Math.ExpPairs.Ivic

Estimates of the Riemann zeta-function in a critical strip, according to Ivić A. "The Riemann zeta-function: theory and applications", Mineola, New York: Dover Publications, 2003, and Lelechenko A. V. "Dirichlet divisor problem on Gaussian integers" in Proceedings of the 6th international conference on analytic number theory and spatial tesselations, Kyiv, 2018, vol. 1, p. 76-86.

Documentation

zetaOnS :: Rational -> OptimizeResult

Compute \( \mu(\sigma) \) such that \( \zeta(\sigma+it) \ll |t|^{\mu(\sigma)} \). See equation (7.57) in Ivić, 2003.

reverseZetaOnS :: Rational -> OptimizeResult

An attempt to reverse zetaOnS.

mOnS :: Rational -> OptimizeResult

Compute maximal \( m(\sigma) \) such that \( \int_1^T | \zeta(\sigma+it) |^{m(\sigma)} dt \ll T^{1+\varepsilon} \). See equation (8.97) in Ivić, 2003.

reverseMOnS :: Rational -> RationalInf -> Rational

Try to reverse mOnS: for a given precision and \( m \) compute \( \sigma \). Implemented as a binary search, so its performance is very poor. Since mOnS is not monotonic, the result is not guaranteed to be neither minimal nor maximal possible, but usually is close enough.

For integer \( m \ge 4 \) this function corresponds to the multidimensional Dirichlet problem and returns \( \sigma \) from error term \( O(x^{\sigma+\varepsilon}) \). See Ch. 13 in Ivić, 2003.

checkAbscissa :: [(Rational, Rational)] -> Rational -> Bool

Check whether \( \int_1^T \prod_i |\zeta(n_i\sigma+it|^{m_i} dt \ll T^{1+\varepsilon} \) for a given list of pairs \( [(n_1, m_1), ...] \) and fixed \( \sigma \).

findMinAbscissa :: Rational -> [(Rational, Rational)] -> Rational

Find for a given precision and list of pairs \( [(n_1, m_1), ...] \) the minimal \( \sigma \) such that \( \int_1^T \prod_i |\zeta(n_i\sigma+it|^{m_i} dt \ll T^{1+\varepsilon} \).

mBigOnHalf :: Rational -> OptimizeResult

For a given \( A \) compute minimal \( M(A) \) such that \( \int_1^T |\zeta(1/2+it)|^A \ll T^{M(A)+\varepsilon} \) See Ch. 8 in Ivić, 2003 and Th. 1 in Lelechenko, 2018.

reverseMBigOnHalf :: Rational -> OptimizeResult

Try to reverse mBigOnHalf: for a given \( M(A) \) find maximal possible \( A \). Sometimes, when mBigOnHalf gets especially lucky exponent pair, reverseMBigOnHalf can miss real \( A \) and returns lower value.

kolpakova2011 :: Integer -> Double

An estimate of the symmetric multidimensional divisor function from Kolpakova O. V., "New estimates of the remainder in an asymptotic formula in the multidimensional Dirichlet divisor problem", Mathematical Notes, vol. 89, p. 504-518, 2011.