The Riemann zeta function is related to the and by
It is also known that the nontrivial zeros are symmetrically placed about the , a result which follows from the functional equation and the symmetry about the line . For if is a nontrivial zero, then is also a zero (by the functional equation), and then is another zero. But and are symmetrically placed about the line , since , and if , then . The Riemann hypothesis is equivalent to , where is the (Csordas 1994). It is also equivalent to the assertion that for some constant ,
The of the Riemann zeta function for is defined by
The first term in this last equation is always negative. The second term, denoted , is real and has the same zeros as the zeta function at , real. Thus locating zeros on the critical line of the (complex) zeta function reduces to locating zeros on the real line of the real function . The term in the previous equation may be expressed
In the novel (Jacoby 2008), the main character Theodore "Mead" Fegley (who is only 18 and a college senior) tries to prove the Riemann Hypothesis for his senior year research project. He also uses a Cray Supercomputer to calculate several billion zeroes of the Riemann zeta function. In several dream sequences within the book, Mead has conversations with Bernhard Riemann about the problem and mathematics in general.
Weisstein, E. W. "Books about Riemann Zeta Function." .
The Riemann zeta function may be computed analytically for using either or with the appropriate . An unexpected and important formula involving a product over the was first discovered by Euler in 1737,
and "Riemann Zeta Function." From --A Wolfram Web Resource.
from which values of the zeta function at can be computed from its values at . Using this equation one sees immediately that the zeta function is zero at the negative even integers. Multiplying this equation through by and applying standard factorial identities shows that
Riemann zeta function - Wikipedia
To date the methods above have been used to verify RH to large heights of the critical strip. Van de Lune et al. have shown that the first one and half billion non-trivial zeros lie on the critical line, while has demonstrated the validity of RH in large regions of the critical strip near the th zero. As part of my Master's project, I used these methods on my PC to show that the first twelve million or so zeros of the zeta function lie on the critical line (big surprise!)
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.
Riemann Zeta Function -- from Wolfram MathWorld
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 .
Locating large values of the Riemann zeta ..
André Weil proved the Riemann hypothesis to be true for field functions (Weil 1948, Eichler 1966, Ball and Coxeter 1987). In 1974, Levinson (1974ab) showed that at least 1/3 of the must lie on the (Le Lionnais 1983), a result which has since been sharpened to 40% (Vardi 1991, p. 142). It is known that the zeros are symmetrically placed about the line . This follows from the fact that, for all complex numbers ,