On certain exponential sums over primes

Let f(x) be a real valued polynomial in x of degree k?4 with leading coefficient α. In this paper, we prove a non-trivial upper bound for the quantity

     

Primitive densities of certain sets of primes

While it has already been demonstrated that the set of twin primes (primes that differ by 2) is scarce in the $\text{Σ}\text{1}\text{p}$ (all twin primes) converges whereas $\text{Σ}\text{1}\text{p}$ (all primes) diverges, this paper proves in Theorems 1 and 2 the scarcity of twin primes (and, in general, of primes p which differ by any even integer as well as primes p for which yp + z is prime, y positive, z nonzero, (y, z) = 1) in a novel and natural way — by showing that the natural density of such primes compared to the set of all primes is 0, that is, $\text{lim}{}_{\mathrm{n\to \infty }}\text{(}\text{π′(n)}\text{π(n)}\text{) = 0}$, where π′(n) is the number of, say, twin primes between 1 and n for any n, and π(n) is the number of all primes between 1 and n. Theorem 3 then establishes that if a set of primes is scarce in the sense that the sum of the reciprocals of such primes converges, they are also scarce in the natural density sense outlined above.     

On Zsigmondy primes

We present simple proofs of Walter Feit's results on large Zsigmondy primes.

     

Distribution of a certain partition function modulo powers of primes

In this paper, we study a certain partition function a(n) defined by Σ _n≥0 a(n)q ⁿ := Π _n=1(1 − q ⁿ )⁻¹(1 − q ²ⁿ )⁻¹. We prove that given a positive integer j ≥ 1 and a prime m ≥ 5, there are infinitely many congruences of the type a(An + B) ≡ 0 (mod m ^j ). This work is inspired by Ono’s ground breaking result in the study of the distribution of the partition function p(n).     

On the sum of distinct primes or squares of primes

In 1965 Erd?s introduced $f_2(s)$: $f_2(s)$ is the smallest integer such that every $l>f_2(s)$ is the sum of s distinct primes or squares of primes where a prime and its square are not both used. We prove that for all sufficiently large s, $f_2(s)?p_2+p_3+?+p_{s+1}+3106$, and the set of s with the equality has the density 1.     

On generalized Ramanujan primes

Let \(\Delta = \sum _{m=0}^\infty q^{(2m+1)^2} \in \mathbf {F}_2[[q]]\) be the reduction mod 2 of the \(\Delta \) series. A modular form of level 1, \(f=\sum _{n\geqslant 0} c(n) \,q^n\), with integer coefficients, is congruent modulo \(2\) to a polynomial in \(\Delta \). Let us set \(W_f(x)=\sum _{n\leqslant x,\ c(n)\text { odd }} 1\), the number of odd Fourier coefficients of \(f\) of index \(\leqslant x\). The order of magnitude of \(W_f(x)\) (for \(x\rightarrow \infty \)) has been determined by Serre in the seventies. Here, we give an asymptotic equivalent for \(W_f(x)\). Let \(p(n)\) be the partition function and \(A_0(x)\) (resp. \(A_1(x)\)) be the number of \(n\leqslant x\) such that \(p(n)\) is even (resp. odd). In the preceding papers, the second-named author has shown that \(A_0(x)\geqslant 0.28 \sqrt{x\;\log \log x}\) for \(x\geqslant 3\) and \(A_1(x)>\frac{4.57 \sqrt{x}}{\log x}\) for \(x\geqslant 7\). Here, it is proved that \(A_0(x)\geqslant 0.069 \sqrt{x}\;\log \log x\) holds for \(x>1\) and that \(A_1(x) \geqslant \frac{0.037 \sqrt{x}}{(\log x)^{7/8}}\) holds for \(x\geqslant 2\). The main tools used to prove these results are the determination of the order of nilpotence of a modular form of level-\(1\) modulo \(2\), and of the structure of the space of those modular forms as a module over the Hecke algebra, which have been given in a recent work of Serre and the second-named author.     

On products of primes and almost primes in arithmetic progressions

We show that for any integers a and m with m ≥ 1 and gcd(a,m) = 1, there is a solution to the congruence pr ≡ a (modm) where p is prime, r is a product of at most k = 17 prime factors and p, r ≤ m. This is a relaxed version of the still open question, studied by P. Erd?s, A. M. Odlyzko and A. Sárközy, that corresponds to k = 1 (that is, to products of two primes).     

On admissible constellations of consecutive primes

Admissible constellations of primes are patterns which, like the twin primes, no simple divisibility relation would prevent from being repeated indefinitely in the series of primes. All admissible constellations, formed ofconsecutive primes, beginning with a prime <1000, are established, and some properties of such constellations in general are conjectured.Dedicated to Peter Naur on the occasion of his 60th birthday     

On primes represented by cubic polynomials

In this paper, we prove a theorem on the distribution of primes in cubic progressions on average.     

On sums of squares of primes II

In this paper we continue our study, begun in G. Harman and A.V. Kumchev (2006) [10], of the exceptional set of integers, not restricted by elementary congruence conditions, which cannot be represented as sums of three or four squares of primes. We correct a serious oversight in our first paper, but make further progress on the exponential sums estimates needed, together with an embellishment of the previous sieve technique employed. This leads to an improvement in our bounds for the maximal size of the exceptional sets.     

On the difference between consecutive primes

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