We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
To send content items to your account,
please confirm that you agree to abide by our usage policies.
If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account.
Find out more about sending content to .
To send content items to your Kindle, first ensure no-reply@cambridge.org
is added to your Approved Personal Document E-mail List under your Personal Document Settings
on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part
of your Kindle email address below.
Find out more about sending to your Kindle.
Note you can select to send to either the @free.kindle.com or @kindle.com variations.
‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi.
‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Let
$k$
be an arbitrary positive integer and let
$\unicode[STIX]{x1D6FE}(n)$
stand for the product of the distinct prime factors of
$n$
. For each integer
$n\geqslant 2$
, let
$a_{n}$
and
$b_{n}$
stand respectively for the maximum and the minimum of the
$k$
integers
$\unicode[STIX]{x1D6FE}(n+1),\unicode[STIX]{x1D6FE}(n+2),\ldots ,\unicode[STIX]{x1D6FE}(n+k)$
. We show that
$\liminf _{n\rightarrow \infty }a_{n}/b_{n}=1$
. We also prove that the same result holds in the case of the Euler function and the sum of the divisors function, as well as the functions
$\unicode[STIX]{x1D714}(n)$
and
$\unicode[STIX]{x1D6FA}(n)$
, which stand respectively for the number of distinct prime factors of
$n$
and the total number of prime factors of
$n$
counting their multiplicity.
Let
$\unicode[STIX]{x1D70F}(\cdot )$
be the classical Ramanujan
$\unicode[STIX]{x1D70F}$
-function and let
$k$
be a positive integer such that
$\unicode[STIX]{x1D70F}(n)\neq 0$
for
$1\leqslant n\leqslant k/2$
. (This is known to be true for
$k<10^{23}$
, and, conjecturally, for all
$k$
.) Further, let
$\unicode[STIX]{x1D70E}$
be a permutation of the set
$\{1,\ldots ,k\}$
. We show that there exist infinitely many positive integers
$m$
such that
$|\unicode[STIX]{x1D70F}(m+\unicode[STIX]{x1D70E}(1))|<|\unicode[STIX]{x1D70F}(m+\unicode[STIX]{x1D70E}(2))|<\cdots <|\unicode[STIX]{x1D70F}(m+\unicode[STIX]{x1D70E}(k))|$
. We also obtain a similar result for Hecke eigenvalues of primitive forms of square-free level.
We show that two distinct singular moduli
$j(\unicode[STIX]{x1D70F}),j(\unicode[STIX]{x1D70F}^{\prime })$
, such that for some positive integers
$m$
and
$n$
the numbers
$1,j(\unicode[STIX]{x1D70F})^{m}$
and
$j(\unicode[STIX]{x1D70F}^{\prime })^{n}$
are linearly dependent over
$\mathbb{Q}$
, generate the same number field of degree at most two. This completes a result of Riffaut [‘Equations with powers of singular moduli’, Int. J. Number Theory, to appear], who proved the above theorem except for two explicit pairs of exceptions consisting of numbers of degree three. The purpose of this article is to treat these two remaining cases.
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 L-functions of elliptic curves and modular forms. In particular, we show that |τ(n)| ⩽ n11/2(logn)−1/2+o(1) for a set of n of asymptotic density 1, where τ(n) is the Ramanujan τ function while the standard argument yields log 2 instead of −1/2 in the power of the logarithm. Another consequence of our result is that in the number of representations of n by a binary quadratic form one has slightly more than square-root cancellations for almost all integers n.
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 and might be within reach unconditionally using the currently established potential automorphy.
$$\begin{eqnarray}\mathfrak{P}_{n}=\mathop{\prod }_{\substack{ p \\ s_{p}(n)\geqslant p}}p,\end{eqnarray}$$
where
$p$
runs over primes and
$s_{p}(n)$
is the sum of the base
$p$
digits of
$n$
. For all
$n$
we prove that
$\mathfrak{P}_{n}$
is divisible by all “small” primes with at most one exception. We also show that
$\mathfrak{P}_{n}$
is large and has many prime factors exceeding
$\sqrt{n}$
, with the largest one exceeding
$n^{20/37}$
. We establish Kellner’s conjecture that the number of prime factors exceeding
$\sqrt{n}$
grows asymptotically as
$\unicode[STIX]{x1D705}\sqrt{n}/\text{log}\,n$
for some constant
$\unicode[STIX]{x1D705}$
with
$\unicode[STIX]{x1D705}=2$
. Further, we compare the sizes of
$\mathfrak{P}_{n}$
and
$\mathfrak{P}_{n+1}$
, leading to the somewhat surprising conclusion that although
$\mathfrak{P}_{n}$
tends to infinity with
$n$
, the inequality
$\mathfrak{P}_{n}>\mathfrak{P}_{n+1}$
is more frequent than its reverse.
In this paper, we prove some conjectures of K. Stolarsky concerning the first and third moments of the Beatty sequences with the golden section and its square.
Depression and obesity are highly prevalent, and major impacts on public health frequently co-occur. Recently, we reported that having depression moderates the effect of the FTO gene, suggesting its implication in the association between depression and obesity.
Aims
To confirm these findings by investigating the FTO polymorphism rs9939609 in new cohorts, and subsequently in a meta-analysis.
Method
The sample consists of 6902 individuals with depression and 6799 controls from three replication cohorts and two original discovery cohorts. Linear regression models were performed to test for association between rs9939609 and body mass index (BMI), and for the interaction between rs9939609 and depression status for an effect on BMI. Fixed and random effects meta-analyses were performed using METASOFT.
Results
In the replication cohorts, we observed a significant interaction between FTO, BMI and depression with fixed effects meta-analysis (β=0.12, P = 2.7 × 10−4) and with the Han/Eskin random effects method (P = 1.4 × 10−7) but not with traditional random effects (β = 0.1, P = 0.35). When combined with the discovery cohorts, random effects meta-analysis also supports the interaction (β = 0.12, P = 0.027) being highly significant based on the Han/Eskin model (P = 6.9 × 10−8). On average, carriers of the risk allele who have depression have a 2.2% higher BMI for each risk allele, over and above the main effect of FTO.
Conclusions
This meta-analysis provides additional support for a significant interaction between FTO, depression and BMI, indicating that depression increases the effect of FTO on BMI. The findings provide a useful starting point in understanding the biological mechanism involved in the association between obesity and depression.
We classify all polynomials
$P(X)\in \mathbb{Q}[X]$
with rational coefficients having the property that the quotient
$(\unicode[STIX]{x1D706}_{i}-\unicode[STIX]{x1D706}_{j})/(\unicode[STIX]{x1D706}_{k}-\unicode[STIX]{x1D706}_{\ell })$
is a rational number for all quadruples of roots
$(\unicode[STIX]{x1D706}_{i},\unicode[STIX]{x1D706}_{j},\unicode[STIX]{x1D706}_{k},\unicode[STIX]{x1D706}_{\ell })$
with
$\unicode[STIX]{x1D706}_{k}\neq \unicode[STIX]{x1D706}_{\ell }$
.
We investigate the monotonic characteristics of the generalised binomial coefficients (phinomials) based upon Euler’s totient function. We show, unconditionally, that the set of integers for which this sequence is unimodal is finite and, assuming the generalised Riemann hypothesis, we find all the exceptions.
In 2006, F. Luca and I. E. Shparlinski (Proc. Indian Acad. Sci. (Math. Sci.)116(1) (2006), 1–8) proved that there are only finitely many pairs (n, m) of positive integers which satisfy the Diophantine equation |τ(n!)|=m!, where τ is the Ramanujan function. In this paper, we follow the same approach of Luca and Shparlinski (Proc. Indian Acad. Sci. (Math. Sci.)116(1) (2006), 1–8) to determine all solutions of the above equation. The proof of our main theorem uses linear forms in two logarithms and arithmetic properties of the Ramanujan function.
We give an upper bound for the number of elliptic Carmichael numbers
$n\,\le \,x$
that were recently introduced by J. H. Silverman in the case of an elliptic curve without complex multiplication (non
$\text{CM}$
). We also discuss several possible further improvements.
Let
$b\,>\,1$
be an integer. We prove that for almost all
$n$
, the sum of the digits in base
$b$
of the numerator of the Bernoulli number
${{B}_{2n}}$
exceeds
$c$
log
$n$
, where
$c\,:=\,c\left( b \right)\,>\,0$
is some constant depending on
$b$
.
The gene product of the ABCB1 gene, the P-glycoprotein, functions as a custodian molecule in the blood–brain barrier and regulates the access of most antidepressants into the brain. Previous studies showed that ABCB1 polymorphisms predicted the response to antidepressants that are substrates of the P-gp, while the response to nonsubstrates was not influenced by ABCB1 polymorphisms. The aim of the present study was to evaluate the clinical application of ABCB1 genotyping in antidepressant pharmacotherapy.
Methods
Data came from 58 depressed inpatients participating in the Munich Antidepressant Response Signature (MARS) project, whose ABCB1 gene test results were implemented into the clinical decision making process. Hamilton Depression Rating Scale (HAM-D) scores, remission rates, and duration of hospital stay were documented with dose and kind of antidepressant treatment.
Results
Patients who received ABCB1 genotyping had higher remission rates [χ2(1) = 6.596, p = 0.005, 1-sided] and lower Hamilton sores [t(111) = 2.091, p = 0.0195, 1-sided] at the time of discharge from hospital as compared to patients without ABCB1 testing. Among major allele homozygotes for ABCB1 single nucleotide polymorphisms (SNPs) rs2032583 and rs2235015 (TT/GG genotype), an increase in dose was associated with a shorter duration of hospital stay [rho(28) = –0.441, p = 0.009, 1-sided], whereas other treatment strategies (eg, switching to a nonsubstrate) showed no significant associations with better treatment outcome.
Discussion
The implementation of ABCB1 genotyping as a diagnostic tool influenced clinical decisions and led to an improvement of treatment outcome. Patients carrying the TT/GG genotype seemed to benefit from an increase in P-gp substrate dose.
Conclusion
Results suggest that antidepressant treatment of depression can be optimized by the clinical application of ABCB1 genotyping.
Let
$L(s, E)= {\mathop{\sum }\nolimits}_{n\geq 1} {a}_{n} {n}^{- s} $
be the
$L$
-series corresponding to an elliptic curve
$E$
defined over
$ \mathbb{Q} $
and
$\mathbf{u} = \mathop{\{ {u}_{m} \} }\nolimits_{m\geq 0} $
be a nondegenerate binary recurrence sequence. We prove that if
${ \mathcal{M} }_{E} $
is the set of
$n$
such that
${a}_{n} \not = 0$
and
${ \mathcal{N} }_{E} $
is the subset of
$n\in { \mathcal{M} }_{E} $
such that
$\vert {a}_{n} \vert = \vert {u}_{m} \vert $
holds with some integer
$m\geq 0$
, then
${ \mathcal{N} }_{E} $
is of density
$0$
as a subset of
${ \mathcal{M} }_{E} $
.
We study positive integers
$n$
such that
$n\phi (n)\equiv 2\hspace{0.167em} {\rm mod}\hspace{0.167em} \sigma (n)$
, where
$\phi (n)$
and
$\sigma (n)$
are the Euler function and the sum of divisors function of the positive integer
$n$
, respectively. We give a general ineffective result showing that there are only finitely many such
$n$
whose prime factors belong to a fixed finite set. When this finite set consists only of the two primes
$2$
and
$3$
we use continued fractions to find all such positive integers
$n$
.
We recall that a polynomial f(X) ∈ K[X] over a field K is called stable if all its iterates are irreducible over K. We show that almost all monic quadratic polynomials f(X) ∈ ℤ[X] are stable over ℚ. We also show that the presence of squares in so-called critical orbits of a quadratic polynomial f(X) ∈ ℤ[X] can be detected by a finite algorithm; this property is closely related to the stability of f(X). We also prove there are no stable quadratic polynomials over finite fields of characteristic 2 but they exist over some infinite fields of characteristic 2.
We give upper and lower bounds on the count of positive integers n ≤ x dividing the nth term of a non-degenerate linearly recurrent sequence with simple roots.
Unfortunately, there are two inaccuracies in the argument of [CLS]. First, the statements of Lemmas 3, 4, 6, and 7 of [CLS] hold only under the additional condition gcd(m, ME) = 1 for some integer ME ≥ 1 depending only on E. Second, the divisibility condition (3·6) in [CLS] implies that tb(ℓ) | nE(p)−1 (rather than tb(ℓ) | nE(p), as it was erroneously claimed on p. 519 in [CLS]). In particular, instead of the divisibility ℓtb(ℓ) | nE(p) (see the last displayed formula on p. 519 in [CLS]), we conclude that for every prime ℓ | L there is an integer aℓ such that
(0.1)
However, the final result is correct and can easily be recovered. To do so, we remark that under the condition gcd(m,ME) =1, we have full analogues of Lemmas 6, 7, 9, and 10 of [CLS] for the function Π(x;m,a) defined as the number of primes p ≤ x with nE(p) ≡ a (mod m) (rather than just for Π(x;m) = Π(x;m,0) as in [CLS]). Define ρ*(n) as the largest square-free divisor of n which is relatively prime to ME. We then derive from (0.1) above that
Therefore
(0.2)
Since
we see that (0.2) above implies the bound (3·7) from [CLS], and the result now follows without any further changes.