Approximation by special values of Dirichlet series. (English) Zbl 1472.11228
Summary: In this note, we will show that real numbers can be strongly approximated by linear combinations of special values of Dirichlet series. We extend the approximation results of E. Alkan [Commun. Number Theory Phys. 7, No. 3, 515–550 (2013; Zbl 1312.11070); Proc. Am. Math. Soc. 143, No. 9, 3743–3752 (2015; Zbl 1327.11058)] in an effective way to all non-zero Dirichlet series with a better approximation. Using the fundamental works of Szemerédi and Green-Tao on arithmetic progressions, we prove that one can approximate real numbers with special values of Dirichlet series coming from sets of positive upper density or the set of prime numbers.
References:
| [1] | Alkan, Emre, Approximation by special values of harmonic zeta function and log-sine integrals, Commun. Number Theory Phys., 7, 3, 515-550 (2013) · Zbl 1312.11070 · doi:10.4310/CNTP.2013.v7.n3.a5 |
| [2] | Alkan, Emre, Special values of the Riemann zeta function capture all real numbers, Proc. Amer. Math. Soc., 143, 9, 3743-3752 (2015) · Zbl 1327.11058 · doi:10.1090/S0002-9939-2015-12649-4 |
| [3] | Alkan, Emre; G\"{o}ral, Haydar, Trigonometric series and special values of \(L\)-functions, J. Number Theory, 178, 94-117 (2017) · Zbl 1418.11117 · doi:10.1016/j.jnt.2017.02.015 |
| [4] | Ap\'{e}ry, Roger, Irrationalit\'{e} de \(\zeta2\) et \(\zeta3\), Ast\'{e}risque, 61, 11-13 (1979) · Zbl 0401.10049 |
| [5] | Apostol, Tom M., Introduction to analytic number theory, xii+338 pp. (1976), Springer-Verlag, New York-Heidelberg · Zbl 1154.11300 |
| [6] | Chudnovsky, D. V.; Chudnovsky, G. V., The computation of classical constants, Proc. Nat. Acad. Sci. U.S.A., 86, 21, 8178-8182 (1989) · Zbl 0683.65008 · doi:10.1073/pnas.86.21.8178 |
| [7] | Elaissaoui, Lahoucine; Guennoun, Zine El Abidine, Evaluation of log-tangent integrals by series involving \(\zeta(2n+1)\), Integral Transforms Spec. Funct., 28, 6, 460-475 (2017) · Zbl 1381.11066 · doi:10.1080/10652469.2017.1312366 |
| [8] | Green, Ben; Tao, Terence, The primes contain arbitrarily long arithmetic progressions, Ann. of Math. (2), 167, 2, 481-547 (2008) · Zbl 1191.11025 · doi:10.4007/annals.2008.167.481 |
| [9] | Ramanujan, S., Modular equations and approximations to \(\pi \) [Quart. J. Math. {\bf45} (1914), 350-372]. Collected papers of Srinivasa Ramanujan, 23-39 (2000), AMS Chelsea Publ., Providence, RI · JFM 45.1249.01 |
| [10] | Roth, K. F., On certain sets of integers, J. London Math. Soc., 28, 104-109 (1953) · Zbl 0050.04002 · doi:10.1112/jlms/s1-28.1.104 |
| [11] | Szemer\'{e}di, E., On sets of integers containing no four elements in arithmetic progression, Acta Math. Acad. Sci. Hungar., 20, 89-104 (1969) · Zbl 0175.04301 · doi:10.1007/BF01894569 |
| [12] | Szemer\'{e}di, E., On sets of integers containing no \(k\) elements in arithmetic progression, Acta Arith., 27, 199-245 (1975) · Zbl 0303.10056 · doi:10.4064/aa-27-1-199-245 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
