Copyright | (c) Andrew Lelechenko 2014-2015 |
---|---|
License | GPL-3 |
Maintainer | andrew.lelechenko@gmail.com |
Stability | experimental |
Portability | POSIX |
Safe Haskell | None |
Language | Haskell2010 |
Provides functions to compute estimates Riemann zeta-function ζ in a critical strip, given in Ivić A. `The Riemann zeta-function: Theory and applications', Mineola, New York: Dover Publications, 2003.
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 µ(σ) such that |ζ(σ+it)| ≪ |t|^µ(σ) . See equation (7.57) in Ivić2003.
reverseZetaOnS :: Rational -> OptimizeResult Source #
An attempt to reverse zetaOnS
.
mOnS :: Rational -> OptimizeResult Source #
Compute maximal m(σ) such that ∫_1^T |ζ(σ+it)|^m(σ) dt ≪ T^(1+ε). See equation (8.97) in Ivić2003. Further justification will be published elsewhere.
reverseMOnS :: Rational -> RationalInf -> Rational Source #
Try to reverse mOnS
: for a given precision and m compute minimal possible σ.
Implementation is usual try-and-divide search, so performance is very poor.
Sometimes, when mOnS
gets especially lucky exponent pair, reverseMOnS
can miss
real σ and returns significantly bigger value.
For integer m>=4 this function corresponds to the multidimensional Dirichlet problem and returns σ from error term O(x^{σ+ε}). See Ch. 13 in Ivić2003.
checkAbscissa :: [(Rational, Rational)] -> Rational -> Bool Source #
Check whether ∫_1^T Π_i |ζ(n_i*σ+it)|^m_i dt ≪ T^(1+ε) for a given list of pairs [(n_1, m_1), ...] and fixed σ.
findMinAbscissa :: Rational -> [(Rational, Rational)] -> Rational Source #
Find for a given precision and list of pairs [(n_1, m_1), ...] the minimal σ such that ∫_1^T Π_i|ζ(n_i*σ+it)|^m_i dt ≪ T^(1+ε).
mBigOnHalf :: Rational -> OptimizeResult Source #
Compute minimal M(A) such that ∫_1^T |ζ(1/2+it)|^A dt ≪ T^(M(A)+ε). See Ch. 8 in Ivić2003. Further justification will be published elsewhere.
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, 2011.