Bazzanella, Danilo Some conditional results on primes between consecutive squares. (English) Zbl 1260.11058 Funct. Approximatio, Comment. Math. 45, No. 2, 255-263 (2011). “Between two consecutive squares there is always (at least) one prime number” is a well-known conjecture; however, even assuming the Riemann Hypothesis (RH), its proof is out of reach.We may consider the “exceptional set” for the asymptotic formula \(\psi(x+h)-\psi(x)\sim h\) (\(x\to \infty\)) i.e. \[ E_{\delta}(N,h):=\{ N\leq x\leq 2N : |\psi(x+h)-\psi(x)|\geq \delta h\} \] where \(h=h(x)\) is an increasing function, \(h\to \infty\) as \(x\to \infty\), and \[ \psi(x):=\sum_{p^m\leq x}\log p \] is the (classical) weighted count of primes \(p\leq x\) (with the prime-powers \(p^m\), \(m>1\), giving a negligible contribute).In the case of primes \(n^2<p\leq(n+1)^2\) one takes \(h(n):=(n+1)^2-n^2=2n+1\) and, then, the exceptional set for the expected number of these primes is linked to \(E_{\delta}(N,h)\), see Lemma 2.2.The author, then, derives from conditional bounds on \(E_{\delta}(N,h)\), i.e., say, in almost all short intervals (for primes), results for the exceptional set regarding the initial conjecture (better, for the number of primes in between squares).In fact, he gives three theorems ensuring that the intervals \([n^2,(n+1)^2] \subset [1,N]\) contain the expected number of primes, with at most \(O(N^{\alpha})\) exceptions (here \(0<\alpha<1\), of course), under three different hypotheses. Of course, the exponent \(\alpha\) depends on the conditional estimate assumed; in these cases, it is always about the Riemann zeta-function.Let, as usual, denote \(N(\sigma,T)\) the number of zeros \(\rho=\beta+i\gamma\) of \(\zeta\), satisfying \(\sigma\leq \beta\leq 1\) and \(|\gamma|\leq T\); also, they (the present author and A. Perelli on [J. Number Theory 80, No. 1, 109–124 (2000; Zbl 0972.11087)] define \(N^*(\sigma,T)\) as the number of ordered sets of \(\zeta\) zeros \(\rho_j=\beta_j+i\gamma_j\) \((1\leq j\leq 4)\) for which same limitations for \(N(\sigma,T)\) hold, together with \(|\gamma_1+\gamma_2-\gamma_3-\gamma_4|\leq 1\) and they assume heuristically \[ N^*(\sigma,T)\ll {{N(\sigma,T)^4}\over T}T^{\varepsilon}, \quad T\to \infty {(\ast)} \] (as usual, \(\ll\) is the Vinogradov notation and \(\varepsilon>0\)’s arbitrarily small), i.e. (1.1) in the paper. This is reasonable (though still unproved!), since \(N^*(\sigma,T)\gg N(\sigma,T)^4/T\) is trivial.There are two conditional assumptions regarding \(\zeta\) which are very famous (and both weaker than RH): namely, the Lindelöf Hypothesis (LH) and the weaker Density Hypothesis (DH), see for example the paper by P. Turán [Acta Arith. 4, 31–56 (1958; Zbl 0108.07203)], free online (in which he quotes Ingham’s : LH \(\Rightarrow \) DH; also, RH is stronger than LH, as RH \(\Rightarrow \) LH but it is still unknown if LH \(\Rightarrow \) RH).Under LH the author (see Thm. 1.2) proves that we have \(\alpha=\varepsilon\) (see above, i.e., say \(O(N^{\varepsilon})\) exceptions for primes between squares).Under DH and \((\ast)\) above (i.e., (1.1) in the paper) he proves (see Thm. 1.3) the same \(\alpha=\varepsilon\).Joining Ingham-Huxley density estimate (see (1.2) in the paper) to (1.1), he gets (1.3) (see the paper), under which hypothesis he gets (in Thm. 1.1) the exponent \(\alpha=1/5+\varepsilon\) for the exceptions.Apart from “classical technology” (explicit formulae for primes, zero-density estimates for \(\zeta\) and Heath-Brown’s method for differences of primes) he applies the Lemma 2.1 on the structure of \(E_{\delta}(N,h)\), the exceptional set defined above; then, its consequences for primes between squares (setting \(h:=2\sqrt{x}+1\)) are combined to a Lemma (2.3 in the paper) due to Yu (see the paper for ref.), that under LH bounds a kind of fourth moment for primes in short intervals (i.e., (2.3) in the paper) that modifies the expected number of primes, i.e. \(y/T\), of a “negligible” quantity \(o(y/T)\) (precisely, (2.4) in Lemma 2.3). Reviewer: Giovanni Coppola (Avellino) Cited in 1 ReviewCited in 2 Documents MSC: 11N05 Distribution of primes Keywords:distribution of prime numbers; primes between squares Citations:Zbl 0972.11087; Zbl 0108.07203 × Cite Format Result Cite Review PDF Full Text: DOI Euclid References: [1] D. Bazzanella, Primes between consecutive squares , Arch. Math. 75 (2000), 29-34. · Zbl 1047.11087 · doi:10.1007/s000130050469 [2] D. Bazzanella and A. Perelli, The exceptional set for the number of primes in short intervals , J. Number Theory 80 (2000), 109-124. · Zbl 0972.11087 · doi:10.1006/jnth.1999.2429 [3] H. Davenport, Multiplicative Number Theory , Second edition, Graduate Texts in Mathematics 74. Springer-Verlag, New York (1980). · Zbl 0453.10002 [4] D. R. Heath-Brown, The difference between consecutive primes II , J. London Math. Soc. (2) 19 (1979), 207-220. · Zbl 0394.10021 · doi:10.1112/jlms/s2-19.2.207 [5] A. Ivić, The Riemann Zeta-Function , John Wiley & Sons, New York (1985). [6] H. L. Montgomery and R. C. Vaughan, The large sieve , Mathematika 20 (1973), 119-134. · Zbl 0296.10023 · doi:10.1112/S0025579300004708 [7] G. Yu, The differences between consecutive primes , Bull. London Math. Soc. 28 (3) (1996), 242-248. · Zbl 0861.11052 · doi:10.1112/blms/28.3.242 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.