exp-pairs-0.2.1.0: Linear programming over exponent pairs

Math.ExpPairs.Ivic

Description

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.

Synopsis

# Documentation

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

An attempt to reverse zetaOnS.

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.

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 Source #

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 $$\sigma$$.

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

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.

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.

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.