On variants of the larger sieve

Gallagher's larger sieve is a powerful tool, when dealing with sequences of integers that avoid many residue classes. We present and discuss various variants of Gallagher's larger sieve. This revised version was published online in June 2006 with corrections to the Cover Date.     

An almost-prime sieve

We compare and contrast three methods for estimating the number of integers in an interval of length x which have fewer than k distinct prime factors less than z, with special attention to the case k = 2. An iterative method based on the case k = 1 is simplest. If z is sufficiently small compared to x one may use a kind of Brun sieve. Selberg's sieve method gives a good estimate for k = 2 but leads into technical difficulties as k increases.     

Some applications of the double large sieve

We sharpen a procedure of Cao and Zhai (J Théorie Nombres Bordeaux,11: 407–423, 1999) to estimate the sum $$\begin{aligned} \sum _{m\sim M} \sum _{n\sim N} a_m b_n \, e\left(\frac{F m^\alpha n^\beta }{M^\alpha N^\beta }\right) \end{aligned}$$ with $|a_m|,\ |b_n| \le 1$ . We apply this to give bounds for the discrepancy (mod 1) of the sequence $\{p^c: p\le X\}$ where $p$ is a prime variable, in the range $\frac{130}{79}\le c \le \frac{11}{5}$ . An alternative strategy is used for the range $1.48 \le c \le \frac{130}{79}$ . We use further exponential sum estimates to show that for large $R>0$ , and a small constant $\eta >0$ , the inequality $$\begin{aligned} \left| p_1^c+p_2^c+p_3^c+p_4^c+p_5^c - R\right| < R^{-\eta } \end{aligned}$$ holds for many prime tuples, provided $2<c\le 2.041$ . This improves work of Cao and Zhai (Monatsh Math, 150:173–179, 2007) and a theorem claimed by Shi and Liu (Monatsh Math, published online, 2012).     

A footnote to the large sieve

This footnote contains a simplified proof of a qualitative version of a result due to Montgomery and Vaughan. A remark on the estimate of the associated bilinear form is also included.     

Parity of the partition function

Although much is known about the partition function, little is known about its parity. For the polynomials D(x):=(Dx²+1)/24, where , we show that there are infinitely many m (resp. n) for which p(D(m)) is even (resp. p(D(n)) is odd) if there is at least one such m (resp. n). We bound the first m and n (if any) in terms of the class number h(−D). For prime D we show that there are indeed infinitely many even values. To this end we construct new modular generating functions using generalized Borcherds products, and we employ Galois representations and locally nilpotent Hecke algebras.     

On Bombieri's asymptotic sieve

We improve the error term in the Bombieri asymptotic sieve when the summation is restricted to integers having at most two prime factors. This results in a refined bilinear decomposition for the characteristic function of the primes that enables us to get a best possible estimate for the trigonometric polynomial over primes.     

A smoothed GPY sieve

Combining the arguments developed in the works of D. A. Goldston,S. W. Graham, J. Pintz, and C. Y. Yildirim [Preprint, 2005,arXiv: math.NT/506067] and Y. Motohashi [Number theory in progress– A. Schinzel Festschrift (de Gruyter, 1999) 1053–1064]we introduce a smoothing device to the sieve procedure of Goldston,Pintz, and Yildirim (see [Proc. Japan Acad. 82A (2006) 61–65]for its simplified version). Our assertions embodied in Lemmas3 and 4 of this article imply that a natural extension of aprime number theorem of E. Bombieri, J. B. Friedlander, andH. Iwaniec [Theorem 8 in Acta Math. 156 (1986) 203–251]should give rise infinitely often to bounded differences betweenprimes, that is, a weaker form of the twin prime conjecture.     

On the restricted partition function

For a vector \(\mathbf a = (a_1,\ldots ,a_r)\) of positive integers, we prove formulas for the restricted partition function \(p_{\mathbf a}(n): = \) the number of integer solutions \((x_1,\dots ,x_r)\) to \(\sum _{j=1}^r a_jx_j=n\) with \(x_1\ge 0, \ldots , x_r\ge 0\) and its polynomial part.     

Modular class primes in the Sundaram sieve

The purpose of this paper is to consider analogues of the twin-prime conjecture in various classes within modular rings.     

On Bombieri's asymptotic sieve

If a sequence of non-negative real numbers has ``best possible' distribution in arithmetic progressions, Bombieri showed that one can deduce an asymptotic formula for the sum for . By constructing appropriate sequences, we show that any weakening of the well-distribution property is not sufficient to deduce the same conclusion.

     

Dissecting the Stanley partition function

Let p(n) denote the number of unrestricted partitions of n. For i=0, 2, let p_i(n) denote the number of partitions π of n such that . Here denotes the number of odd parts of the partition π and π^′ is the conjugate of π. Stanley [Amer. Math. Monthly 109 (2002) 760; Adv. Appl. Math., to appear] derived an infinite product representation for the generating function of p₀(n)-p₂(n). Recently, Swisher [The Andrews–Stanley partition function and p(n), preprint, submitted for publication] employed the circle method to show that

(i)

and that for sufficiently large n

(ii)

In this paper we study the even/odd dissection of the Stanley product, and show how to use it to prove (i) and (ii) with no restriction on n. Moreover, we establish the following new result:

Two proofs of this surprising inequality are given. The first one uses the Göllnitz–Gordon partition theorem. The second one is an immediate corollary of a new partition inequality, which we prove in a combinatorial manner. Our methods are elementary. We use only Jacobi's triple product identity and some naive upper bound estimates.     

On the Yao-Yao partition theorem

The Yao-Yao partition theorem states that for any probability measure μ on having a density which is continuous and bounded away from 0, it is possible to partition into 2ⁿ regions of equal measure for μ in such a way that every affine hyperplane of avoids at least one of the regions. We give a constructive proof of this result and extend it to slightly more general measures. Received: 21 August 2008     

Multiplicative properties of the partition function

A lower bound for the number of multiplicatively independent values ofp(n) forN ≤ n <N + R is given. The proof depends on the Hardy-Ramanujan formula and is of an elementary nature.     

Computing the integer partition function

In this paper we discuss efficient algorithms for computing the values of the partition function and implement these algorithms in order to conduct a numerical study of some conjectures related to the partition function. We present the distribution of for for primes up to and small powers of and .