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.
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