On the divisor function and the Riemann zeta-function in short intervals
We obtain, for T ε ≤ U = U ( T )≤ T 1/2− ε , asymptotic formulas for where Δ( x ) is the error term in the classical divisor problem, and E ( T ) is the error term in the mean square formula for . Upper bounds of the form O ε ( T 1+ ε U 2 ) for the above integrals with biquadrates instead of square...
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| Published in | The Ramanujan journal Vol. 19; no. 2; pp. 207 - 224 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Boston
Springer US
01.07.2009
|
| Subjects | |
| Online Access | Get full text |
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| Summary: | We obtain, for
T
ε
≤
U
=
U
(
T
)≤
T
1/2−
ε
, asymptotic formulas for
where Δ(
x
) is the error term in the classical divisor problem, and
E
(
T
) is the error term in the mean square formula for
. Upper bounds of the form
O
ε
(
T
1+
ε
U
2
) for the above integrals with biquadrates instead of square are shown to hold for
T
3/8
≤
U
=
U
(
T
)
≪
T
1/2
. The connection between the moments of
E
(
t
+
U
)−
E
(
t
) and
is also given. Generalizations to some other number-theoretic error terms are discussed. |
|---|---|
| ISSN: | 1382-4090 1572-9303 |
| DOI: | 10.1007/s11139-008-9142-0 |