Leírás
Kivonat: In the classical case of natural primes, Littlewood asked
for explicit, effective oscillation results of the remainder term in
the Prime Number Theorem, given a concrete nontrivial zero of the
Riemann zeta function. The problem was answered by Turán, and an
essentially optimal final answer was reached by Pintz via Turán's
Power Sum method. Another classical question was posed by Ingham, who
asked what (possibly best) result follows if a concrete zero-free
region is known. Conversely, the question of sharpness of any result
in the Ingham direction can be formulated as follows. What magnitude
of oscillation follows if we know that in some region there are
infinitely many zeros? Again, in the classical natural number case
Turán and Pintz succeeded in answering the question.
However, everybody believes that the Riemann Hypothesis holds, whence
there seems to be little relevance of the above analysis for natural
numbers. In contrast, it is already known since the ground-breaking
paper of Diamond, Montgomery and Vorhauer, that for Beurling number
systems RH may well fail, and in fact nothing better than the XIXth
century de la Vallée Poussin-Landau zero-free region and error term
can in general be asserted. Therefore, the Littlewood and Ingham
questions seem to be even more relevant for the Beurling theory.
The lectures survey the author's recent work on this circle of
problems. On our way we establish the analogue of the Riemann-von
Mangoldt formula, and give three results on the distribution of zeros
of the Beurling zeta function, including an unexpected Carlson type
density result. In the Littlewood problem we obtain a sharp result
with a surprising constant, and show its sharpness by use of a recent
marvelous achievement of Broucke and Vindas, improving the original
Diamond-Montgomery-Vorhauer construction. We also obtain a sharp pair
of results in the Ingham problem and its converse, and analyze the
connection of the two forms of these results, (one belonging to Ingham
and one to Pintz), which lead to an interesting geometric description.
For Zoom access please contact Andras Biro (biro.andras[a]renyi.hu).