Additional note on two problems in the analytic theory of numbers. (Q1465357)
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scientific article; zbMATH DE number 2606008
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Additional note on two problems in the analytic theory of numbers. |
scientific article; zbMATH DE number 2606008 |
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Additional note on two problems in the analytic theory of numbers. (English)
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1920
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Hardy hatte kürzlich (F. d. M. 46, 262 (JFM 46.0262.*), 1916-18) für die Funktion \[ P(x) =\sum_{u^2+v^2\leqq x}1-\pi x \] bewiesen: \[ \frac 1x \int_1^x | P(\tau)| d\tau = O(x^{ \frac 14+\varepsilon}). \] Heute zeigt er, daß\ der bekannte Satz \[ P(x) = O(x^{\frac 13+\varepsilon }) \] aus den beim Beweise vorgekommenen Hilfsbetrachtungen folgt. Entsprechendes für \[ \Delta (x) =\sum_{n\leqq x}\left[ \frac xn\right]-(x \log x +(2C-1)x). \]
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