In this article, we provide explicit bounds for the prime counting functions
θ
(
x
)
\theta (x)
for all ranges of
x
x
. The bounds for the error term for
θ
(
x
)
−
x
\theta (x)- x
are of the shape
ε
x
\varepsilon x
and
c
k
x
(
log
x
)
k
\frac {c_k x}{(\log x)^k}
, for
k
=
1
,
…
,
5
k=1,\ldots ,5
. Tables of values for
ε
\varepsilon
and
c
k
c_k
are provided.
In this paper we investigate the moments and the distribution of L(1, χ D ), where χ D varies over quadratic characters associated to square-free polynomials D of degree n over F q , as n → ∞. Our first result gives asymptotic formulas for the complex moments of L(1, χ D ) in a large uniform range. Previously, only the first moment has been computed due to work of Andrade and Jung. Using our asymptotic formulas together with the saddle-point method, we show that the distribution function of L(1, χ D ) is very close to that of a corresponding probabilistic model. In particular, we uncover an interesting feature in the distribution of large (and small) values of L(1, χ D ), that is not present in the number field setting. We also obtain Ω-results for the extreme values of L(1, χ D ), which we conjecture to be best possible. Specializing n = 2g + 1 and making use of one case of Artin's class number formula, we obtain similar results for the class number h D associated to F q (T )[ √ D]. Similarly, specializing to n = 2g + 2 we can appeal to the second case of Artin's class number formula and deduce analogous results for h
In this article, we provide explicit bounds for the prime counting functions θ(x) in all ranges of x. The bounds for the error term for θ(x) − x are of the shape ǫx and c k x (log x) k , for k = 1, . . . , 5. Tables of values for ǫ and c k are provided.2000 Mathematics Subject Classification. 11N05, 11M06, 11M26. Key words and phrases. prime number theorem, ψ(x), θ(x), explicit formula, zeros of Riemann zeta function. 1 There is a typo in the definition of A in [5] on p. 376. We have given a corrected definition of A. 2 These are the elementary bounds we are aware of. If readers know of others please let us know.
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