Number Theory

# Analytic Number Theory [lecture notes] by Jan-Hendrik Evertse PDF By Jan-Hendrik Evertse

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1) A(t)g (t)dt k=1 xk xr k=1 = A(xr )g(xr ) − A(t)g (t)dt. x1 In case that x1 = M ,xr = N we are done. if x1 > M , then A(t) = 0 for M t < x1 x and thus, M1 A(t)g (t)dt = 0. If xr < N , then A(t) = A(xr ) for xr t N , hence N A(t)g (t)dt = A(N )g(N ) − A(xr )g(xr ). 1) this implies our Theorem. 2. Let f : Z>0 → C be an arithmetic function with the property that there exists a constant C > 0 such that | N C for every N 1. Then n=1 f (n)| ∞ −s Lf (s) = n=1 f (n)n converges for every s ∈ C with Re s > 0.

In the 1930’s, Wiener and Ikehara proved a general so-called Tauberian theorem (from functional analysis) which implies the Prime Number Theorem in a very simple manner. In 1948, Erd˝os and Selberg independently found an elementary proof, “elementary” meaning that the proof avoids complex analysis or functional analysis, but definitely not that the proof is easy! In 1980, Newman gave a new, simple proof of the Prime Number 28 Theorem, based on complex analysis. Korevaar observed that Newman’s approach can be used to prove a simpler version of the Wiener-Ikehara Tauberian theorem with a not so difficult proof based on complex analysis alone and avoiding functional analysis.

Iz cos z = (e + e −iz (−1)n )/2 = n=0 ∞ sin z = (eiz − e−iz )/2i = ∞ z 2n , (2n)! R = ∞, cos z = − sin z. z 2n+1 , (2n + 1)! R = ∞, sin z = cos z. (−1)n n=0 α n (1 + z)α = z , R = 1, n n=0 α α(α − 1) · · · (α − n + 1) where α ∈ C, = . n n! ∞ (−1)n−1 n log(1 + z) = ·z , R = 1, n n=1 36 ((1 + z)α ) = α(1 + z)α−1 log (1 + z) = (1 + z)−1 . 2 Cauchy’s Theorem and some applications For the necessary definitions concerning paths, line integrals and topology we refer to the Prerequisites. In the remainder of this course, a path will always be a piecewise continuously differentiable path.