A note on arithmetic Diophantine series

Alexander E. Patkowski

Czechoslovak Mathematical Journal (2021)

  • Volume: 71, Issue: 4, page 1149-1155
  • ISSN: 0011-4642

Abstract

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We consider an asymptotic analysis for series related to the work of Hardy and Littlewood (1923) on Diophantine approximation, as well as Davenport. In particular, we expand on ideas from some previous work on arithmetic series and the RH. To accomplish this, Mellin inversion is applied to certain infinite series over arithmetic functions to apply Cauchy's residue theorem, and then the remainder of terms is estimated according to the assumption of the RH. In the last section, we use simple properties of the fractional part function and its Fourier series to state some identities involving different arithmetic functions. We then discuss some of their individual properties, such as convergence, as well as implications related to known work.

How to cite

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Patkowski, Alexander E.. "A note on arithmetic Diophantine series." Czechoslovak Mathematical Journal 71.4 (2021): 1149-1155. <http://eudml.org/doc/298191>.

@article{Patkowski2021,
abstract = {We consider an asymptotic analysis for series related to the work of Hardy and Littlewood (1923) on Diophantine approximation, as well as Davenport. In particular, we expand on ideas from some previous work on arithmetic series and the RH. To accomplish this, Mellin inversion is applied to certain infinite series over arithmetic functions to apply Cauchy's residue theorem, and then the remainder of terms is estimated according to the assumption of the RH. In the last section, we use simple properties of the fractional part function and its Fourier series to state some identities involving different arithmetic functions. We then discuss some of their individual properties, such as convergence, as well as implications related to known work.},
author = {Patkowski, Alexander E.},
journal = {Czechoslovak Mathematical Journal},
keywords = {arithmetic series; Riemann zeta function; Möbius function},
language = {eng},
number = {4},
pages = {1149-1155},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A note on arithmetic Diophantine series},
url = {http://eudml.org/doc/298191},
volume = {71},
year = {2021},
}

TY - JOUR
AU - Patkowski, Alexander E.
TI - A note on arithmetic Diophantine series
JO - Czechoslovak Mathematical Journal
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 71
IS - 4
SP - 1149
EP - 1155
AB - We consider an asymptotic analysis for series related to the work of Hardy and Littlewood (1923) on Diophantine approximation, as well as Davenport. In particular, we expand on ideas from some previous work on arithmetic series and the RH. To accomplish this, Mellin inversion is applied to certain infinite series over arithmetic functions to apply Cauchy's residue theorem, and then the remainder of terms is estimated according to the assumption of the RH. In the last section, we use simple properties of the fractional part function and its Fourier series to state some identities involving different arithmetic functions. We then discuss some of their individual properties, such as convergence, as well as implications related to known work.
LA - eng
KW - arithmetic series; Riemann zeta function; Möbius function
UR - http://eudml.org/doc/298191
ER -

References

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  2. Davenport, H., 10.1093/qmath/os-8.1.8, Q. J. Math., Oxf. Ser. 8 (1937), 8-13. (1937) Zbl0016.20105DOI10.1093/qmath/os-8.1.8
  3. Hardy, G. H., Littlewood, J. E., 10.1007/BF02422942, Acta Math. 41 (1917), 119-196 9999JFM99999 46.0498.01. (1917) MR1555148DOI10.1007/BF02422942
  4. Hardy, G. H., Littlewood, J. E., Some problems of Diophantine approximation: The analytic properties of certain Dirichlet's series associated with the distribution of numbers to modulus unity, Trans. Camb. Philos. Soc. 27 (1923), 519-534 9999JFM99999 49.0131.01. (1923) 
  5. Iwaniec, H., Kowalski, E., 10.1090/coll/053, Colloquium Publications. American Mathematical Society 53. American Mathematical Society, Providence (2004). (2004) Zbl1059.11001MR2061214DOI10.1090/coll/053
  6. Li, H. L., Ma, J., Zhang, W. P., 10.1007/s10114-010-8387-x, Acta Math. Sin., Engl. Ser. 26 (2010), 1125-1132. (2010) Zbl1221.11060MR2644050DOI10.1007/s10114-010-8387-x
  7. Luther, W., 10.1016/0021-9045(86)90053-5, J. Approximation Theory 48 (1986), 303-321. (1986) Zbl0626.42008MR0864753DOI10.1016/0021-9045(86)90053-5
  8. Paris, R. B., Kaminski, D., 10.1017/CBO9780511546662, Encyclopedia of Mathematics and Its Applications 85. Cambridge University Press, Cambridge (2001). (2001) Zbl0983.41019MR1854469DOI10.1017/CBO9780511546662
  9. Segal, S. L., 10.4064/aa-28-4-345-348, Acta Arith. 28 (1976), 345-348. (1976) Zbl0319.10050MR0387222DOI10.4064/aa-28-4-345-348
  10. Titchmarsh, E. C., The Theory of the Riemann Zeta-Function, Oxford Science Publications. Oxford University Press, Oxford (1986). (1986) Zbl0601.10026MR0882550

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