Analytic Number Theory

By Henryk Iwaniec, Emmanuel Kowalski. All the Math You Missed, 2021

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NEW RESULTS ON SUM‐PRODUCT TYPE GROWTH OVER FIELDS (2017, background): “The orthogonality of multiplicative characters, which we denote by χ, implies (see [19] for details)”
On the concentration of points of polynomial maps and applications (2012, methodology): “Here we use the approach of [5], combined with estimates on the Vinogradov mean value theorem (see [10, 11, 18] to estimate N(H; K, L).”
Oscillatory integrals with uniformity in parameters (2017, background): “ber theory. 1. Introduction Exponential integrals occur in many problems in analytic number theory, including moments of L-functions, lattice-point counting, and the circle method (see e.g. [3], [5], [6]). For some more advanced applications (e.g. in [2]) it is necessary to develop an asymptotic exp”
Non‐vanishing of the symmetric square L‐function at the central point (2008, neutral): “, andJ k−1(x) ∼ ( π 2 x) −1/2 cos(x− π 2 k+ π 4 ) for x ≥ 2+ǫ. The long term behavior of the J-Bessel function manifests itself when we average over k. The following result is taken from chapter 5 of [11] where it is derived by Poisson summation. Lemma 2.6. We have for t ≥ 0, X k≡0 mod 2 2ikh k −1 K”
More than 41% of the zeros of the zeta function are on the critical line (2010, methodology): “The Voronoi formula (see Exercise 7 of [IK] p.”