Short interval results for squarefree numbers

J. Number Theory, 35, 128--149.], [Filaseta & Trifonov, 1996 †Filaseta, M., & Trifonov, O. 1996

The distribution of fractional parts with applications to gap results in number theory

Proc. Lond. Math. Soc., III. Ser., 73(2), 241--278.], [Huxley, 1996 †Huxley, M.N. 1996

Area, Lattice Points and Exponential Sums

Oxford Science Pub.], [Huxley & Sargos, 1995 †Huxley, M.N., & Sargos, P. 1995

Integer points close to a plane curve of class

Acta Arith., 69(4), 359--366.], [Huxley & Sargos, 2006 †Huxley, M.N., & Sargos, P. 2006

Integer points in the neighborhood of a plane curve of class

Funct. Approximatio, Comment. Math., 35, 91--115.], [Huxley & Trifonov, 1996 †Huxley, M.N., & Trifonov, O. 1996

The square-full numbers in an interval

Math. Proc. Camb. Phil. Soc., 119, 201--208.] and . We deal with either getting an asymptotic formula of the shape

The method of trigonometrical sums in the theory of numbers

Mineola, NY: Dover Publications Inc. Translated from the Russian, revised and annotated by K. F. Roth and Anne Davenport, Reprint of the 1954 translation.]. The proof follows from a clever use of the mean-value theorem (see Theorem~5.6 of [Bordellès, 2012 †Bordellès, O. 2012

Arithmetic Tales

Universitext. Springer London Heidelberg New York Dordrecht.] for instance).

Theorem (First derivative test)

This result is useful whenLetf∈C1[N,2N] such that there existλ1>0 andc1⩾1 such that, for allx∈[N,2N] , we haveThenλ1⩽∣∣f′(x)∣∣⩽c1λ1. R(f,N,δ)⩽2c1Nλ1+4c1Nδ+2δλ1+1.

Area, Lattice Points and Exponential Sums

Oxford Science Pub.] succeeded in passing from the first derivative to the second derivative. This reduction step enables him to apply this theorem to a function being approximatively of the same order of magnitude as

Theorem (Second derivative test)

Both hypotheses above are often satisfied in practice, so that this result may be considered as the first useful tool of the theory. A proof of this Theorem may be found in Theorem 5.8 of [Bordellès, 2012 †Bordellès, O. 2012Letf∈C2[N,2N] such that there existλ2>0 andc2⩾1 such that, for allx∈[N,2N] , we haveThenλ2⩽∣∣f′′(x)∣∣⩽c2λ2andNλ2⩾c−12. R(f,N,δ)⩽6{(3c2)1/3Nλ1/32+(12c2)1/2Nδ1/2+1}.

Arithmetic Tales

Universitext. Springer London Heidelberg New York Dordrecht.]. Using a

Integer points close to a plane curve of class

Acta Arith., 69(4), 359--366.] and [Huxley & Sargos, 2006 †Huxley, M.N., & Sargos, P. 2006

Integer points in the neighborhood of a plane curve of class

Funct. Approximatio, Comment. Math., 35, 91--115.] succeeded in proving the following fundamental result. A proof of an explicit version may be found in Theorem 5.12 of [Bordellès, 2012 †Bordellès, O. 2012

Arithmetic Tales

Universitext. Springer London Heidelberg New York Dordrecht.].

Theorem (

The next result leads us to estimateLetk⩾3 be an integer andf∈Ck[N,2N] such that there existλk>0 andck⩾1 such that, for allx∈[N,2N] , we haveLetλk⩽∣∣f(k)(x)∣∣⩽ckλk. δ∈(0,14) . ThenwhereR(f,N,δ)⩽αkNλ2k(k+1)k+βkNδ2k(k−1)+8k3(δλk)1/k+2k2(5e3+1) αk=2k2c2k(k+1)kandβk=4k2(5e3c2k(k−1)k+1).

Van der Corput's Method of Exponential Sums

London Math. Soc. Lect. Note, no. 126. Cambridge University Press.] for instance), which have been extensively studied in the 20th century by many specialists, such as van der Corput, Weyl or Vinogradov. Nevertheless, even using the finest exponent pairs to date, the result generally does not significantly improve on the previous estimates seen above. A simple proof of the following inequality may be found in [Filaseta, 1990 †Filaseta, M. 1990

Short interval results for squarefree numbers

J. Number Theory, 35, 128--149.].

Theorem (

Letf:[N,2N]⟶R be any function andδ∈(0,14) . SetK=⌊(8δ)−1⌋+1 . Then, for any positive integerH⩽K , we haveR(f,N,δ)⩽4NH+4H∑h=1H∣∣∣∣∑N⩽n⩽2Ne(hf(n))∣∣∣∣.

This last part is somewhat out of the scope of the TME-EMT project, but may help the reader in orienting him/herself in the litterature.

When

Über die Gitterpunkte auf konvexen Kurven

Math. Z., 24, 500--518.] who proved that a strictly convex arc

The number of lattice points on a convex curve

J. Number Theory, 6, 128--135.] proved that if

The number of integral points on arcs and ovals

Duke Math. J., 59, 337--357.] who showed among other things the following estimate.

Theorem (1989)

LetN⩾1 ,k⩾4 be integers and defineK=(k+22) . LetI be an interval with lengthN andf∈CK(I) satisfying|f′(x)|⩽1 ,f′′(x)>0 and such that the number of solutions of the equationf(K)(x)=0 is⩽m . Then there exists a constantc0=c0(k)>0 such thatR(f,N,0)⩽c0(m+1)N1/2+3/(k+3).

Last updated on July 23rd, 2012, by Olivier Bordellès