Copyright | (c) Andrew Lelechenko 2014-2020 |
---|---|

License | GPL-3 |

Maintainer | andrew.lelechenko@gmail.com |

Safe Haskell | None |

Language | Haskell2010 |

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

- zetaOnS :: Rational -> OptimizeResult
- reverseZetaOnS :: Rational -> OptimizeResult
- mOnS :: Rational -> OptimizeResult
- reverseMOnS :: Rational -> RationalInf -> Rational
- checkAbscissa :: [(Rational, Rational)] -> Rational -> Bool
- findMinAbscissa :: Rational -> [(Rational, Rational)] -> Rational
- mBigOnHalf :: Rational -> OptimizeResult
- reverseMBigOnHalf :: Rational -> OptimizeResult
- kolpakova2011 :: Integer -> Double

# Documentation

zetaOnS :: Rational -> OptimizeResult Source #

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

reverseZetaOnS :: Rational -> OptimizeResult Source #

An attempt to reverse `zetaOnS`

.

mOnS :: Rational -> OptimizeResult Source #

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 Source #

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 \( \sigma \).

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

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 Source #

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 Source #

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 Source #

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.