Oscillations in the Goldbach conjecture

Mossinghoff, Michael J.; Trudgian, Timothy S.

Journal de théorie des nombres de Bordeaux, Tome 34 (2022) no. 1, pp. 295-307.

Soit $R\left(n\right)={\sum }_{a+b=n}\Lambda \left(a\right)\Lambda \left(b\right)$, où $\Lambda \left(·\right)$ est la fonction de von Mangoldt. La fonction $R\left(n\right)$ est souvent étudiée en relation avec la conjecture de Goldbach. Sous l’hypothèse de Riemann (RH), on sait que ${\sum }_{n\le x}R\left(n\right)=x^2/2-4x^{3/2}G\left(x\right)+O\left(x^{1+ϵ}\right)$, où $G\left(x\right)=\Re {\sum }_{\gamma >0}\frac{x^{i\gamma }}{(\frac{1}{2}+i\gamma )(\frac{3}{2}+i\gamma )}$ 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 $G\left(x\right)<-0.02297$ et $G\left(x\right)>0.02103$ 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.

Let $R\left(n\right)={\sum }_{a+b=n}\Lambda \left(a\right)\Lambda \left(b\right)$, where $\Lambda \left(·\right)$ is the von Mangoldt function. The function $R\left(n\right)$ is often studied in connection with Goldbach’s conjecture. On the Riemann hypothesis (RH) it is known that ${\sum }_{n\le x}R\left(n\right)=x^2/2-4x^{3/2}G\left(x\right)+O\left(x^{1+ϵ}\right)$, where $G\left(x\right)=\Re {\sum }_{\gamma >0}\frac{x^{i\gamma }}{(\frac{1}{2}+i\gamma )(\frac{3}{2}+i\gamma )}$ 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 $G\left(x\right)<-0.02297$ and $G\left(x\right)>0.02103$ 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.

Classification : 11P32, 11J20, 11M26, 11Y35
Mots clés : Goldbach conjecture, Hardy–Littlewood conjectures, oscillations, Riemann hypothesis, simultaneous approximation

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