Pretentious multiplicative functions and the prime number theorem for arithmetic progressions

Abstract

Abstract. Building on the concept of pretentious multiplicative functions, we give a new and largely elementary proof of the best result known on the counting function of primes in arithmetic progressions.

A new proof of Halász’s theorem, and its consequences

“…Now Lemma 2.4 of Koukoulopoulos [15] implies that 1 for all z ≥ y ≥ 2 n≤z 1 y (n)n −2it = z 1−2it 1 − 2it p≤y 1 − 1 p + O z 1−c/ log(y+|t|) log y . and we now input the asymptotic formula above.…”

On multiplicative functions which are small on average

“…It is evident from the above result that our methods are of comparable strength with more classical arguments that use the analyticity of L(s, f ) to the left of the line ℜ(s) = 1 such as the ones in [Dav00]. Indeed, in [Kou13] it was shown how to combine the methods of this paper with estimates for exponential sums due to Korobov and Vinogradov to give a new proof of the best error term known in the prime number theorem for arithmetic progressions. However, since we always work with conditions of the form (1.5), we are forced to use different methods than analytic continuation and the residue theorem.…”

“…Next, we have the triangle inequality [GS] for the distance function defined in Section 1. The following result is Lemma 3.3 in [Kou13].…”

“…In this section we give some estimates about the Riemann ζ function. First, we have the following lemma, which is a simple corollary of Lemma 2.4 in [Kou13].…”

On Karamata’s proof of the Landau–Ingham Tauberian theorem

“…(Cf. the review [3] and the very recent paper [19], which provides a "a new and largely elementary proof of the best result known on the counting function of primes in arithmetic progressions".) It is not clear whether this is necessarily so.…”