Alessandro Zaccagnini

A Diophantine problem with a prime and three squares of primes

We prove that if λ1, λ2, λ3 and λ4 are non-zero real numbers, not all of the same sign, λ1/λ2 is irrational, and ϖ is any real number then, for any e>0, the inequality |λ1p1+λ2p22+λ3p32+λ4p42+ϖ|⩽(maxjpj)−1/18+e has infinitely many solutions in prime variables p1,…,p4.

The number of Goldbach representations of an integer

The first result of this kind was proved in 1991 by Fujii who subsequently improved it (see [4]-[5]-[6]) until reaching the error term O ( (N logN) ) . Then Granville [8]-[9] gave an alternative proof of the same result and, finally, Bhowmik and Schlage-Puchta [2] were able to reach the error term O ( N logN ) ; in [2] they also proved that the error term is Ω(N log logN). Our result improves the upper bound in Bhowmik and Schlage-Puchta [2] by a factor logN . In fact, this seems to be the limit of the method in the current state of the circlemethod technology: see the remark after the proof. If one admits the presence of some suitable weight in our average, this loss can be avoided. For example, using the Fejér weight we could work with L(N ;α) = ∑N n=−N(N−|n|)e(nα) = |T (N ;α)| instead of T (N ;α) in (23). The key property is that, for 1/N < |α| ≤ 1/2, the function L(N ;α) decays as α instead of |α| and so the dissection argument in (26) is now more efficient and does not cause any loss of logs. Such a phenomenon is well-known from the literature about the existence of Goldbach numbers in short intervals, see, e.g., Languasco and Perelli [12]. In fact we will obtain Theorem 1 as a consequence of a weighted result. Letting ψ(x) = ∑ m≤x Λ(m), we have

On the exceptional set for the sum of a prime and a k -th power

Let k ≤ 2 be an integer, and set E k (X) = |{ n ≤ X, n ≠ m k , n not a sum of a prime and a k -th power}|. We prove that there exists δ = δ( k ) > 0 such that E k (X) E k ( X )≪ k X 1−δ .

On the constant in the Mertens product for arithmetic progressions. II: Numerical values

We give explicit numerical values with 100 decimal digits for the constant in the Mertens product over primes in the arithmetic progressions

A Cesàro average of Goldbach numbers

a \bmod q

A Cesàro average of Hardy–Littlewood numbers

, for

On a ternary Diophantine problem with mixed powers of primes

q \in \{3

Sums of many primes

, ...,

Computing the Mertens and Meissel–Mertens Constants for Sums over Arithmetic Progressions

100\}

SUM OF ONE PRIME AND TWO SQUARES OF PRIMES IN SHORT INTERVALS

and