Abstract:We prove that | d≤x μ(d)/d| log x ≤ 1/69 when x ≥ 96 955 and deduce from that:for every x > q ≥ 1. We also give better constants when x/q is larger. Furthermore we prove that |1 − d≤x μ(d) log(x/d)/d| ≤ 3 14 / log x and several similar bounds, from which we also prove corresponding bounds when summing the same quantity, but with the additional condition (d, q) = 1. We prove similar results for d≤x μ(d) log 2 (x/d)/d, among which we mention the bound | d≤x μ(d) log 2 (x/d)/d − 2 log x + 2γ 0 | ≤ 5 24 / log x, w… Show more
“…Rather than expanding on this subject, the author prefers to concentrate here on one application. Here is an identity proved in [37] by following [4] and [3]. For every x ě 1, we have…”
Section: 2mentioning
confidence: 99%
“…[37]). When D ě 463 421, we hav졡ˇÿdďD µpdq{dˇˇˇˇď 0.0144 log D´0.1 plog Dq 2 .When D ě 97 000, we haveˇˇˇˇÿ…”
We mix some of the novelties that have occured recently in the field of explicit multiplicative number theory, together with some questions that have not been answered yet and with several new results.
“…Rather than expanding on this subject, the author prefers to concentrate here on one application. Here is an identity proved in [37] by following [4] and [3]. For every x ě 1, we have…”
Section: 2mentioning
confidence: 99%
“…[37]). When D ě 463 421, we hav졡ˇÿdďD µpdq{dˇˇˇˇď 0.0144 log D´0.1 plog Dq 2 .When D ě 97 000, we haveˇˇˇˇÿ…”
We mix some of the novelties that have occured recently in the field of explicit multiplicative number theory, together with some questions that have not been answered yet and with several new results.
“…Inferring a quantitative error term (here, simply a bound) for M (x) from the one of ψ(x) − x has received attention. Let us mention the work [7] of Kienast,[16] of Schoenfeld and [5] of El Marraki, although the last two authors do not present their investigation in this perspective; and lately the paper [13]. The answers are up to now rather unsatisfactory.…”
mentioning
confidence: 99%
“…First, as noticed by Landau, inferring an error term for M (x) once we have one for m(x) is routine. Secondly, a path using identities has been investigated by Balazard [3] (see [2] for a French translation) and in [13].…”
mentioning
confidence: 99%
“…Indeed, the latest best bound for 1/ζ(s) is due to Trudgian [17] and would put a constant at least of size 10 7 in front of x −1/(8 log log x) . When the emphasis is on this aspect, the path of identities taken in [3] and [13] remains a better choice.…”
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