100 research outputs found
An alternative to Riemann-Siegel type formulas
Simple unsmoothed formulas to compute the Riemann zeta function, and
Dirichlet -functions to a power-full modulus, are derived by elementary
means (Taylor expansions and the geometric series). The formulas enable
square-root of the analytic conductor complexity, up to logarithmic loss, and
have an explicit remainder term that is easy to control. The formula for zeta
yields a convexity bound of the same strength as that from the Riemann-Siegel
formula, up to a constant factor. Practical parameter choices are discussed.Comment: 16 page
Fast methods to compute the Riemann zeta function
The Riemann zeta function on the critical line can be computed using a
straightforward application of the Riemann-Siegel formula, Sch\"onhage's
method, or Heath-Brown's method. The complexities of these methods have
exponents 1/2, 3/8 (=0.375), and 1/3 respectively. In this paper, three new
fast and potentially practical methods to compute zeta are presented. One
method is very simple. Its complexity has exponent 2/5. A second method relies
on this author's algorithm to compute quadratic exponential sums. Its
complexity has exponent 1/3. The third method employs an algorithm, developed
in this paper, to compute cubic exponential sums. Its complexity has exponent
4/13 (approximately, 0.307).Comment: Presentation simplifie
Numerical study of the derivative of the Riemann zeta function at zeros
The derivative of the Riemann zeta function was computed numerically on
several large sets of zeros at large heights. Comparisons to known and
conjectured asymptotics are presented.Comment: 13 pages, 5 figures; minor typos fixe
- …