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Disproof of the Mertens conjecture
 J. REINE ANGEW. MATH
, 1985
"... The Mertens conjecture states that ⎪ M(x) ⎪ < x 1 ⁄ 2 for all x> 1, where M(x) = Σ μ(n), n ≤ x and μ(n) is the Möbius function. This conjecture has attracted a substantial amount of interest in its almost 100 years of existence because its truth was known to imply the truth of the Riemann hypothesi ..."
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Cited by 29 (3 self)
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The Mertens conjecture states that ⎪ M(x) ⎪ < x 1 ⁄ 2 for all x> 1, where M(x) = Σ μ(n), n ≤ x and μ(n) is the Möbius function. This conjecture has attracted a substantial amount of interest in its almost 100 years of existence because its truth was known to imply the truth of the Riemann hypothesis. This paper disproves the Mertens conjecture by showing that lim sup M(x) x x → ∞ − 1 ⁄ 2> 1. 06. The disproof relies on extensive computations with the zeros of the zeta function, and does not provide an explicit counterexample.
How to Stretch Random Functions: The Security of Protected Counter Sums
 Journal of Cryptology
, 1999
"... . Let f be an unpredictable random function taking (b + c)bit inputs to bbit outputs. This paper presents an unpredictable random function f 0 taking variablelength inputs to bbit outputs. This construction has several advantages over chaining, which was proven unpredictable by Bellare, Ki ..."
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Cited by 19 (7 self)
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. Let f be an unpredictable random function taking (b + c)bit inputs to bbit outputs. This paper presents an unpredictable random function f 0 taking variablelength inputs to bbit outputs. This construction has several advantages over chaining, which was proven unpredictable by Bellare, Kilian, and Rogaway, and cascading, which was proven unpredictable by Bellare, Canetti, and Krawczyk. The highlight here is a very simple proof of security. 1.
Number Theory, Dynamical Systems and Statistical Mechanics
, 1998
"... We shortly review recent work interpreting the quotient ζ(s − 1)/ζ(s) of Riemann zeta functions as a dynamical zeta function. The corresponding interaction function (Fourier transform of the energy) has been shown to be ferromagnetic, i.e. positive. On the additive group we set inductively Gk: = (Z/ ..."
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Cited by 8 (2 self)
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We shortly review recent work interpreting the quotient ζ(s − 1)/ζ(s) of Riemann zeta functions as a dynamical zeta function. The corresponding interaction function (Fourier transform of the energy) has been shown to be ferromagnetic, i.e. positive. On the additive group we set inductively Gk: = (Z/2Z) k, with Z/2Z = ({0, 1}, +). h0: = 1, hk+1(σ, 0): = hk(σ) and hk+1(σ, 1): = hk(σ) + hk(1 − σ), (1) where σ = (σ1,..., σk) ∈ Gk and 1 − σ: = (1 − σ1,..., 1 − σk) is the inverted configuration. The sequences hk(σ) of integers, written in lexicographic order, coincide with the denominators of the modified Farey sequence. We now formally interpret σ ∈ Gk as a configuration of a spin chain with k spins and energy function Hk: = ln(hk). Thus we may interpret
The numbertheoretical spin chain and the Riemann zeroes
 Comm. Math. Phys
, 1998
"... Abstract It is an empirical observation that the Riemann zeta function can be well approximated in its critical strip using the NumberTheoretical Spin Chain. A proof of this would imply the Riemann Hypothesis. Here we relate that question to the one of spectral radii of a family of Markov chains. T ..."
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Cited by 7 (1 self)
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Abstract It is an empirical observation that the Riemann zeta function can be well approximated in its critical strip using the NumberTheoretical Spin Chain. A proof of this would imply the Riemann Hypothesis. Here we relate that question to the one of spectral radii of a family of Markov chains. This in turn leads to the question whether certain graphs are Ramanujan. The general idea is to explain the pseudorandom features of certain numbertheoretical functions by considering them as observables of a spin chain of statistical mechanics. In an Appendix we relate the free energy of that chain to the Lewis Equation of modular theory. 1 Introduction The Euler
The Collatz 3n+1 Conjecture is Unprovable
, 2006
"... Disclaimer: This article was authored by Craig Alan Feinstein in his private capacity. No official support or endorsement by the U.S. Government is intended or should be inferred. In this note, we consider the following function: Definition 1: Let T: N → N be a function such that T(n) = 3n+1 2 if n ..."
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Disclaimer: This article was authored by Craig Alan Feinstein in his private capacity. No official support or endorsement by the U.S. Government is intended or should be inferred. In this note, we consider the following function: Definition 1: Let T: N → N be a function such that T(n) = 3n+1 2 if n is odd and T(n) = n 2 if n is even. The Collatz 3n + 1 Conjecture states that for each n ∈ N, there exists a k ∈ N such that T (k)(n) = 1, where T (k)(n) is the function T iteratively applied k times to n. As of September 4, 2003, the conjecture has been verified for all positive integers up to 224 × 250 ≈ 2.52 × 1017 (Roosendaal, 2003+). Furthermore, one can give a heuristic probabilistic argument (Crandall, 1978) that since every iterate of the function T decreases on average by a multiplicative factor of about ( 3 2)1/2 ( 1 2)1/2 = ( 3
Complexity Theory for Simpletons
, 2005
"... Abstract: In this article, we shall describe some of the most interesting topics in the subject of Complexity Theory for a general audience. Anyone with a solid foundation in high school mathematics (with some calculus) and an elementary understanding of computer programming will be able to follow t ..."
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Abstract: In this article, we shall describe some of the most interesting topics in the subject of Complexity Theory for a general audience. Anyone with a solid foundation in high school mathematics (with some calculus) and an elementary understanding of computer programming will be able to follow this article. First, we shall describe the P versus NP problem and its significance. Next, we shall describe two other famous mathematics problems, the Collatz 3n + 1 Conjecture and the Riemann Hypothesis, and show how the notion of “computational irreducibility ” is important for understanding why no one has, as of yet, solved these two problems. Disclaimer: This article was authored by Craig Alan Feinstein in his private capacity. No official support or endorsement by the U.S. Government is intended or should be inferred. 1 1