Computing π(x): An Analytic Method
نویسندگان
چکیده
منابع مشابه
A theoretical study on halogen-π interactions: X-C2-Y…C8H8 complexes
M06-2X functional was employed to study halogen-π interactions in X-C2-Y…C8H8 complexes (X, Y=H, F, Cl, and Br). In fact, interactions of mono- or di-halogenated acetylenes and planar cyclooctatetraene as an anti-aromatic π system were considered. Relationship between binding energies of the complexes and charge transfer effects was investigated. Also, electronic charge density values were calc...
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Let π = ⊗πv and π ′ = ⊗π ′ v be two irreducible, automorphic, cuspidal representations of GLm (AK). Using the logarithmic zero-free region of Rankin-Selberg L-function, Moreno established the analytic strong multi-plicity one theorem if at least one of them is self-contragredient, i.e. π and π ′ will be equal if they have finitely many same local components πv, π ′ v , for which the norm of pla...
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The paper describes a systematic computational study of the prime counting function π(x) and three of its analytic approximations: the logarithmic integral li(x) := ∫ x 0 dt log t , li(x)− 2 li( √ x), and R(x) := ∞k=1 μ(k)li(x1/k)/k, where μ is the Möbius function. The results show that π(x) < li(x) for 2 ≤ x ≤ 1014, and also seem to support several conjectures on the maximal and average errors...
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Let π = ⊗πv and π ′ = ⊗π ′ v be two irreducible, automorphic, cuspidal representations of GLm (AK). Using the logarithmic zero-free region of Rankin-Selberg L-function, Moreno established the analytic strong multi-plicity one theorem if at least one of them is self-contragredient, i.e. π and π ′ will be equal if they have finitely many same local components πv, π ′ v , for which the norm of pla...
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We explain how the Meissel-Lehmer-Lagarias-Miller-Odlyzko method for computing π(x) can be used for computing efficiently π(x, k, l), the number of primes congruent to l modulo k up to x. As an application, we computed the number of prime numbers of the form 4n ± 1 less than x for several values of x up to 1020 and found a new region where π(x, 4, 3) is less than π(x, 4, 1) near x = 1018.
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