Math.ExpPairs.Ivic

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.

Documentation

zetaOnS :: Rational -> OptimizeResult

Compute µ(σ) such that |ζ(σ+it)| ≪ |t|^µ(σ) . See equation (7.57) in Ivić2003.

reverseZetaOnS :: Rational -> OptimizeResult

An attempt to reverse zetaOnS.

mOnS :: Rational -> OptimizeResult

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

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

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

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

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

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, 2011.