Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

On a certain sum in number theory. II
HTML articles powered by AMS MathViewer

by Břetislav Novák
Trans. Amer. Math. Soc. 195 (1974), 357-364
DOI: https://doi.org/10.1090/S0002-9947-1974-0435002-1

Abstract:

We derive “exact order” of the function \[ \sum \limits _{k \leq \sqrt x } {{k^\rho }{{\min }^\beta }\left ( {\frac {{\sqrt x }}{k},\frac {1}{{{P_k}}}} \right )}.\] Here $\rho$ and $\beta$ are nonnegative real numbers and, for given real ${\delta _1},{\delta _2}, \cdots ,{\delta _r},{P_k} = {\max _j}\langle k{\delta _j}\rangle$ where $\langle t\rangle$, for real t, denotes distance of t from the nearest integer. Using our results, we obtain the solution of the basic problem in the theory of lattice points with weight in rational many-dimensional ellipsoids.
References
  • Bohuslav Diviš, Über Gitterpunkte in mehrdimensionalen Ellipsoiden. I, II, Czechoslovak Math. J. 20(95) (1970), 130–139; ibid. 20 (95) (1970), 149–159 (German, with English summary). MR 255503
  • G. H. Hardy and J. E. Littlewood, Some problems of Diophantine approximation: The lattice-points of a right-angle triangle, (second memoir), Abh. Math. Sem. Univ. Hamburg 1 (1929), 212-249.
  • B. Novák, Verallgemeinerung eines Peterssonschen Satzes und Gitterpunkte mit Gewichten, Acta Arith. 13 (1967/68), 423–454 (German). MR 229577, DOI 10.4064/aa-13-4-423-454
  • B. Novák, On lattice points with weight in high-dimensional ellipsoids, Acta Arith. 14 (1967/68), 371–397. MR 230683, DOI 10.4064/aa-14-4-371-397
  • Břetislav Novák, Mean value theorems in the theory of lattice points with weight. II, Comment. Math. Univ. Carolinae 11 (1970), 53–81. MR 262197
  • Břetislav Novák, On certain sum in number theory, Comment. Math. Univ. Carolinae 12 (1971), 669–685. MR 308054
  • Břetislav Novák, Mittelwertsätze der Gitterpunktlehre. II, Časopis Pěst. Mat. 96 (1971), 245–261, 300 (German, with Czech summary). MR 0291097
  • Břetislav Novák, Über eine Methode der $\Omega$-Abschätzungen, Czechoslovak Math. J. 21(96) (1971), 257–279 (German). MR 292761
  • Břetislav Novák, Über Gitterpunkte in mehrdimensionalen Ellipsoiden, Czechoslovak Math. J. 22(97) (1972), 495–507 (German). MR 308055
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 10J25
  • Retrieve articles in all journals with MSC: 10J25
Bibliographic Information
  • © Copyright 1974 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 195 (1974), 357-364
  • MSC: Primary 10J25
  • DOI: https://doi.org/10.1090/S0002-9947-1974-0435002-1
  • MathSciNet review: 0435002