In this article, we give a characterization of nontrivial zeros of the Riemann zeta function using two real integrals. Using this characterization we can provide simple proof of the fact that the Riemann zeta function has no nontrivial real zeros. We also establish tha...
…) 为一阶极点,故 ζ(s) 以且仅以 s=−2n 为其一阶零点,这些零点称为平凡零点(trivial zeros),然而 ζ(s) 可能有其它的零点均为复零点,称为非平凡零点(non-trivial zeros),它们位于临界带 0≤σ≤1 内.事实上这些非平凡零点对实轴及 σ=12 对称(留给读者证明). 由式(9) 知(10)ζ(s)=A(s...
Guillera, J. (2013, July). Some sums over the non-trivial zeros of the Riemann zeta function. Retrieved from http://arxiv.org/ abs/1307.5723v7J. Guillera. Some sums over the non-trivial zeros of the Riemann zeta function. http://arxiv.org/abs/1307.5723v7, July 2013....
Zeros of the Riemann zeta function come in two different types. So-called "trivial zeros" occur at all negative even integers , , , ..., and "nontrivial zeros" occur at certain values of satisfying (1) for in the "critical strip" . In general, a nontrivial zero of is denoted ,...
(The Riemann zeta function is defined for all complex numbers—numbers of the form x + iy, where i = Square root of√−1—except for the line x = 1.) Riemann knew that the function equals zero for all negative even integers −2, −4, −6, … (so-called trivial zeros), ...
复平面上使 Riemann ζ 函数取值为零的点被称为 Riemann ζ 函数的零点。s=-2n (n 为正整数) 是 Riemann ζ 函数的零点。这些零点被称为 Riemann ζ 函数的平凡零点 (trivial zeros)。除了这些平凡零点外,Riemann ζ 函数其它零点被称为非平凡零点 (non-trivial zeros)。
Zeros of come in (at least) two different types. So-called "trivial zeros" occur at all negative even integers , , , ..., and "nontrivial zeros" at certain (26) for in the "critical strip" . The Riemann hypothesis asserts that the nontrivial Riemann zeta function zeros of all ha...
On the zeros of Riemann's zeta-function on the critical line We combine the mollifier method with a zero detection method of Atkinson to prove in a new way that a positive proportion of the nontrivial zeros of the Ri... Baluyot,AC Siegfred - 《Journal of Number Theory》 被引量: 24发...
Numerical work has verified that the first 300×109 nontrivial zeros of ζ(z) are simple and indeed fall on the critical line. See J. Van de Lune, H. J. J. Te Riele, and D. T. Winter, “On the zeros of the Riemann zeta function in the critical strip. IV," Math. Comput. 47...
In summary, the Zeta Riemann Function is a mathematical function defined as the sum of a series involving positive integers and complex numbers. The trivial zeros of this function are the negative even numbers. This is a consequence of the functional equation of the Zeta-function, which also ...