Titlebook: Arithmetic Tales; Olivier Bordellès Textbook 20121st edition Springer-Verlag London 2012 algebraic number fields.asymptotics for arithmeti

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Arithmetic Functions, Further Developments is devoted to a complete study of Dirichlet series from an arithmetic viewpoint and we also provide some estimates for other types of summation, such as multiplicative functions over short intervals or additive functions. Finally, a brief account of Selberg’s sieve and the large sieve is also given.

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Writing Simple .NET Applications,provide a proof of the PNT as a consequence of deep estimates of . near the line .=1 and summation formulae. It is also the opportunity to provide explicit estimates of the classic results, yet quite rare in the literature, and to explore some of the consequences of the famous Riemann hypothesis.

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,Bézout and Gauss,n particular Diophantine problems. The section Further Developments investigates the number of integer solutions of certain linear Diophantine equations, i.e. the number of certain restricted partitions of an integer.

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Prime Numbers,o paved the way for all branches of modern number theory. After recalling the basic tools essentially due to Euclid, we investigate Chebyshev’s reasoning in his attempt to give a proof of the Prime Number Theorem. The latter will finally be shown with the theory of functions building on Riemann’s id

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Arithmetic Functions, of integer factorizations. The text is aimed at introducing the Dirichlet convolution product, thus giving a ring structure to the set of arithmetic functions, and then establishing some useful summation results for multiplicative functions with the help of the Möbius inversion formula. The section

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Integer Points Close to Smooth Curves,asic results and some refinements of the theory. Some criteria are investigated and the theorem of Huxley and Sargos is studied in detail. In the section Further Developments, we prove a particular case of a general theorem given by Filaseta and Trifonov improving on the distribution of squarefree n

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Exponential Sums,al sums. In the early 1920s and 1930s, three different schools of thought investigated this problem. Following the lines of van der Corput, we provide the first criteria based upon the second and third derivatives of the studied function, and we apply them to the Dirichlet divisor problem. Many ques

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Algebraic Number Fields,ertain Diophantine equations. After recalling basic concepts from algebra and providing some polynomial irreducibility tools, the ring of integers . of an algebraic number field . is investigated. Next, the ., as Kummer called them, are introduced to restore unique factorization. The last section sh