![]() ![]() On the other hand, the randomness heuristics from Supplement 4 suggest that should be able to be taken as small as, and perhaps even if one is particularly optimistic about the accuracy of these probabilistic models. On the Riemann Hypothesis Maier and Montgomery lowered the threshold to for an absolute constant (the bound is more classical, following from Exercise 33 of Notes 2). By using the techniques based on zero density estimates discussed in Notes 6, it was shown by Motohashi and that one can also establish \eqref. However it is significantly more difficult to understand what happens when grows much slower than this. From (2) one hasĪs if is such that for some fixed. Henceforth we shall focus our discussion more on the Liouville function, and turn our attention to averages on shorter intervals. The Liouville function behaves almost identically to the Möbius function, in that estimates for one function almost always imply analogous estimates for the other:Įxercise 1 Without using the prime number theorem, show that (1) is also equivalent to (Throughout these notes we will try to normalise most of the sums and integrals appearing here as averages that are trivially bounded by note that other normalisations are preferred in some of the literature cited here.) For instance, as we established in Theorem 58 of Notes 1, the prime number theorem is equivalent to the assertion thatĪs. Where and are both large, but is significantly smaller than. Given a multiplicative function, we are often interested in the asymptotics of long averages such asįor large values of, as well as short sums The space of -bounded multiplicative functions is also closed under multiplication and complex conjugation. Dirichlet characters (or “non-Archimedean” characters) (which are essentially pullbacks of Fourier characters on a multiplicative cyclic group with the discrete ( non-Archimedean) metric). ![]() “ Archimedean” characters (which I call Archimedean because they are pullbacks of a Fourier character on the multiplicative group, which has the Archimedean property).Some key examples of such functions include: In this section we focus on the asymptotic behaviour of -bounded multiplicative functions. Let us call an arithmetic function -bounded if we have for all. ![]()
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