# The theory of the riemann zeta function

## The Theory of the Riemann Zeta-Function by E.C. Titchmarsh

The Riemann zeta-function embodies both additive and multiplicative structures in a single function, making it our most important tool in the study of prime numbers. This volume studies all aspects of the theory, starting from first principles and probing the functions own challenging theory, with the famous and still unsolved Riemann hypothesis at its heart. The second edition has been revised to include descriptions of work done in the last forty years and is updated with many additional references; it will provide stimulating reading for postgraduates and workers in analytic number theory and classical analysis.## Donate to arXiv

But the most beautiful in all mathematics is the zeta function. There is no doubt about it. These connections between the seemingly disparate worlds of quantum mechanics and number theory are tantalising. Riemann zeta function: resources " The zeta function is probably the most challenging and mysterious object of modern mathematics, in spite of its utter simplicity. Gutzwiller , Chaos in Classical and Quantum Mechanics Springer-Verlag " We may — paraphrasing the famous sentence of George Orwell — say that 'all mathematics is beautiful, yet some is more beautiful than the other'.

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Riemann zeta function , function useful in number theory for investigating properties of prime numbers. For values of x larger than 1, the series converges to a finite number as successive terms are added. If x is less than 1, the sum is again infinite. The zeta function was known to the Swiss mathematician Leonhard Euler in , but it was first studied extensively by the German mathematician Bernhard Riemann. In Riemann published a paper giving an explicit formula for the number of primes up to any preassigned limit—a decided improvement over the approximate value given by the prime number theorem. Riemann conjectured that all of the nontrivial zeros are on the critical line, a conjecture that subsequently became known as the Riemann hypothesis.

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