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Mathematics on the Irrationality of Certain Series
| Content Provider | Semantic Scholar |
|---|---|
| Author | Erdös, Paul |
| Copyright Year | 2004 |
| Abstract | Extending previous results of CHOWLA 1 1) proved that for every integer t > 1 the series Zj ; >-and 00 r(n))) I n=1 n=1 are irrational, where d(n) denotes the number of divisors of n and r(n) denotes the number of solutions of n = x 2 + y 2. In my above paper I remarked that I cannot prove that any of the series 00 °° T (n) I e(n) I v(n) 00 n=1 n=1 n=1 are irrational, where qq(n) is Euler's 92 function, 6(n) the sum of the divisors of n and v(n) the number of distinct prime factors of n. On the other hand by the methods used in the above paper I can prove without difficulty that the two series 00 1 00 1 I to+v(n)' I to-v(n) n1 n=1 are irrational, but I failed to prove the same for the two series to+d 1 011 (n)' I told(n) n=1 n=1 The main difficulty seems to be that I cannot prove that for infinitely many n (1) max (m+ d(mn (1) can be proved with v(m) instead of d(m) (2). I cannot prove anything about the series °0 1 ~ 00 1 ' 0~ O 1 I to+,p(n)' Lr to+0(n)' L to+v " n=1 n=1 n=1 where pn is the greatest prime factor of n (if in (1) d(m) is replaced by p(n), r(n) or pn (1) becomes false) . |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | https://users.renyi.hu/~p_erdos/1957-07.pdf |
| Alternate Webpage(s) | http://www.math-inst.hu/~p_erdos/1957-07.pdf |
| Alternate Webpage(s) | http://www.renyi.hu/~p_erdos/1957-07.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
