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The Riemann zeta function can also be defined in terms of by

Biane, P.; Pitman, J.; and Yor, M. "Probability Laws Related to the Jacobi Theta and Riemann Zeta Functions, and Brownian Excursions." 38, 435-465, 2001.

The Riemann zeta function satisfies the

The Riemann Zeta Search Project started at 2013. The aim of the Riemann Zeta Search Project is locating peak values of the zeta function on the critical line in order to have a better understanding of the distribution of prime […]

The Riemann zeta function can be split up into

The Riemann Zeta Search Project started at 2013. The aim of the Riemann Zeta Search Project is locating peak values of the zeta function on the critical line in order to have a better understanding of the distribution of prime […]

Prime numbers are the building blocks of mathematics and affects every part of life. It is well known that there are infinitely many primes, as proved by Euclid around 300 BC. In the 18th century Gauss conjectured that the number of primes up to x is asymptotically . Many new results has been achieved in the past century with respect to the prime numbers, but the distribution of primes remained unknown. The Riemann zeta function was introduced and studied by Leonhard Euler in the first half of the eighteenth century without using complex analysis. Let where . The Riemann zeta function is defined by

The Riemann zeta function is related to the and by

pdf A. Kovács, N Tihanyi, Efficient computing of n-dimensional simultaneous Diophantine approximation problems, Acta Univ. Sapientia Informatica, 5, 1 (2013) 16–34 pdf N. Tihanyi, Fast method for locating peak values of the Riemann-zeta function on the critical line, IEEE publication on […]

pdf A. Kovács, N Tihanyi, Efficient computing of n-dimensional simultaneous Diophantine approximation problems, Acta Univ. Sapientia Informatica, 5, 1 (2013) 16–34 pdf N. Tihanyi, Fast method for locating peak values of the Riemann-zeta function on the critical line, IEEE publication on […]

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The of the Riemann zeta function for is defined by

Srivastava, H. M. "Some Simple Algorithms for the Evaluations and Representations of the Riemann Zeta Function at Positive Integer Arguments." 246, 331-351, 2000.

de Branges, L. "Riemann Zeta Functions." May 24, 2004. .

Sondow, J. "Analytic Continuation of Riemann's Zeta Function and Values at Negative Integers via Euler's Transformation of Series." 120, 421-424, 1994.

Keiper, J. "The Zeta Function of Riemann." 4,5-7, 1995.

Borwein, J. M. and Bradley, D. M. "Searching Symbolically for Apéry-Like Formulae for Values of the Riemann Zeta Function." 30, 2-7, 1996.

Weisstein, E. W. "Books about Riemann Zeta Function." .

which is an extremely important function of mathematics and physics. In 1859 Bernhard Riemann conjectured that all nontrivial zeros of the Riemann zeta function have real part . This is the famous Riemann-hypothesis, one of the most important unsolved problem in the theory of prime numbers. The Riemann zeta function can be calculated on the critical line by using the Riemann-Siegel -function. The function can be calculated in time complexity of by

and "Riemann Zeta Function." From --A Wolfram Web Resource.

Abramowitz, M. and Stegun, I. A. (Eds.). "Riemann Zeta Function and Other Sums of Reciprocal Powers." §23.2 in New York: Dover, pp. 807-808, 1972.

Riemann zeta function - Wikipedia

The inverse of the Riemann zeta function , plotted above, is the asymptotic density of th-powerfree numbers (i.e., numbers, numbers, etc.). The following table gives the number of th-powerfree numbers for several values of .

called the Riemann Zeta function.

van de Lune, J.; te Riele, H. J. J.; and Winter, D. T. "On the Zeros of the Riemann Zeta Function in the Critical Strip. IV." 46, 667-681, 1986.

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