Multiple Positivity and the Riemann Zeta-Function

q-Riemann zeta function

We consider the modified q-analogue of Riemann zeta function which is defined by ζq(s)= ∑∞ n=1(qn(s−1)/[n]s), 0< q < 1, s ∈ C. In this paper, we give q-Bernoulli numbers which can be viewed as interpolation of the above q-analogue of Riemann zeta function at negative integers in the same way that Riemann zeta function interpolates Bernoulli numbers at negative integers. Also, we will treat some...

Mollifying the Riemann Zeta-function

Supercomputers and the Riemann Zeta Function

The Riemann Hypothesis, which specifies the location of zeros of the Riemann zeta function, and thus describes the behavior of primes, is one of the most famous unsolved problems in mathematics, and extensive efforts have been made over more than a century to check it numerically for large sets of cases. Recently a new algorithm, invented by the speaker and A. Scho . . nhage, has been implement...

Noncommutative Geometry and the Riemann Zeta Function

According to my first teacher Gustave Choquet one does, by openly facing a well known unsolved problem, run the risk of being remembered more by one’s failure than anything else. After reaching a certain age, I realized that waiting “safely” until one reaches the end-point of one’s life is an equally selfdefeating alternative. In this paper I shall first look back at my early work on the classi...

Alan Turing and the Riemann Zeta Function

Turing encountered the Riemann zeta function as a student, and developed a life-long fascination with it. Though his research in this area was not a major thrust of his career, he did make a number of pioneering contributions. Most have now been superseded by later work, but one technique that he introduced is still a standard tool in the computational analysis of the zeta and related functions...