LEADER 04424nam 22005535 450 001 996466416403316 005 20240215150112.0 010 $a9783030731694$b(electronic bk.) 010 $z9783030731687 024 7 $a10.1007/978-3-030-73169-4 035 $a(MiAaPQ)EBC6904035 035 $a(Au-PeEL)EBL6904035 035 $a(CKB)21348624400041 035 $a(DE-He213)978-3-030-73169-4 035 $a(PPN)261518879 035 $a(EXLCZ)9921348624400041 100 $a20220303d2022 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aExcursions in Multiplicative Number Theory$b[electronic resource] /$fby Olivier Ramaré 205 $a1st ed. 2022. 210 1$aCham :$cSpringer International Publishing :$cImprint: Birkhäuser,$d2022. 215 $a1 online resource (342 pages) 225 1 $aBirkhäuser Advanced Texts Basler Lehrbücher,$x2296-4894 311 08$aPrint version: Ramaré, Olivier Excursions in Multiplicative Number Theory Cham : Springer International Publishing AG,c2022 9783030731687 320 $aIncludes bibliographical references and index. 327 $aApproach: Multiplicativity -- Arithmetic Convolution -- A Calculus on Arithmetical Functions -- Analytical Dirichlet Series -- Growth of Arithmetical Functions -- An "Algebraical" Multiplicative Function -- Möbius Inversions -- The Convolution Walk -- Handling a Smooth Factor -- The Convolution Method -- Euler Products and Euler Sums -- Some Practice -- The Hyperbola Principle -- The Levin-Fanleib Walk -- The Mertens Estimates -- The Levin-Fanleib Theorem -- Variations on a Theme of Chebyshev -- Primes in progressions -- A famous constant -- Euler Products with Primes in AP -- Chinese Remainder and Multiplicativity -- The Mellin Walk -- The Riemann zeta-function -- The Mellin Transform -- Proof Theorem ? -- Roughing up: Removing a Smoothening -- Proving the Prime Number Theorem -- Higher Ground: Applications / Extensions -- The Selberg Formula -- Rankin's Trick and Brun's Sieve -- Three Arithmetical Exponential Sums -- Convolution method / Möbius function -- The Large Sieve Inequality -- Montgomery's Sieve. 330 $aThis textbook offers a unique exploration of analytic number theory that is focused on explicit and realistic numerical bounds. By giving precise proofs in simplified settings, the author strategically builds practical tools and insights for exploring the behavior of arithmetical functions. An active learning style is encouraged across nearly three hundred exercises, making this an indispensable resource for both students and instructors. Designed to allow readers several different pathways to progress from basic notions to active areas of research, the book begins with a study of arithmetic functions and notions of arithmetical interest. From here, several guided ?walks? invite readers to continue, offering explorations along three broad themes: the convolution method, the Levin?Fa?nle?b theorem, and the Mellin transform. Having followed any one of the walks, readers will arrive at ?higher ground?, where they will find opportunities for extensions and applications, such as the Selberg formula, Exponential sums with arithmetical coefficients, and the Large Sieve Inequality. Methodology is emphasized throughout, with frequent opportunities to explore numerically using computer algebra packages Pari/GP and Sage. Excursions in Multiplicative Number Theory is ideal for graduate students and upper-level undergraduate students who are familiar with the fundamentals of analytic number theory. It will also appeal to researchers in mathematics and engineering interested in experimental techniques in this active area. 410 0$aBirkhäuser Advanced Texts Basler Lehrbücher,$x2296-4894 606 $aNumber theory 606 $aNumber Theory 606 $aTeoria de nombres$2thub 606 $aMultiplicació$2thub 608 $aLlibres electrònics$2thub 615 0$aNumber theory. 615 14$aNumber Theory. 615 7$aTeoria de nombres 615 7$aMultiplicació 676 $a512.7 700 $aRamare?$b Olivier$01209737 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 912 $a996466416403316 996 $aExcursions in Multiplicative Number Theory$92791488 997 $aUNISA