


• error estimate


• prime


• riemann ?-function


• riemann hypothesis

Sujets

• Riemann zeta function
• Riemann hypothesis

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 28
• Langue: English

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MEAN REPRESENTATION NUMBER OF INTEGERS AS THE SUM OF PRIMES

GAUTAMI BHOWMIK AND JANCHRISTOPH SCHLAGEPUCHTA

Abstract.Assuming the Riemann Hypothesis we obtain asymptotic esti mates for the mean value of the number of representations of an integer as a sum of two primes.By proving a corresponding Ωterm, we show that our result is essentially the best possible.

1.Introduction and Results When studying the Goldbach conjecture that every even integer larger than 2 is the sum of two primes it is natural to consider the corresponding problem for the von Mangoldt function Λ.Instead of showing that an even integernis the sum of P two primes, one aims at showing thatG(nΛ() =k 1 )Λ(k 2 ) is sufficiently k 1 +k 2 =n large, more precisely,G(n)n> CIt is knownimplies the Goldbach conjecture. since long that this result is true for almost allnis easy to see that if. Itfis an increasing function such that the Tchebychev function Ψ(x) =x+O(f(x)), then the mean value ofG(n) satisfies the relation X 2 G(n) =x /2 +O(xf(x)). n≤x If we consider the contribution of only one zero of the Riemann zeta functionζ, 2 an error term of sizeO(f(x) ) appears, which, under the current knowledge on zero free regions ofζ, would not be significantly better thanO(xf(x)). Fujii[3] studied the error term of this mean value under the Riemann Hypothesis (RH) and obtained X 2 3/2 G(n) =x /2 +O(x) n≤x which he later improved [4] to X 2 4/3 (1)G(n) =x /2 +H(x) + (O(xlogx) ) n≤x P 1+ρ x withH(x) =−where the summation runs over all nontrivial zeros of2 , ρ ρ (1+ρ) ζfact,the oscillatory term. InH(x) is present even without assuming RH, however, it is necessary for the error estimate above. 1991Mathematics Subject Classification.11P32, 11P55. 1