Let , where is the von Mangoldt function. The function is often studied in connection with Goldbach’s conjecture. On the Riemann hypothesis (RH) it is known that , where 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 and 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 , où est la fonction de von Mangoldt. La fonction est souvent étudiée en relation avec la conjecture de Goldbach. Sous l’hypothèse de Riemann (RH), on sait que , où 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 et 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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Accepted:
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Mots-clés : Goldbach conjecture, Hardy–Littlewood conjectures, oscillations, Riemann hypothesis, simultaneous approximation
Michael J. Mossinghoff 1; Timothy S. Trudgian 2

@article{JTNB_2022__34_1_295_0, author = {Michael J. Mossinghoff and Timothy S. Trudgian}, title = {Oscillations in the {Goldbach} conjecture}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {295--307}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {34}, number = {1}, year = {2022}, doi = {10.5802/jtnb.1202}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1202/} }
TY - JOUR AU - Michael J. Mossinghoff AU - Timothy S. Trudgian TI - Oscillations in the Goldbach conjecture JO - Journal de théorie des nombres de Bordeaux PY - 2022 SP - 295 EP - 307 VL - 34 IS - 1 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1202/ DO - 10.5802/jtnb.1202 LA - en ID - JTNB_2022__34_1_295_0 ER -
%0 Journal Article %A Michael J. Mossinghoff %A Timothy S. Trudgian %T Oscillations in the Goldbach conjecture %J Journal de théorie des nombres de Bordeaux %D 2022 %P 295-307 %V 34 %N 1 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1202/ %R 10.5802/jtnb.1202 %G en %F JTNB_2022__34_1_295_0
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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