Geodesic
Distribution of zeta zeros and the oscillation of the error term of the prime number theorem
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic and combinatorial number theory, Tome 296 (2017), pp. 207-219

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An 84-year-old classical result of Ingham states that a rather general zero-free region of the Riemann zeta function implies an upper bound for the absolute value of the remainder term of the prime number theorem. In 1950 Turán proved a partial conversion of the mentioned theorem of Ingham. Later the author proved sharper forms of both Ingham's theorem and its conversion by Turán. The present work shows a very general theorem which describes the average and the maximal order of the error terms by a relatively simple function of the distribution of the zeta zeros. It is proved that the maximal term in the explicit formula of the remainder term coincides with high accuracy with the average and maximal order of the error term. 
@article{TM_2017_296_a15,
     author = {J. Pintz},
     title = {Distribution of zeta zeros and the oscillation of the error term of the prime number theorem},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {207--219},
     publisher = {mathdoc},
     volume = {296},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2017_296_a15/}
}
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J. Pintz. Distribution of zeta zeros and the oscillation of the error term of the prime number theorem. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic and combinatorial number theory, Tome 296 (2017), pp. 207-219. http://geodesic.mathdoc.fr/item/TM_2017_296_a15/