|
There exist no Ramanujan congruences $\mod691^2$
A. A. Panchishkin M. V. Lomonosov Moscow State University, USSR
Abstract:
Let $\tau(n)$ be Ramanujan's function,
$$
x\prod_{m=1}^\infty(1-x^m)^{24}=\sum_{n=1}^\infty\tau(n)x^n.
$$
In this paper it is shown that the Ramanujan congruence $\tau(n)\equiv\sum_{d/n}d^{11}\bmod691$ cannot be improved $\bmod691^2$. The following result is proved: for arbitrary $r$, $s\bmod691$ the set of primes such that $p\equiv r\bmod691$, $\tau(p)\equiv p^{11}+1+691\cdot s\bmod691^2$ has positive density.
Received: 30.04.1974
Citation:
A. A. Panchishkin, “There exist no Ramanujan congruences $\mod691^2$”, Mat. Zametki, 17:2 (1975), 255–263; Math. Notes, 17:2 (1975), 148–153
Linking options:
https://www.mathnet.ru/eng/mzm7244 https://www.mathnet.ru/eng/mzm/v17/i2/p255
|
Statistics & downloads: |
Abstract page: | 454 | Full-text PDF : | 118 | First page: | 1 |
|