On an estimate of Walfisz and Saltykov for an error term related to the Euler function

Y.-F. S. Pétermann

Journal de théorie des nombres de Bordeaux (1998)

  • Volume: 10, Issue: 1, page 203-236
  • ISSN: 1246-7405

Abstract

top
The technique developed by A. Walfisz in order to prove (in 1962) the estimate H ( x ) ≪ ( log x ) 2 / 3 ( log log x ) 4 / 3 for the error term H ( x ) = ∑ n ≤ x φ ( n ) n - 6 π 2 x related to the Euler function is extended. Moreover, the argument is simplified by exploiting works of A.I. Saltykov and of A.A. Karatsuba. It is noted in passing that the proof proposed by Saltykov in 1960 of H ( x ) ≪ ( log x ) 2 / 3 ( log log x ) 1 + ϵ is erroneous and once corrected “only” yields Walfisz’ result. The generalizations obtained can be applied to error terms related to various classical - and less classical - arithmetical functions, as for instance to ( φ ( n ) / n ) r , ( σ ( n ) / n ) r and ( σ ( n ) / φ ( n ) ) r for every real value of r , and also to σ ( r ) ( n ) , the sum of the exponential divisors d of n with p α ¬ ∥ d if p 2 α ∥ n and α > 1 .

How to cite

top

Pétermann, Y.-F. S.. "On an estimate of Walfisz and Saltykov for an error term related to the Euler function." Journal de théorie des nombres de Bordeaux 10.1 (1998): 203-236. <http://eudml.org/doc/248159>.

@article{Pétermann1998,
abstract = {The technique developed by A. Walfisz in order to prove (in 1962) the estimate $H(x) \ll (\log x)^\{2/3\} (\log \log x)^\{4/3\}$ for the error term $H(x) = \sum _\{n \le x\} \frac\{\phi (n)\}\{n\} - \frac\{6\}\{\pi ^2\}x$ related to the Euler function is extended. Moreover, the argument is simplified by exploiting works of A.I. Saltykov and of A.A. Karatsuba. It is noted in passing that the proof proposed by Saltykov in 1960 of $H(x) \ll (\log x)^\{2/3\}(\log \log x)^\{1 + \epsilon \}$ is erroneous and once corrected “only” yields Walfisz’ result. The generalizations obtained can be applied to error terms related to various classical - and less classical - arithmetical functions, as for instance to $(\phi (n)/n)^r, (\sigma (n)/n)^r \text\{ and \} (\sigma (n)/ \phi (n))^r$ for every real value of $r$, and also to $\sigma ^\{(r)\}(n)$, the sum of the exponential divisors $d$ of $n$ with $p^\alpha \!\!\! \lnot \{\parallel \} d \text\{ if \} p^\{2\alpha \} \Vert n \text\{ and \} \alpha &gt; 1$.},
author = {Pétermann, Y.-F. S.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {asymptotic formulae for the Euler function with remainder term; sum of divisors; estimation of exponential sums with primes; multiplicative arithmetical functions; Walfisz method},
language = {eng},
number = {1},
pages = {203-236},
publisher = {Université Bordeaux I},
title = {On an estimate of Walfisz and Saltykov for an error term related to the Euler function},
url = {http://eudml.org/doc/248159},
volume = {10},
year = {1998},
}

TY - JOUR
AU - Pétermann, Y.-F. S.
TI - On an estimate of Walfisz and Saltykov for an error term related to the Euler function
JO - Journal de théorie des nombres de Bordeaux
PY - 1998
PB - Université Bordeaux I
VL - 10
IS - 1
SP - 203
EP - 236
AB - The technique developed by A. Walfisz in order to prove (in 1962) the estimate $H(x) \ll (\log x)^{2/3} (\log \log x)^{4/3}$ for the error term $H(x) = \sum _{n \le x} \frac{\phi (n)}{n} - \frac{6}{\pi ^2}x$ related to the Euler function is extended. Moreover, the argument is simplified by exploiting works of A.I. Saltykov and of A.A. Karatsuba. It is noted in passing that the proof proposed by Saltykov in 1960 of $H(x) \ll (\log x)^{2/3}(\log \log x)^{1 + \epsilon }$ is erroneous and once corrected “only” yields Walfisz’ result. The generalizations obtained can be applied to error terms related to various classical - and less classical - arithmetical functions, as for instance to $(\phi (n)/n)^r, (\sigma (n)/n)^r \text{ and } (\sigma (n)/ \phi (n))^r$ for every real value of $r$, and also to $\sigma ^{(r)}(n)$, the sum of the exponential divisors $d$ of $n$ with $p^\alpha \!\!\! \lnot {\parallel } d \text{ if } p^{2\alpha } \Vert n \text{ and } \alpha &gt; 1$.
LA - eng
KW - asymptotic formulae for the Euler function with remainder term; sum of divisors; estimation of exponential sums with primes; multiplicative arithmetical functions; Walfisz method
UR - http://eudml.org/doc/248159
ER -

References

top
  1. [1] U. Balakrishnan and Y.-F.S. Pétermann, The Dirichlet series of ζ(s)ζα (s + 1) f (s + 1): On an error term associated with its coefficients, Acta Arith.75 (1996), 39-69. Zbl0846.11054
  2. [2] A. Ivi, The Riemann zeta-function, John Wiley and Sons1985. Zbl0556.10026MR792089
  3. [3] E. Grosswald.The average order of an arithmetical function, Duke Math. J.23 (1956), 41-44. Zbl0070.27501MR74459
  4. [4] A.A. Karatsuba, Estimates for trigonometric sums by Vinogradov's method, and some applications, Proc. Steklov Inst. Math. (A.M.S English translation, 1973) 112 (1971), 251-265. Zbl0259.10040
  5. [5] M.N. Korobov, Estimates of trigonometrical sums and their applications (in Russian), Uspekhi Mat. Nauk.13 (4) (1958), 185-192. Zbl0086.03803MR106205
  6. [6] Y.-F.S. Pétermann and Jie Wu, On the sum of exponential divisors of an integer, Acta Math. Hungar.77 (1997), 159-175. Zbl0902.11037MR1485593
  7. [7] A.I. Saltykov, On Euler's function (in Russian), Vestnik Moskovskogo Universiteta, Seriya I: Matematika, Mekhanika, no vol. number, fasc. number 6 (1960), 34-50. Zbl0099.03702MR125088
  8. [8] G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres. Institut Elie Cartan131990. Zbl0788.11001
  9. [9] E.C. Titchmarsh, The theory of the Riemann zeta-function, Oxford, Clarendon Press1951; second edition revised by D.R. Heath-Brown, ibid1986. Zbl0042.07901MR882550
  10. [10] A. Walfisz, Über die Wirksamkeit einiger Abschätzungen trigonometrischer Summen, Acta Arith.4 (1958), 108-180. Zbl0084.27304MR103860
  11. [11] A. Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie. VEB Deutscher Verlag der Wissenschaften, Berlin1963. Zbl0146.06003MR220685

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.