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

Autor:	Hany M. Farag
Rok vydání:	2008
Předmět:	Riemann Xi function Arithmetic zeta function symbols.namesake Riemann hypothesis Algebra and Number Theory Explicit formulae Mathematical analysis symbols Dirichlet eta function Dirichlet series Prime zeta function Mathematics Riemann zeta function
Zdroj:	International Journal of Number Theory. :653-662
ISSN:	1793-7310 1793-0421
DOI:	10.1142/s1793042108001596
Popis:	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.
Databáze:	OpenAIRE
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