On the Zeros of the Riemann Zeta Function in the Critical Strip
نویسندگان
چکیده
We describe a computation which shows that the Riemann zeta function f(s) has exactly 75,000,000 zeros of the form a + it in the region 0 < t < 32,585,736.4; all these zeros are simple and lie on the line o = Hi. (A similar result for the first 3,500,000 zeros was established by Rosser, Yohe and Schoenfeld.) Counts of the number of Gram blocks of various types and the number of failures of "Rosser's rule" are
منابع مشابه
Distribution of the zeros of the Riemann Zeta function
One of the most celebrated problem of mathematics is the Riemann hypothesis which states that all the non trivial zeros of the Zeta-function lie on the critical line <(s) = 1/2. Even if this famous problem is unsolved for so long, a lot of things are known about the zeros of ζ(s) and we expose here the most classical related results : all the non trivial zeros lie in the critical strip, the num...
متن کاملA positive answer to the Riemann hypothesis: A new result predicting the location of zeros
In this paper, a positive answer to the Riemann hypothesis is given by using a new result that predict the exact location of zeros of the alternating zeta function on the critical strip.
متن کاملAt Least Two Fifths of the Zeros of the Riemann Zeta Function Are on the Critical Line
Of central importance in number theory is the distribution of the complex zeros of f(s), all of which are in the critical strip 0 < a < 1 and are symmetrically located about the real axis and about the critical line a = 1/2. Riemann conjectured in 1859 that all of these zeros are on the critical line; this conjecture, which is still unproved, is known as the Riemann Hypothesis. The number of ze...
متن کاملFinite Euler Products and the Riemann Hypothesis
Abstract. We show that if the Riemann Hypothesis is true, then in a region containing most of the right-half of the critical strip, the Riemann zeta-function is well approximated by short truncations of its Euler product. Conversely, if the approximation by products is good in this region, the zeta-function has at most finitely many zeros in it. We then construct a parameterized family of non-a...
متن کاملA Proof for the Density Hypothesis
The Riemann zeta function ζ(s) is defined by ζ(s) = ∑∞ n=1 1 ns for R(s) > 1 and may be extended to a regular function on the whole complex plane excluding its unique pole at s = 1. The Riemann hypothesis is a conjecture made by Riemann in 1859 asserting that all non-trivial zeros for ζ(s) lie on the line R(s) = 12 , which has a broad application in every branch of mathematics. The density hypo...
متن کامل