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Fujii obtained a formula for the average number of Goldbach representations with lower-order terms expressed as a sum over the zeros of the Riemann zeta function and a smaller error term. This assumed the Riemann Hypothesis. We obtain an unconditional version of this result and obtain applications conditional on various conjectures on zeros of the Riemann zeta function.
Recently, when studying intricate connections between Ramanujan’s theta functions and a class of partition functions, Banerjee and Dastidar [‘Ramanujan’s theta functions and parity of parts and cranks of partitions’, Ann. Comb., to appear] studied some arithmetic properties for
$c_o(n)$
, the number of partitions of n with odd crank. They conjectured a congruence modulo
$4$
satisfied by
$c_o(n)$
. We confirm the conjecture and evaluate
$c_o(4n)$
modulo
$8$
by dissecting some q-series into even powers. Moreover, we give a conjecture on the density of divisibility of odd cranks modulo 4, 8 and 16.
A partition of a positive integer n is called
$\ell $
-regular if none of its parts is divisible by
$\ell $
. Denote by
$b_{\ell }(n)$
the number of
$\ell $
-regular partitions of n. We give a complete characterisation of the arithmetic of
$b_{23}(n)$
modulo
$11$
for all n not divisible by
$11$
in terms of binary quadratic forms. Our result is obtained by establishing a relation between the generating function for these values of
$b_{23}(n)$
and certain modular forms having complex multiplication by
${\mathbb Q}(\sqrt {-69})$
.
In this paper, we show that every pair of sufficiently large even integers can be represented as a pair of eight prime cubes and k powers of
$2$
. In particular, we prove that
$k=335$
is admissible, which improves the previous result.
A partition
$\lambda $
of n is said to be nearly self-conjugate if the Ferrers graph of
$\lambda $
and its transpose have exactly
$n-1$
cells in common. The generating function of the number of such partitions was first conjectured by Campbell and recently confirmed by Campbell and Chern (‘Nearly self-conjugate integer partitions’, submitted for publication). We present a simple and direct analytic proof and a combinatorial proof of an equivalent statement.
Let N be a sufficiently large integer. We prove that, with at most
$O(N^{23/48+\varepsilon })$
exceptions, all even positive integers up to N can be represented in the form
$p_1^2+p_2^3+p_3^3+p_4^3+p_5^3+p_6^4$
, where
$p_1,p_2,p_3,p_4,p_5,p_6$
are prime numbers.
In this paper, we investigate pigeonhole statistics for the fractional parts of the sequence $\sqrt {n}$. Namely, we partition the unit circle $ \mathbb {T} = \mathbb {R}/\mathbb {Z}$ into N intervals and show that the proportion of intervals containing exactly j points of the sequence $(\sqrt {n} + \mathbb {Z})_{n=1}^N$ converges in the limit as $N \to \infty $. More generally, we investigate how the limiting distribution of the first $sN$ points of the sequence varies with the parameter $s \geq 0$. A natural way to examine this is via point processes—random measures on $[0,\infty )$ which represent the arrival times of the points of our sequence to a random interval from our partition. We show that the sequence of point processes we obtain converges in distribution and give an explicit description of the limiting process in terms of random affine unimodular lattices. Our work uses ergodic theory in the space of affine unimodular lattices, building upon work of Elkies and McMullen [Gaps in $\sqrt {n}$ mod 1 and ergodic theory. Duke Math. J.123 (2004), 95–139]. We prove a generalisation of equidistribution of rational points on expanding horocycles in the modular surface, working instead on nonlinear horocycle sections.
In 2012, Andrews and Merca proved a truncated theorem on Euler's pentagonal number theorem. Motivated by the works of Andrews and Merca, Guo and Zeng deduced truncated versions for two other classical theta series identities of Gauss. Very recently, Xia et al. proved new truncated theorems of the three classical theta series identities by taking different truncated series than the ones chosen by Andrews–Merca and Guo–Zeng. In this paper, we provide a unified treatment to establish new truncated versions for the three identities of Euler and Gauss based on a Bailey pair due to Lovejoy. These new truncated identities imply the results proved by Andrews–Merca, Wang–Yee, and Xia–Yee–Zhao.
A number of recent papers have estimated ratios of the partition function
$p(n-j)/p(n)$
, which appear in many applications. Here, we prove an easy-to-use effective bound on these ratios. Using this, we then study the second shifted difference of partitions,
$f(\,j,n) := p(n) -2p(n-j) +p(n-2j)$
, and give another easy-to-use estimate of
$f(\,j,n)$
. As applications of these, we prove a shifted convexity property of
$p(n)$
, as well as giving new estimates of the k-rank partition function
$N_k(m,n)$
and non-k-ary partitions along with their differences.
Let
$p_t(a,b;n)$
denote the number of partitions of n such that the number of t-hooks is congruent to
$a \bmod {b}$
. For
$t\in \{2, 3\}$
, arithmetic progressions
$r_1 \bmod {m_1}$
and
$r_2 \bmod {m_2}$
on which
$p_t(r_1,m_1; m_2 n + r_2)$
vanishes were established in recent work by Bringmann, Craig, Males and Ono [‘Distributions on partitions arising from Hilbert schemes and hook lengths’, Forum Math. Sigma10 (2022), Article no. e49] using the theory of modular forms. Here we offer a direct combinatorial proof of this result using abaci and the theory of t-cores and t-quotients.
There has been recent interest in a hybrid form of the celebrated conjectures of Hardy–Littlewood and of Chowla. We prove that for any $k,\ell \ge 1$ and distinct integers $h_2,\ldots ,h_k,a_1,\ldots ,a_\ell $, we have:
for all except $o(H)$ values of $h_1\leq H$, so long as $H\geq (\log X)^{\ell +\varepsilon }$. This improves on the range $H\ge (\log X)^{\psi (X)}$, $\psi (X)\to \infty $, obtained in previous work of the first author. Our results also generalise from the Möbius function $\mu $ to arbitrary (non-pretentious) multiplicative functions.
Recent works at the interface of algebraic combinatorics, algebraic geometry, number theory and topology have provided new integer-valued invariants on integer partitions. It is natural to consider the distribution of partitions when sorted by these invariants in congruence classes. We consider the prominent situations that arise from extensions of the Nekrasov–Okounkov hook product formula and from Betti numbers of various Hilbert schemes of n points on
${\mathbb {C}}^2$
. For the Hilbert schemes, we prove that homology is equidistributed as
$n\to \infty $
. For t-hooks, we prove distributions that are often not equidistributed. The cases where
$t\in \{2, 3\}$
stand out, as there are congruence classes where such counts are zero. To obtain these distributions, we obtain analytic results of independent interest. We determine the asymptotics, near roots of unity, of the ubiquitous infinite products
Let
$\mathcal {C}_n =\left [\chi _{\lambda }(\mu )\right ]_{\lambda , \mu }$
be the character table for
$S_n,$
where the indices
$\lambda $
and
$\mu $
run over the
$p(n)$
many integer partitions of
$n.$
In this note, we study
$Z_{\ell }(n),$
the number of zero entries
$\chi _{\lambda }(\mu )$
in
$\mathcal {C}_n,$
where
$\lambda $
is an
$\ell $
-core partition of
$n.$
For every prime
$\ell \geq 5,$
we prove an asymptotic formula of the form
where
$\sigma _{\ell }(n)$
is a twisted Legendre symbol divisor function,
$\delta _{\ell }:=(\ell ^2-1)/24,$
and
$1/\alpha _{\ell }>0$
is a normalization of the Dirichlet L-value
$L\left (\left ( \frac {\cdot }{\ell } \right ),\frac {\ell -1}{2}\right ).$
For primes
$\ell $
and
$n>\ell ^6/24,$
we show that
$\chi _{\lambda }(\mu )=0$
whenever
$\lambda $
and
$\mu $
are both
$\ell $
-cores. Furthermore, if
$Z^*_{\ell }(n)$
is the number of zero entries indexed by two
$\ell $
-cores, then, for
$\ell \geq 5$
, we obtain the asymptotic
Andrews [Generalized Frobenius Partitions, Memoirs of the American Mathematical Society, 301 (American Mathematical Society, Providence, RI, 1984)] defined two families of functions,
$\phi _k(n)$
and
$c\phi _k(n),$
enumerating two types of combinatorial objects which he called generalised Frobenius partitions. Andrews proved a number of Ramanujan-like congruences satisfied by specific functions within these two families. Numerous other authors proved similar results for these functions, often with a view towards a specific choice of the parameter
$k.$
Our goal is to identify an infinite family of values of k such that
$\phi _k(n)$
is even for all n in a specific arithmetic progression; in particular, we prove that, for all positive integers
$\ell ,$
all primes
$p\geq 5$
and all values
$r, 0 < r < p,$
such that
$24r+1$
is a quadratic nonresidue modulo
$p,$
for all
$n\geq 0.$
Our proof of this result is truly elementary, relying on a lemma from Andrews’ memoir, classical q-series results and elementary generating function manipulations. Such a result, which holds for infinitely many values of
$k,$
is rare in the study of arithmetic properties satisfied by generalised Frobenius partitions, primarily because of the unwieldy nature of the generating functions in question.
Hausel and Rodriguez-Villegas (2015, Astérisque 370, 113–156) recently observed that work of Göttsche, combined with a classical result of Erdös and Lehner on integer partitions, implies that the limiting Betti distribution for the Hilbert schemes
$(\mathbb {C}^{2})^{[n]}$
on
$n$
points, as
$n\rightarrow +\infty ,$
is a Gumbel distribution. In view of this example, they ask for further such Betti distributions. We answer this question for the quasihomogeneous Hilbert schemes
$((\mathbb {C}^{2})^{[n]})^{T_{\alpha ,\beta }}$
that are cut out by torus actions. We prove that their limiting distributions are also of Gumbel type. To obtain this result, we combine work of Buryak, Feigin, and Nakajima on these Hilbert schemes with our generalization of the result of Erdös and Lehner, which gives the distribution of the number of parts in partitions that are multiples of a fixed integer
$A\geq 2.$
Furthermore, if
$p_{k}(A;n)$
denotes the number of partitions of
$n$
with exactly
$k$
parts that are multiples of
$A$
, then we obtain the asymptotic
We investigate norms of spectral projectors on thin spherical shells for the Laplacian on tori. This is closely related to the boundedness of resolvents of the Laplacian and the boundedness of $L^{p}$ norms of eigenfunctions of the Laplacian. We formulate a conjecture and partially prove it.
has appreciably fewer solutions in the subcritical range
$s < \tfrac 12k(k+1)$
than its homogeneous counterpart, provided that
$a_{\ell } \neq 0$
for some
$\ell \leqslant k-1$
. Our methods use Vinogradov’s mean value theorem in combination with a shifting argument.
We prove uniform bounds for the Petersson norm of the cuspidal part of the theta series. This gives an improved asymptotic formula for the number of representations by a quadratic form. As an application, we show that every integer
$n \neq 0,4,7 \,(\textrm{mod}\ 8)$
is represented as
$n= x_1^2 + x_2^2 + x_3^3$
for integers
$x_1,x_2,x_3$
such that the product
$x_1x_2x_3$
has at most 72 prime divisors.