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Showing 1 to 20 of 30 for “"Riemann zeta function"”.
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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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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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The prime number theorem: Analytic and elementary proofs
… theorem and various results relating to the Riemann zeta function play a vital role. A weaker result than the prime number theorem is used for the proof, namely Chebyshev's theorem. The second proof is elementary in the sense that it involves no complex analysis. Instead, mainly …
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On the distribution of central values of Hecke L-functions
<p>Questions regarding the behavior of the Riemann zeta function on the critical line 1/2 + <em>it</em> can be naturally interpreted as questions regarding the family of <em>L</em>-functions over Q associated to the archimedian characters<em> ψ (k)</em> =<em> k -it</em> at the center point 1/2. …
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Lercho ir Selbergo dzeta funkcijų reikšmių pasiskirstymai /
… for the value distribution of Lerch and Selberg zeta-functions and computer calculations performed using the computational software program MATHEMATICA. The dissertation consists of the introduction, 3 chapters, the conclusions and the references. The results of the thesis are published in three …
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A Bound for the Irrationality Measure of & zeta(3)
… an upper bound for the irrationality measure of &zeta(3), where &zeta denotes the Riemann zeta function. Results are presented in three steps: a linear combination of 1 and &zeta(3) depending on certain parameters is expressed as a triple integral, a description of the group of permutations of the …
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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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Higher Derivatives of the Hurwitz Zeta Function
<p>The Riemann zeta function ζ(s) is one of the most fundamental functions in number theory. Euler demonstrated that ζ(s) is closely connected to the prime numbers and Riemann gave proofs of the basic analytic properties of the zeta function. Values of the zeta function and its derivatives have …
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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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Extensions of Selberg-Delange method
… in Chapter $4$, is a new identity for Multiple Zeta Values. The Selberg-Delange method is a method that is widely used to determine the asymptotic behavior of the sum of arithmetic functions whose corresponding Dirichlet's series can be written in the term of the Riemann zeta function, …
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Analysis in Fractional Calculus and Asymptotics Related to Zeta Functions
… The most commonly used is the classical Riemann-Liouville model, but others may be preferred in different situations, and now new fractional models are being proposed and developed each year. This creates many opportunities for new research, since each time a model is proposed, its …
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L-functions in Number Theory
As a generalization of the Riemann zeta function, L-function has become one of the central objects in Number Theory. The theory of L-functions, which produces a large family of consequences and conjectures in a unified way, concerns their zeros and poles, functional equations, special values and …
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Distribution of sequences related to L-functions
… of linear combinations of derivatives of $L$-functions. We consider a collection of such combinations and prove asymptotic formulas for the supremum of the real parts of their zeros. Moreover, an investigation of an inverse-type question related to the case of the Riemann zeta function is …
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The stochastic operator approach to random matrix theory
… quantum mechanics, and even the zeros of the Riemann zeta function. However, their eigenvectors are Haar-distributed-completely random. Therefore, these classical random matrices are rarely considered as operators. The stochastic operator approach to random matrix theory, introduced here, …
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Explicit Formula for the Mean Square of Dirichlet L-Functions to Prime Power Moduli
<p>Individual values of L-functions inside the critical strip 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 …
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Visibility of lattice points in high dimensional spaces and extreme values of combinations of Dirichlet L-functions
… we examine a `short' partial sum of the Riemann Zeta function, $\zeta_N(s) = \sum_{n\le N}\frac{1}{n^s}$. We obtain large values of $\zeta_N\left(\frac{1}{2}+it\right)$ for $t\in[\sqrt{T}, T]$ when $N\le (\log T)^\lambda$ with $0<\lambda<\sqrt{e}$. We also adapt the methods by K. …
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Distribution of Farey Series and Free Path Lengths for a Certain Billiard in the Unit Square
… sums into expressions involving the Mobius function. In 1924 Franel produced a quantitative form of this equivalence as an identity for the sum of the squares of the values of the local discrepancy function of the Mobius function and related this to the real parts of zeros of the Riemann …
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