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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 well-understanding sign changes of these coefficients you can provide me would be greatly appreciated ? Many thanks, Khadija

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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 Matomaki-Radziwill 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.

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  • $\begingroup$ 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 Lau-Wu. I talked about the result of Matomaki-Radziwill, 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 $\prod_{p\leq x}(1-1/p)\gg 1/\log x$ by Mertens' theorems). You have to read the paper to understand how they got this result, this is what the paper is about. $\endgroup$
    – GH from MO
    Aug 31 '15 at 8:33

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