Pretentious Multiplicative Functions and an Inequality for the Zeta-Function

We note how several central results in multiplicative number theory may be rephrased naturally in terms of multiplicative functions f that pretend to be another multiplicative function g. We formalize a ‘distance’ which gives a measure of such pretentiousness, and as one consequence obtain a curious inequality for the zeta-function. A common theme in several problems in multiplicative number theory involves identifying multiplicative functions f that pretend to be another multiplicative function g. Indeed, this theme may be found as early as in the proof of the prime number theorem; in particular in showing that ζ(1 + it) 6= 0. For, if ζ(1 + it) equals zero, then we expect the Euler product ∏ p≤P (1 − 1/p1+it)−1 to be small. This means that p−it ≈ −1 for many small primes p; or equivalently, that the multiplicative function n−it pretends to be the multiplicative function (−1)Ω(n). The insight of Hadamard and de la Vallée Poussin is that in such a case n−2it would pretend to be the multiplicative function that is identically 1, and this possibility can be eliminated by noting that ζ(1 + 2it) is regular for t 6= 0. Another example is given by Vinogradov’s conjecture that the least quadratic non-residue (mod p) is¿ p. If this were false, then the Legendre symbol np ) would pretend to be the trivial character for a long range of n. Even more extreme is the possibility that a quadratic Dirichlet L-function has a Landau – Siegel zero (a real zero close to 1), in which case that quadratic character χ would pretend to be the function (−1)Ω(n). In both these examples, it is not known how to eliminate the possibility of such pretentious behavior by characters. A third class of examples is provided by the theory of mean values of multiplicative functions. Let f(n) be a multiplicative function with |f(n)| ≤ 1 for all n, 2000 Mathematics Subject Classification. Primary: 11N25; Secondary: 11L20, 11L40, 11M06, 11N60. Le premier auteur est partiellement soutenu par une bourse de la Conseil de recherches en sciences naturelles et en génie du Canada. The second author is partially supported by the American Institute of Mathematics and the National Science Foundation. This is the final form of the paper. c ©2008 American Mathematical Society 205 206 A. GRANVILLE AND K. SOUNDARARAJAN and consider when the mean value (1) 1 x ∑