On simultaneous approximations to the logarithms of primes
https://doi.org/10.22405/2226-8383-2022-23-5-87-100
Abstract
In the first part of the paper, a modification of elementary Titchmarsh’s method is applied to the proof of the local Kronecker’s theorem. For any finite real sequence ¯𝜆= (𝜆1, . . . , 𝜆𝑟) of linearly independent (over Q) numbers and for any 𝜀 > 0, this method leads to the explicit upper bound of the value ℎ = ℎ(𝜀,¯𝜆) with the following property: for any real sequence ¯𝛼 = (𝛼1, . . . , 𝛼𝑟), any interval of the length ℎ contains a point 𝑡 such that ‖𝑡𝜆𝑠 − 𝛼𝑠‖ <= 𝜀, 1 <= 𝑠 <= 𝑟. Such estimate is weaker than the best known, but it’s proof is quite simple and leads
to the same (in essence) results in the applications.
The second part contains the short memoirs concerning the academician Alexey Nikolaevich
Parshin who passed away on June, 18 this year.
About the Authors
Maxim Aleksandrovich Korolev
Steklov Mathematical Institute of Russian Academy of Sciences
Russian Federation
Russian Federation
doctor of physical and mathematical sciences
Irina Sergeevna Rezvyakova
Steklov Mathematical
Institute of Russian Academy of Sciences
Russian Federation
Russian Federation
candidate of physical and mathematical sciences
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Review
For citations:
Korolev M.A., Rezvyakova I.S. On simultaneous approximations to the logarithms of primes. Chebyshevskii Sbornik. 2022;23(5):87-100. (In Russ.) https://doi.org/10.22405/2226-8383-2022-23-5-87-100
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