By Chaohua Jia, Kohji Matsumoto

Contains a number of survey articles on leading numbers, divisor difficulties, and Diophantine equations, in addition to learn papers on a number of points of analytic quantity concept difficulties.

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2n} with cardinality M = n + 1. i) Show that there are at least two elements in M which are coprime. ii) Prove that there are at least two elements in M such that one divides the other. Hint: Use the pigeonhole principle. * Given any positive function ψ(q) deﬁned on N, prove the existence of an irrational number α for which the inequality α− has inﬁnitely many solutions p q p < ψ(q) q ∈ Q. ∞ Hint: Consider numbers of the form α = n=1 (−1)n 2−mn and try to ﬁnd a sequence of integers mn depending on the values of ψ at the integers 2mn .

1 below. * Show that c p > 2 q q log q ∈ Q with q > 1 and some positive constant c. e− for all p q CHAPTER 4 The irrationality of ζ(3) The Riemann zeta-function ζ(s) is an important object in analytic number theory. The strongest version of the prime number theorem is proved by analyzing the zeta-function as a complex function; in particular, its mysterious zero distribution is an active ﬁeld of research with plenty of open questions. Also the behavior on the real axis is not yet understood very well.

An ] = a0 + [a1 , . . , an ] For n ≤ m we call [a0 , a1 , . . , an ] the nth convergent to [a0 , a1 , . . , am ]. 3) ⎩ q−1 = 0, q0 = 1, and qn = an qn−1 + qn−2 . 1. For 0 ≤ n ≤ m, the functions pn , qn satisfy pn = [a0 , a1 , . . , an ]. qn Proof by induction on n. The cases n = 0 is obvious. The case n = 1 is easily computed by a1 a0 + 1 p1 [a0 , a1 ] = = . a1 q1 Now assume that the formula in question holds for n. 2) 1 . an+1 By the recursion formulae for the pn , qn , the latter expression is equal to [a0 , a1 , .