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101 0 $aeng
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200 10$aReassessing Riemann's paper $eon the number of primes less than a given magnitude /$fWalter Dittrich
205   $aSecond edition.
210  1$aCham, Switzerland :$cSpringer,$d[2018]
210  4$d©2018
215   $a1 online resource (XI, 107 p. 18 illus., 10 illus. in color.) 
225 1 $aSpringerBriefs in History of Science and Technology
311   $a3-030-61048-9 
327   $aPreface -- Introduction -- Short Biography of Bernhard Riemann (1826 ? 1866) -- Towards Euler's Product Formula and Riemann?s Extension of the Zeta Function -- Prime Power Number Counting Function -- Riemann as an Expert in Fourier Transforms -- On the Way to Riemann?s Entire Function ?(s) -- The Product Representation of ?(s) and ?(s) by Riemann (1859) -- Derivation of Von Mangoldt?s Formula for ?(x) -- The Number of Roots in the Critical Strip -- Riemann?s Zeta Function Regularization -- ?-Function Regularization of the Partition Function of the Harmonic Oscillator -- ?-Function Regularization of the Partition Function of the Fermi Oscillator -- The Zeta-Function in Quantum Electrodynamics (QED) -- Summary of Euler-Riemann Formulae -- Appendix.
330   $aIn this book, the author pays tribute to Bernhard Riemann (1826-1866), a mathematician with revolutionary ideas, whose work on the theory of integration, the Fourier transform, the hypergeometric differential equation, etc. contributed immensely to mathematical physics. The text concentrates in particular on Riemann?s only work on prime numbers, including ideas ? new at the time ? such as analytical continuation into the complex plane and the product formula for entire functions. A detailed analysis of the zeros of the Riemann zeta-function is presented. The impact of Riemann?s ideas on regularizing infinite values in field theory is also emphasized. This revised and enhanced new edition contains three new chapters, two on the application of Riemann?s zeta-function regularization to obtain the partition function of a Bose (Fermi) oscillator and one on the zeta-function regularization in quantum electrodynamics. Appendix A2 has been re-written to make the calculations more transparent. A summary of Euler-Riemann formulae completes the book.
410  0$aSpringerBriefs in history of science and technology.
606   $aNumber theory
615  0$aNumber theory.
676   $a512.7
700   $aDittrich$b Walter$046017
801  0$bMiAaPQ
801  1$bMiAaPQ
801  2$bUtOrBLW
906   $aBOOK
912   $a996466409103316
996   $aReassessing Riemann's Paper$91563734
997   $aUNISA