1 results
Random ordering in modulus of consecutive Hecke eigenvalues of primitive forms
- Part of
-
- Journal:
- Compositio Mathematica / Volume 154 / Issue 11 / November 2018
- Published online by Cambridge University Press:
- 18 October 2018, pp. 2441-2461
- Print publication:
- November 2018
-
- Article
-
- You have access
- HTML
- Export citation
-
Let
$\unicode[STIX]{x1D70F}(\cdot )$ be the classical Ramanujan 
$\unicode[STIX]{x1D70F}$-function and let 
$k$ be a positive integer such that 
$\unicode[STIX]{x1D70F}(n)\neq 0$ for 
$1\leqslant n\leqslant k/2$. (This is known to be true for 
$k<10^{23}$, and, conjecturally, for all 
$k$.) Further, let 
$\unicode[STIX]{x1D70E}$ be a permutation of the set 
$\{1,\ldots ,k\}$. We show that there exist infinitely many positive integers 
$m$ such that 
$|\unicode[STIX]{x1D70F}(m+\unicode[STIX]{x1D70E}(1))|<|\unicode[STIX]{x1D70F}(m+\unicode[STIX]{x1D70E}(2))|<\cdots <|\unicode[STIX]{x1D70F}(m+\unicode[STIX]{x1D70E}(k))|$. We also obtain a similar result for Hecke eigenvalues of primitive forms of square-free level.
