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Higher correlations of divisor sums related to primes, II: Variations of . . .
, 2007
"... We calculate the triple correlations for the truncated divisor sum λR(n). The λR(n) behave over certain averages just as the prime counting von Mangoldt function Λ(n) does or is conjectured to do. We also calculate the mixed (with a factor of Λ(n)) correlations. The results for the moments up to the ..."
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Cited by 28 (6 self)
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We calculate the triple correlations for the truncated divisor sum λR(n). The λR(n) behave over certain averages just as the prime counting von Mangoldt function Λ(n) does or is conjectured to do. We also calculate the mixed (with a factor of Λ(n)) correlations. The results for the moments up to the third degree, and therefore the implications for the distribution of primes in short intervals, are the same as those we obtained (in the first paper with this title) by using the simpler approximation ΛR(n). However, when λR(n) is used, the error in the singular series approximation is often much smaller than what ΛR(n) allows. Assuming the Generalized Riemann Hypothesis (GRH) for Dirichlet Lfunctions, we obtain an Ω±result for the variation of the error term in the prime number theorem. Formerly, our knowledge under GRH was restricted to Ωresults for the absolute value of this variation. An important ingredient in the last part of this work is a recent result due to Montgomery and Soundararajan which makes it possible for us to dispense with a large error term in the evaluation of a certain singular series average. We believe that our results on the sums λR(n) and ΛR(n) can be employed in diverse problems concerning primes.
Cruel and unusual behavior of the Riemann zeta function
, 1998
"... We exhibit a sequence cn such that the convergence of P n1 cnz n for jzj ! 1 is equivalent to the Riemann Hypothesis. Numerical investigation of the cn revealed some astonishingly deceptive behavior. Keywords  Riemann zeta function, analyticity. The Riemann zeta function is defined by i(s ..."
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Cited by 1 (0 self)
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We exhibit a sequence cn such that the convergence of P n1 cnz n for jzj ! 1 is equivalent to the Riemann Hypothesis. Numerical investigation of the cn revealed some astonishingly deceptive behavior. Keywords  Riemann zeta function, analyticity. The Riemann zeta function is defined by i(s) = X n1 1 n s (1) if !(s) ? 1, and by analytic continuation for other complex s. i(s) has a simple pole at s = 1, but is analytic everywhere else in the complex s plane. The "critical line" is !(s) = 1=2. The "Riemann Hypothesis," which is one of the most important open problems in mathematics, is the conjecture that all of the zeroes of i(s) in the region !(s) 1=2 lie on the critical line. A tremendous amount of information about i(s) and about various statements implied by, implying, or equivalent to the Riemann Hypothesis, or about the known evidence and partial results on it, may be found in the references. A very simple attack on the Riemann Hypothesis occurred to me. Observe th...
Modular Invariance
, 2009
"... Two methods to prove the Riemann Hypothesis are presented. One is based on the modular properties of Θ (theta) functions and the other on the Hilbert–Polya proposal to find an operator whose spectrum reproduces the ordinates ρn (imaginary parts) of the zeta zeros in the critical line: sn = 1 +iρn. A ..."
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Two methods to prove the Riemann Hypothesis are presented. One is based on the modular properties of Θ (theta) functions and the other on the Hilbert–Polya proposal to find an operator whose spectrum reproduces the ordinates ρn (imaginary parts) of the zeta zeros in the critical line: sn = 1 +iρn. A detailed analysis of a onedimensional Diraclike 2 operator with a potential V (x) is given that reproduces the spectrum of energy levels En = ρn, when the boundary conditions ΨE(x = −∞) =±ΨE(x =+∞) areimposed. Such potential V (x) is derived implicitly from the relation x = x(V) = π (dN (V)/dV), 2 where the functional form of N (V) is given by the fullfledged Riemann–von Mangoldt counting function of the zeta zeros, including the fluctuating as well as the O(E−n) terms. The construction is also extended to selfadjoint Schroedinger operators. Crucial is the introduction of an energydependent cutoff function Λ(E). Finally, the natural quantization of the phase space areas (associated to nonperiodic crystallike structures) in integer multiples of π follows from the Bohr–Sommerfeld quantization conditions of Quantum Mechanics. It allows to find a physical reasoning why the average density of the primes distribution for very large x(O ( 1)) has a onetoone correspondence with log x the asymptotic limit of the inverse average density of the zeta zeros in the critical line suggesting intriguing connections to the renormalization group program.
YikMan Chiang ‡∗
, 2006
"... Abstract. It is proved that the Riemann zeta function does not satisfy any nontrivial algebraic difference equation whose coefficients are meromorphic functions φ with Nevanlinna characteristic satisfying T(r,φ) = o(r) as r → ∞. ..."
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Abstract. It is proved that the Riemann zeta function does not satisfy any nontrivial algebraic difference equation whose coefficients are meromorphic functions φ with Nevanlinna characteristic satisfying T(r,φ) = o(r) as r → ∞.