Multiplicative Number Theory I

Abstract

Prime numbers are the multiplicative building blocks of natural numbers. Understanding their overall influence and especially their distribution gives rise to central questions in mathematics and physics. In particular their finer distribution is closely connected with the Riemann hypothesis, the most important unsolved problem in the mathematical world. Assuming only subjects covered in a standard degree in mathematics, the authors comprehensively cover all the topics met in first courses on multiplicative number theory and the distribution of prime numbers. They bring their extensive and distinguished research expertise to bear in preparing the student for intelligent reading of the more advanced research literature. This 2006 text, which is based on courses taught successfully over many years at Michigan, Imperial College and Pennsylvania State, is enriched by comprehensive historical notes and references as well as over 500 exercises.

A conjecture for the regularized fourth moment of Eisenstein series

“…Also we recall that ξ(2) = π 6 . Using Stirling's approximations |Γ(σ + it)| = e −π|t|/2 |t| σ− 1 2 √ 2π{1 + O(|t| −1 )} and Γ ′ Γ (s) = log s + O(|s| −1 ) valid in a fixed vertical strip when |t| → ∞, and classical estimates for the Riemann zeta-function on the edge of the critical strip [MoV,section 6.3], (4.7) (log t) −2/3 (log log t) −1/3 ≪ ζ(s) ≪ (log t) 2/3 and ζ ′ ζ (s) ≪ (log t) 2/3 (log log t) 1/3 for s = σ + it, 1 − σ ≪ (log t) −2/3 (log log t) −1/3 , one obtains first that ξ ′ ξ (1 ± 2T i) ≪ log T and then that the contribution of the terms in the last two lines in the formula (4.6) is O( log 2 T T 1/2 ). Therefore the contribution on the right-hand side of (4.6) coming from the regularization process is…”

Low-lying zeros of elliptic curve L-functions: Beyond the Ratios Conjecture

“…We can thus shift the contour of integration to the left until the line (s) = 1 4 + ε, since by the Riemann Hypothesis, the zeros of ζ(2s) all have real part at most 1 4 . We apply the estimates [IK,(5.20)], [MV,Thm. 13.23] and Lemma 2.1 together with the rapid decay of Mw(s) on vertical lines (rather than following the proof of [MV,Thm.…”

“…We apply the estimates [IK,(5.20)], [MV,Thm. 13.23] and Lemma 2.1 together with the rapid decay of Mw(s) on vertical lines (rather than following the proof of [MV,Thm. 13.24] directly), to obtain, for some C > 0,…”

Modular symbols, Eisenstein series, and congruences

“…This can be done using standard formulas for Dirichlet L-functions (which can be found in [11], chapter 10, or Section 3.3 of [16] where if j is odd,…”