Dubnium is a synthetic chemical element with the symbol Db and atomic number 105. It is highly radioactive: the most stable known isotope, dubnium-268, has a half-life of about 16 hours. This greatly limits extended research on the element.

Dubnium does not occur naturally on Earth and is produced artificially. The Soviet Joint Institute for Nuclear Research (JINR) claimed the first discovery of the element in 1968, followed by the American Lawrence Berkeley Laboratory in 1970. Both teams proposed their names for the new element and used them without formal approval. The long-standing dispute was resolved in 1993 by an official investigation of the discovery claims by the Transfermium Working Group, formed by the International Union of Pure and Applied Chemistry and the International Union of Pure and Applied Physics, resulting in credit for the discovery being officially shared between both teams. The element was formally named dubnium in 1997 after the town of Dubna, the site of the JINR.

Theoretical research establishes dubnium as a member of group 5 in the 6d series of transition metals, placing it under vanadium, niobium, and tantalum. Dubnium should share most properties, such as its valence electron configuration and having a dominant +5 oxidation state, with the other group 5 elements, with a few anomalies due to relativistic effects. A limited investigation of dubnium chemistry has confirmed this.

https://en.wikipedia.org/wiki/Dubnium

https://en.wikipedia.org/wiki/Hilbert%27s_problems

https://en.wikipedia.org/wiki/Paul_Cohen

https://en.wikipedia.org/wiki/Dirichlet_integral

https://en.wikipedia.org/wiki/Hyperbolic_geometry

The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ζ (zeta), is a mathematical function of a complex variable defined as

{\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots }for {\displaystyle \operatorname {Re} (s)>1}

 and its analytic continuation elsewhere.

The Riemann zeta function plays a pivotal role in analytic number theory, and has applications in physics, probability theory, and applied statistics.

Leonhard Euler first introduced and studied the function over the reals in the first half of the eighteenth century. Bernhard Riemann‘s 1859 article “On the Number of Primes Less Than a Given Magnitude” extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution of prime numbers. This paper also contained the Riemann hypothesis, a conjecture about the distribution of complex zeros of the Riemann zeta function that many mathematicians consider the most important unsolved problem in pure mathematics.

The values of the Riemann zeta function at even positive integers were computed by Euler. The first of them, ζ(2), provides a solution to the Basel problem. In 1979 Roger Apéry proved the irrationality of ζ(3). The values at negative integer points, also found by Euler, are rational numbers and play an important role in the theory of modular forms. Many generalizations of the Riemann zeta function, such as Dirichlet series, Dirichlet L-functions and L-functions, are known.

https://en.wikipedia.org/wiki/Riemann_zeta_function