Annali di Matematica Pura ed Applicata

, Volume 103, Issue 1, pp 199–205

Investigations in the powersum theory III

• S. Dancs
• P. Turán
Article

Summary

As found by the second named author certain lower bounds of generalised powersums$$g\left( v \right) = \sum\limits_{j = 1}^n {b_j z_j^v }$$ have several applications in analytical numbertheory. In this paper, strengthening some results of E. Makai and R. Tijdeman, the inequality
$$\begin{array}{*{20}c} {\max } \\ {m + 1 \leqq v \leqq m + n} \\ {v integer} \\ \end{array} \left| {g\left( v \right)} \right| \geqq \frac{1}{6}\left( {\frac{{\delta _2 - \delta _1 }}{{4\left( {2 + \delta _2 } \right)}}} \right)^{n - 1} \left( {1 - \delta _2 } \right)^{m + 1} \mathop {\max }\limits_{p_1 \leqq h \leqq p_2 } \left| {\sum\limits_{j = 1}^h {b_j } } \right|$$
is proved where
$$0 = \left| {1 - z_1 } \right| \leqq \left| {1 - z_2 } \right| \leqq \ldots \leqq \left| {1 - z_n } \right|,$$
m nonnegative integer, 0<δ12<1 and the indices p1 and p2 are restricted by
$$\begin{array}{*{20}c} {\left| {1 - z_{p_1 } } \right| \leqq \delta _1 } \\ {\left| {1 - z_{p_2 } } \right| \leqq \delta _2 .} \\ \end{array}$$

Keywords

Lower Bound Nonnegative Integer
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

1. (**).
See his book « Eine neue Methode der Analysis und deren Anwendungen », Akad. Kiadó, Budapest, 1953. A rewritten English version will appear in the Interscience Tracts series.Google Scholar
2. (*).
The constant 24e 2 was replaced by 8e in our paper withVera T. Sós,On some new theorems in the theory of diophantine approximations, Acta Math. Hung., T. VI, Fasc. 3–4 (1955), pp. 241–257. Further improvement of the constant, notably in the caseb 1=...=b n=1 would be of importance for some applications in the theory of Riemann zetafunction.Google Scholar
3. (**).
E. Makai,On a minimum problem, Annales Univ. Sci. Budapestiensis, Sect. Math. (1960–61), pp. 177–182.Google Scholar
4. (***).
F. M. Geysel,On generalised sums of complex numbers, M. C. Report U. W. (1968), Math. Centrum Amsterdam.Google Scholar
5. (***).
R. Tijdeman,On the distribution of the values of certain functions, Dissertation, Amsterdam, 1969.Google Scholar
6. (*).
P. Turán,On an inequality of Chebysev, Annales Univ. Sci. Budapest de R. Eötvös nomin. Sect, Math.,11 (1968), pp. 15–16.