Oscillations in the Goldbach conjecture
Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 1, pp. 295-307.

Let R(n)= a+b=n Λ(a)Λ(b), where Λ(·) is the von Mangoldt function. The function R(n) is often studied in connection with Goldbach’s conjecture. On the Riemann hypothesis (RH) it is known that nx R(n)=x 2 /2-4x 3/2 G(x)+O(x 1+ϵ ), where G(x)= γ>0 x iγ (1 2+iγ)(3 2+iγ) and the sum is over the ordinates of the nontrivial zeros of the Riemann zeta function in the upper half-plane. We prove (on RH) that each of the inequalities G(x)<-0.02297 and G(x)>0.02103 holds infinitely often, and establish an improvement on the latter bound under an assumption of linearly independence for zeros of the zeta function. We also show that the bounds we obtain are very close to optimal.

Soit R(n)= a+b=n Λ(a)Λ(b), où Λ(·) est la fonction de von Mangoldt. La fonction R(n) est souvent étudiée en relation avec la conjecture de Goldbach. Sous l’hypothèse de Riemann (RH), on sait que nx R(n)=x 2 /2-4x 3/2 G(x)+O(x 1+ϵ ), où G(x)= γ>0 x iγ (1 2+iγ)(3 2+iγ) et la somme est prise sur les ordonnées des zéros non triviaux de la fonction zêta de Riemann dans le demi-plan supérieur. Nous prouvons (sous l’hypothèse de Riemann) que chacune des inégalités G(x)<-0.02297 et G(x)>0.02103 est vérifiée infiniment souvent, et établissons une amélioration de cette dernière borne sous une hypothèse d’indépendance linéaire pour les zéros de la fonction zêta. Nous montrons également que les bornes obtenues sont très proches de l’optimal.

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DOI: 10.5802/jtnb.1202
Classification: 11P32,  11J20,  11M26,  11Y35
Keywords: Goldbach conjecture, Hardy–Littlewood conjectures, oscillations, Riemann hypothesis, simultaneous approximation
Michael J. Mossinghoff 1; Timothy S. Trudgian 2

1 Center for Communications Research Princeton, NJ, USA
2 School of Science UNSW Canberra at ADFA ACT 2610, Australia
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Michael J. Mossinghoff; Timothy S. Trudgian. Oscillations in the Goldbach conjecture. Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 1, pp. 295-307. doi : 10.5802/jtnb.1202. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1202/

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