319 research outputs found
The error term in the prime number theorem
We make explicit a theorem of Pintz concerning the error term in the prime
number theorem. This gives an improved version of the prime number theorem with
error term roughly square-root of that which was previously known. We apply
this to a long-standing problem concerning an inequality studied by Ramanujan.Comment: To appear in Math. Com
Lower order terms in the 1-level density for families of holomorphic cuspidal newforms
The Katz-Sarnak density conjecture states that, in the limit as the
conductors tend to infinity, the behavior of normalized zeros near the central
point of families of L-functions agree with the N -> oo scaling limits of
eigenvalues near 1 of subgroups of U(N). Evidence for this has been found for
many families by studying the n-level densities; for suitably restricted test
functions the main terms agree with random matrix theory. In particular, all
one-parameter families of elliptic curves with rank r over Q(T) and the same
distribution of signs of functional equations have the same limiting behavior.
We break this universality and find family dependent lower order correction
terms in many cases; these lower order terms have applications ranging from
excess rank to modeling the behavior of zeros near the central point, and
depend on the arithmetic of the family. We derive an alternate form of the
explicit formula for GL(2) L-functions which simplifies comparisons, replacing
sums over powers of Satake parameters by sums of the moments of the Fourier
coefficients lambda_f(p). Our formula highlights the differences that we expect
to exist from families whose Fourier coefficients obey different laws (for
example, we expect Sato-Tate to hold only for non-CM families of elliptic
curves). Further, by the work of Rosen and Silverman we expect lower order
biases to the Fourier coefficients in families of elliptic curves with rank
over Q(T); these biases can be seen in our expansions. We analyze several
families of elliptic curves and see different lower order corrections,
depending on whether or not the family has complex multiplication, a forced
torsion point, or non-zero rank over Q(T).Comment: 38 pages, version 2.2: fixed some typos, included some comments from
Steven Finch which give more rapidly converging expressions for the constants
gamma_{PNT}, gamma_{PNT,1,3} and gamma_{PNT,1,4}, updated reference
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