DIMACS TR: 93-66

The Least Possible Value at Zero of some Nonnegative Cosine Polynomials and Dual Problems

Author: Szilard Gy. REVESZ


The present work is inspired by Edmund Landau's famous book, "Handbuch der Lehre von der Verteilung der Primzahlen", where he posed two extremal questions on cosine polynomials and deduced various estimates on the distribution of primes using known estimates of the extremal quantities. Although since then better theoretical results are available for the error term of the prime number formula, Landau's method is still the best in finding explicit, computable bounds. In particular, Rosser and Schonfeld used the method in their work "Approximate formulas for some functions of prime numbers". In turn, improving the estimates often involves excessive computations in search of better polynomials. However, as the degree increases, complexity of that search increases steadily. We show that the basic function in the extremal problem is equal to another complicated-looking extremal function. Actually the approximation of that quantity was at the heart of the method of van der Waerden, when giving his lower estimate for one of the extremal problems, but now we see that it is theoretically optimal. Thus our analysis reveals why that estimate was so sharp. As we shall present in a forthcoming paper, the new insight into the structure of the problem also helps to improve upon the best known upper and lower estimates of French, Steckin and van der Waerden. The proof of the duality theorem uses functional analysis comparable to linear programming.

Paper available at: ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/1993/93-66.ps
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