DIRICHLET TRUNCATIONS OF THE RIEMANN ZETA FUNCTION IN THE CRITICAL STRIP POSSESS ZEROS NEAR EVERY VERTICAL LINE

We study the zeros of the finite truncations of the alternating Dirichlet series expansion of the Riemann zeta function in the critical strip. We do this with an (admittedly highly) ambitious goal in mind. Namely, that this series converges to the zeta function (up to a trivial term) in the critical strip and our hope is that if we can obtain good estimates for the zeros of these approximations it may be possible to generalize some of the results to zeta itself. This paper is a first step towards this goal. Our results show that these finite approximations have zeros near every vertical line (so no vertical strip in this region is zero-free). Furthermore, we give upper bounds for the imaginary parts of the zeros (the real parts are pinned). The bounds are numerically very large. Our tools are: the inverse mapping theorem (for a perturbative argument), the prime number theorem (for counting primes), elementary geometry (for constructing zeros of a related series), and a quantitative version of Kronecker's theorem to go from one series to another.

Topological Degree Computation

In this chapter we address the problem of computing topological degree of Lipschitz functions. From the knowledge of the topological degree one may ascertain whether there exists a zero of a function inside the domain, a knowledge that is practically and theoretically worthwile. Namely, Kronecker’s theorem states that if the topological degree is not zero then there exists a zero of a function inside the domain. Under more-restrictive assumptions one may also derive equivalence statements, i.e., nonzero degree is equivalent to the existence of a zero. By computing a sequence of domains with nonzero degrees and decreasing diameters one can obtain a region with arbitrarily small diameter that contains at least one zero of the function. Such methods, called generalized bisections, have been implemented and tested by several authors, as described in the annotations to this chapter. These methods have been touted as appropriate when the function is not smooth or cannot be evaluated accurately. For such functions they yield close approximations to roots in many cases for which all available other methods tested have failed (see annotations). The generalized bisection methods based on the degree computation are related to simplicial continuation methods. Their worst case complexity in general classes of functions is unbounded, as results of section 2.1.2 indicate; however, for tested functions they did converge. This suggests the need of average case analysis of such methods. There are numerous applications of the degree computation in nonlinear analysis. In addition to the existence of roots, the degree computation is used in methods for finding directions proceeding from bifurcation points in the solution of nonlinear functional differential equations as well as others as indicated in annotations. Algorithms proposed for the degree computation were tested on relatively small number of examples. The authors concluded that the degree of arbitrary continuous function could be computed. It was observed, however, that the algorithms could require an unbounded number of function evaluations. This is why in our work we restrict the functions to still relatively large class of functions satisfying the Lipschitz condition with a given constant K.