Multiplicative Number Theory I: Classical Theory

By H. Montgomery, Robert C. Vaughan. 2006

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Small gaps between prime numbers: The work of Goldston-Pintz-Yildirim (2006, methodology): “For more detailed accounts we refer the reader to the books of Bombieri [2], Davenport [6], and Montgomery and Vaughan [24].”
Bounds on the suprema of Gaussian processes, and omega results for the sum of a random multiplicative function (2010, background): “Using the estimate |log ζ (σ + it)| ≤ log log|t | + O(1), σ ≥ 1, |t | ≥ 2, which is contained in, for example, Montgomery and Vaughan [14], Theorem 6.7, we can say that for t ≥ 1, e ∑ p f (p) cos(t logp)/p 1/2+1/ logx−log t−log log(t+2)/2 √ logx sup z≥1 |M(z)|√ z + logx sup z≥e√logx |M(z)|√ z .”
Moments of random multiplicative functions, II: High moments (2018, background): “Harmonic Analysis Result 2.6 (See (5.26) in Section 5.1 of [Montgomery and Vaughan 2007]).”
Sharp conditional bounds for moments of the Riemann zeta function (2013, background): “See e.g. Theorem 9.14 in Montgomery and Vaughan’s book [10], noting that p is always a fundamental discriminant, and that χ d ( p ) = χ d ( p ) ǫ ( d, p ) in the notation of that theorem.”