A problem of Chowla revisited

作者：Murty M Ram^*; Murty V Kumar

来源：Journal of Number Theory, 2011, 131(9): 1723-1733.

DOI：10.1016/j.jnt.2011.04.003

• Chowla's problem
• Application of Baker's method
• Special values of Dirichlet L-series

In 1964, S. Chowla asked if there is a non-zero integer-valued function f with prime period p such that f (p) = 0 and
Sigma(p)(n=1)f(n) = 0
and Sigma(infinity)(n=1) f(n)/n = 0? Chowla conjectured that there was no such function. Later that year, Chowla proved a special case of his conjecture and in a subsequent paper, asked if the condition that f(p) = 0 can be dropped. In 1973, Baker, Birch and Wirsing, using the theory of linear forms in logarithms obtained a general theorem, a special case of which implies the conjecture of Chowla. They alluded to the fact that Chowla also settled the conjecture in his special case, but this proof (if different from the one by Baker. Birch and Wirsing) never seems to have been published. We resurrect the 1964 approach of Chowla and indicate how the cyclotomic units discovered by Ramachandra in 1966, can be combined with Baker's theorem to answer Chowla's question. We also obtain a mild generalization of the theorem of Baker, Birch and Wirsing.

• 出版日期2011-9