I want to write a survey on the subject 'Sign changes for coefficients of symmetric power $L$functions'. So, I browse the Web and I got some papers. I read it and I gave special interest to the paper namely 'Sign change of Hecke eigenvalues' of Kaisa Matomaki and Maksym Radziwill. In this paper, I found this sentence "Previously it was only known that there are $≫ \sqrt{x}$ sign changes in the holomorphic case (see [16])". This is the result of the paper of THE NUMBER OF HECKE EIGENVALUES OF SAME SIGNS of Y.K Lau and Jie Wu in which they gave the best possible lower bounds in order of magnitude for the number of positive and negative Hecke eigenvalues which is $\gg x$ for a large enough $x.$ My question is the following: How the authors of 'Sign change of Hecke eigenvalues' obtain this lower bound, I mean $≫\sqrt{x}$ although in the paper of Lau and Wu, we have $≫ x$? May be I misunderstood what they mean by 'sign changes'. Can someone clarify this to me? Also, in case of coefficients high symmetric power $L$functions, any reference that help me wellunderstanding sign changes of these coefficients you can provide me would be greatly appreciated ? Many thanks, Khadija
A sign change of a sequence $(a_n)_{n=1}^\infty$ means an $n$ such that $a_na_{n+1}<0$, i.e. $a_n$ and $a_{n+1}$ are of opposite sign. You can have many positive and negative terms with very few sign changes. For example, up to $x$ you can have $a_n>0$ for $n\leq x/2$ and $a_n<0$ for $n>x/2$, which means $x/2$ positive terms and $x/2$ negative terms but only one sign change.
Note that the result of MatomakiRadziwill is in some sense optimal as they show that the number of sign changes is of the same order of magnitude as the number of nonzero terms.

$\begingroup$ Thanks @GH from MO for your explanation. I want also to understand how they got this result, i.e $(\gg \sqrt{x})$ is there a formula or deduction that can be used to derive this result? $\endgroup$ Aug 31 '15 at 8:29

$\begingroup$ @KhadijaMbarki: The result $\gg\sqrt{x}$ you mention is due to LauWu. I talked about the result of MatomakiRadziwill, which is stronger, e.g. it yields $\gg x/\log x$ sign changes (see Theorem 1.2 in their preprint arXiv:1405.7671v2, and note that $\prod_{p\leq x}(11/p)\gg 1/\log x$ by Mertens' theorems). You have to read the paper to understand how they got this result, this is what the paper is about. $\endgroup$ Aug 31 '15 at 8:33