18 research outputs found
A structure theorem in probabilistic number theory
We prove that if two additive functions (from a certain class) take large
values with roughly the same probability then they must be identical. This is a
consequence of a structure theorem making clear the inter-relation between the
distribution of an additive function on the integers, and its distribution on
the primes.Comment: 10 page
The Riemann-zeta function on vertical arithmetic progressions
We show that the twisted second moments of the Riemann zeta function averaged
over the arithmetic progression with , real,
exhibits a remarkable correspondance with the analogous continuous average and
derive several consequences. For example, motivated by the linear independence
conjecture, we show at least one third of the elements in the arithmetic
progression are not the ordinates of some zero of lying on
the critical line. This improves on earlier work of Martin and Ng. We then
complement this result by producing large and small values of on
arithmetic progressions which are of the same quality as the best
results currently known for with real.Comment: 20 page
Refinements of G\'al's theorem and applications
We give a simple proof of a well-known theorem of G\'al and of the recent
related results of Aistleitner, Berkes and Seip [1] regarding the size of GCD
sums. In fact, our method obtains the asymptotically sharp constant in G\'al's
theorem, which is new. Our approach also gives a transparent explanation of the
relationship between the maximal size of the Riemann zeta function on vertical
lines and bounds on GCD sums; a point which was previously unclear. Furthermore
we obtain sharp bounds on the spectral norm of GCD matrices which settles a
question raised in [2]. We use bounds for the spectral norm to show that series
formed out of dilates of periodic functions of bounded variation converge
almost everywhere if the coefficients of the series are in , with . This was previously known with ,
and is known to fail for . We also develop a sharp Carleson-Hunt-type
theorem for functions of bounded variations which settles another question
raised in [1]. Finally we obtain almost sure bounds for partial sums of dilates
of periodic functions of bounded variations improving [1]. This implies almost
sure bounds for the discrepancy of with an arbitrary growing
sequences of integers.Comment: 16 page
Moments and distribution of central L-values of quadratic twists of elliptic curves
We show that if one can compute a little more than a particular moment for
some family of L-functions, then one has upper bounds of the conjectured order
of magnitude for all smaller (positive, real) moments and a one-sided central
limit theorem holds. We illustrate our method for the family of quadratic
twists of an elliptic curve, obtaining sharp upper bounds for all moments below
the first. We also establish a one sided central limit theorem supporting a
conjecture of Keating and Snaith. Our work leads to a conjecture on the
distribution of the order of the Tate-Shafarevich group for rank zero quadratic
twists of an elliptic curve, and establishes the upper bound part of this
conjecture (assuming the Birch-Swinnerton-Dyer conjecture).Comment: 28 page
A converse to Halasz's theorem
We show that the distribution of large values of an additive function on the
integers, and the distribution of values of the additive function on the primes
are related to each other via a Levy Process. As a consequence we obtain a
converse to an old theorem of Halasz. Halasz proved that if f is an strongly
additive function with f (p) \in {0, 1}, then f is Poisson distributed on the
integers. We prove, conversely, that if f is Poisson distributed on the
integers then for most primes p, f(p) = o(1) or f(p) = 1 + o(1).Comment: 14 pages. Part of my B. Sc. thesis: arxiv:0909.527
The 4.36-th moment of the Riemann zeta-function
Conditionally on the Riemann Hypothesis we obtain bounds of the correct order
of magnitude for the 2k-th moment of the Riemann zeta-function for all positive
real k < 2.181. This provides for the first time an upper bound of the correct
order of magnitude for some k > 2; the case of k = 2 corresponds to a classical
result of Ingham. We prove our result by establishing a connection between
moments with k > 2 and the so-called "twisted fourth moment". This allows us to
appeal to a recent result of Hughes and Young. Furthermore we obtain a
point-wise bound for |zeta(1/2 + it)|^{2r} (with 0 < r < 1) that can be
regarded as a multiplicative analogue of Selberg's bound for S(T). We also
establish asymptotic formulae for moments (k < 2.181) slightly off the
half-line.Comment: 10 pages; Fixed errors in Section 3. Proof of Thm 2 (now Corollary 1)
shorter; Added a few comments following Proposition
On the Typical Size and Cancelations Among the Coefficients of Some Modular Forms
We obtain a nontrivial upper bound for almost all elements of the sequences
of real numbers which are multiplicative and at the prime indices are
distributed according to the Sato--Tate density. Examples of such sequences
come from coefficients of several -functions of elliptic curves and modular
forms. In particular, we show that
for a set of of asymptotic density 1, where is the Ramanujan
function while the standard argument yields instead of
in the power of the logarithm. Another consequence of our result is that in the
number of representations of by a binary quadratic form one has slightly
more than square-root cancellations for almost all integers .
In addition we obtain a central limit theorem for such sequences, assuming a
weak hypothesis on the rate of convergence to the Sato--Tate law. For Fourier
coefficients of primitive holomorphic cusp forms such a hypothesis is known
conditionally assuming the automorphy of all symmetric powers of the form and
seems to be within reach unconditionally using the currently established
potential automorphy.Comment: The second version contains some improvements and extensions of
previous results, suggested by Maksym Radziwill, who is now a co-autho