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Showing 1 to 18 of 18 for “"riemann hypothesis"”.
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On the Theory of Zeta-functions and L-functions
… we provide a body of knowledge that concerns Riemann zeta-function and its generalizations in a cohesive manner. In particular, we have studied and mentioned some recent results regarding Hurwitz and Lerch functions, as well as Dirichlet's L-function. We have also investigated some fundamental …
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Zeros of the Dedekind Zeta-Function
… sums over two sets of nontrivial zeros of the Riemann zeta-function. Assuming the Riemann Hypothesis, he used this formula and Fourier analysis to prove an estimate for the proportion of simple zeros of the Riemann zeta-function. We prove a generalization of his formula for the nontrivial zeros …
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Counterexamples related to the Sato-Tate conjecture
… statistic. This conjecture implies the Riemann hypothesis for all $L$-functions associated with $E$. It is natural to assume that the converse (``generalized Riemann hypothesis implies discrepancy estimate'') holds, as is suggested by analogy with Artin $L$-functions. We construct, for …
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On Colin de Verdière's Proof of the Meromorphic Continuation of Eisenstein Series
… result as it relates to ideas relating to the Riemann hypothesis and offer a timeline of other results that influenced this eventual proof.
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AN EXPOSITION OF THE RIEMANN ZETA FUNCTION
This thesis is an exposition of the Riemann zeta function. Included are techniques of analytic continuation and relationships to special functions. Some generalizations of the Riemann zeta function are outlined, as well as the calculation of zeta constants and the development of some identities. …
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THE EFFECTIVE SATO-TATE CONJECTURE AND DENSITIES PERTAINING TO LEHMER-TYPE QUESTIONS
… f(z) are automorphic and satisfy the Generalized Riemann Hypothesis. We use this result to compute the density of positive integers n for which the n-th Fourier coefficient of f(z) is nonzero.
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Some conditional results involving arithmetic functions
… von Mangoldt and Mobius functions under the Riemann hypothesis, and we also present a strong converse. Second, we investigate a discrete negative moment of the zeta function, obtaining a lower bound that supports a previously conjectured estimate. Both results rely on the use of arithmetic …
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Limit Theorems for L-functions in Analytic Number Theory
… for the real part of the logarithm of the Riemann zeta function on the critical line in the multivariate case. This gives an alternate proof of a result of Bourgade. An upshot of the method is to determine a rate of convergence in the sense of the Dudley distance. This is the same rate …
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Distribution of sequences related to L-functions
… inverse-type question related to the case of the Riemann zeta function is included. In the second part of this thesis, we expand the class of Dirichlet series whose monotonicity properties are known. In particular, we describe a large class of Dirichlet series that are not logarithmically …
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A relatively small turing machine whose behavior is independent of set theory
… Turing machine R that halts if and only if the Riemann hypothesis is false. In the process of creating G, R, and Z, I define a higher-level language, TMD, which is much more convenient than direct state manipulation, and explain in great detail the process of compiling this language down to a …
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Explicit Formula for the Mean Square of Dirichlet L-Functions to Prime Power Moduli
… have great arithmetic significance (such as the Riemann hypothesis) but are presently not well-understood. In several important problems in analytic number theory, including how primes are distributed, estimates such as power moments provide critical input about L-functions. We derive an explicit …
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Moments Of Products Of L-Functions
… spacing. Finally, assuming the Generalized Riemann Hypothesis and some standard conjectures, we prove upper bounds for moments of arbitrary products of automorphic L-functions and for Dedekind zeta-functions of Galois number fields on the critical line. As an application, we use these bounds …
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Distribution of Farey Series and Free Path Lengths for a Certain Billiard in the Unit Square
… related this to the real parts of zeros of the Riemann zeta-function. In 1971 Huxley obtained a vast generalization of Franel's theorem for the zeros of a fixed Dirichlet L-function. In light of the difficulties surrounding the Riemann Hypothesis, in 1973 Niederreiter showed that it is, …
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Topics in Multiplicative and Probabilistic Number Theory
… unconditionally and assuming the Generalized Riemann Hypothesis, and show that our conditional results are nearly best possible unconditionally. We also prove a quantitative bivariate Erdo&huml;s-Kac theorem on the joint distribution of pairs of values of certain additive functions, and use …
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Topics in number theory
… is concerned mainly with topics in the theory of Riemann’s zeta function, but it also includes some contributions to prime number theory and the study of the Möbius function. New conditions are stated for the validity of the quasi-Riemann hypothesis RH (σ₀), that ζ(s) ≠ O for σ > σ₀. The orders …
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Partition asymptotics; zeros of zeta functions; and Apéry-like numbers
… with B. C. Berndt and A. Zaharescu. PART II The Riemann Hypothesis implies that the zeros of all the derivatives of the Riemann-_ function lie on the critical line. Results on the proportion of zeros on the critical line of derivatives of _(s) have been investigated before by B. Conrey, and I. …