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Of what use are the zeros of the Riemann zeta function? We can use sums involving zeta zeros to count the primes up to $x$. Perrons formula leads to sums over zeta zeros that can count the squarefree integers up to $x$, or tally Eulers $phi$ function and other arithmetical functions. This is largely a presentation of experimental results.
We prove an analogue of Kroneckers second limit formula for a continuous family of indefinite zeta functions. Indefinite zeta functions were introduced in the authors previous paper as Mellin transforms of indefinite theta functions, as defined by Zw
In this paper we give the q-extension of Euler numbers which can be viewed as interpolating of the q-analogue of Euler zeta function ay negative integers, in the same way that Riemann zeta function interpolates Bernoulli numbers at negative integers.
In 2008 I thought I found a proof of the Riemann Hypothesis, but there was an error. In the Spring 2020 I believed to have fixed the error, but it cannot be fixed. I describe here where the error was. It took me several days to find the error in a ca
The Hardy hypothesis, as an analogue to the Riemann hypothesis for the Riemann zeta function, is a conjecture proposed by Hardy in 1940, that all of the nontrivial zeros for the Ramanujan zeta function have a real part equal to 6. In this paper, we p
The Collatz Conjecture (also known as the 3x+1 Problem) proposes that the following algorithm will, after a certain number of iterations, always yield the number 1: given a natural number, multiply by three and add one if the number is odd, halve the