We give explicit formulae for all of the terms in the asymptotic expansion of
the mean fourth power of the Riemann zeta-function on the critical line

Conrey, J. Brian

Romanian

arXiv

. We give explicit formulae for all of the terms in the asymptotic expansion of the mean fourth power of the Riemann zeta-function on the critical line. Heath-Brown has evaluated the 4th power moment of the Riemann zeta-function on the critical line via the formula I 2 (T ) =  Z T 0  ji(1=2 + it)j  4  dt = TP 4  `  log  T  2ß  &apos;  + O(T  7=8+ffl  ) where P 4 is a 4th degree polynomial. Here we give an explicit formula for P 4 . In particular, we show that the coefficients of P 4 are in the field  Q(fl 0 ; fl 1 ; fl 2 ; fl 3 ; i(2); i  0  (2); i  00  (2)) where the fl&apos;s are the coefficients in the Laurent expansion i(s) = 1  s \Gamma 1 + fl 0 + fl 1 (s \Gamma 1) + : : : of i(s) at s = 1.  Theorem. We have  P 4 (x) = g 0 (x) + g 1 (x)  where  g 0 (x) = Res  s=0  2x  s  i(s + 1)  4  s(s + 1)i(2s + 2)  and g 1 (x) =  `  d ds  &apos; 2  (xe  2fl 0  )  s  (  1 2  i(s + 1)  2  \Gamma  1  s  i(2s + 1) \Gamma i(2s + 2)) (s + 1)i(s + 2)  fi fi fi fi fi  s=0  : Ivic has also written down formulae for t..

J. B. Conrey

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A note on the fourth power moment of the Riemann zeta-function

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.50.5124

A note on the fourth power moment of the Riemann zeta-function

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