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Zap. Nauchn. Sem. POMI, 2016, Volume 445, Pages 250–267 (Mi znsl6279)  

Extreme values of Epstein zeta-functions

O. M. Fomenko

St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, St. Petersburg, Russia

Abstract: Let $Q(u_1,u_2,…,u_l)$ be a positive definite quadratic form in $l(\geq2)$ variables and with integer coefficients. Put
$$ \zeta_Q(s)=\sum'(Q(u_1,u_2,…,u_l))^{-s} $$
where the accent indicates that the summation is over all integer $l$-tuples $(u_1,u_2,…,u_l)$ with the exception of $(0,0,…,0)$. It is known that $\zeta_Q(s)(s-\frac l2)$ is an entire function.
We treat $\Omega$-theorems for $\zeta_Q(s)l\leq3)$ and for some $\zeta_Q(s)(l=2)$. Let $l\leq4$ and $F_Q(s)=\zeta_Q(s+\frac l2-1)$. As $t$ tends to infinity, we have
$$ \log|F_Q(\frac12+it)|=\Omega_+((\frac{\log t}{\log\log t})^{1/2}), $$
and
$$ \log |F_Q(\sigma_0+it)|=\Omega_+(\frac{(\log t)^{1-\sigma_0}}{\log\log t}) $$
for fixed $\sigma_0\in(\frac12,1)$.

Key words and phrases: Epstein zeta-function, quadratic form, extremal value.

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English version:
Journal of Mathematical Sciences (New York), 2017, 222:5, 690–702

Bibliographic databases:

UDC: 511.466+517.863
Received: 09.03.2016

Citation: O. M. Fomenko, “Extreme values of Epstein zeta-functions”, Analytical theory of numbers and theory of functions. Part 31, Zap. Nauchn. Sem. POMI, 445, POMI, St. Petersburg, 2016, 250–267; J. Math. Sci. (N. Y.), 222:5 (2017), 690–702

Citation in format AMSBIB
\Bibitem{Fom16}
\by O.~M.~Fomenko
\paper Extreme values of Epstein zeta-functions
\inbook Analytical theory of numbers and theory of functions. Part~31
\serial Zap. Nauchn. Sem. POMI
\yr 2016
\vol 445
\pages 250--267
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl6279}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3511163}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2017
\vol 222
\issue 5
\pages 690--702
\crossref{https://doi.org/10.1007/s10958-017-3325-4}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85015671680}


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