On \(4/n= 1/x+ 1/y+1/z\) and Iwaniec' half dimensional sieve
The Erdős-Straus conjecture: For every prime \(q>1\) the diophantine equation \[ {\textstyle {4\over q}= {1\over x} + {1\over y} + {1\over z}} \] has a solution in positive integers \(x\), \(y\), \(z\), is still open. If there is a solution, then either exactly one of \(x\), \(y\), \(z\) or exactly two of these integers have the divisor \(q\). The author only considers the second case, the solubility of \[ {\textstyle {4\over q} = {1\over w} + {1\over {gq}} + {1\over {hq}}}, \tag{*} \] and he defines for integers \(t\) the counting function \[ V(x;k,\ell;t)= \#\{ q\leq x;\;q\equiv\ell\bmod k; \text{ (*) is insoluble with } {\textstyle{{gh}\over w}}=t\}. \] His first result is: If \((k,\ell)=1\), and if \((p\not\equiv 3\bmod 4\) or \(p\nmid (4t\ell+1)\)) for all primes \(p\mid k\), then \[ V(x;k,\ell;t)\geq C_ 1(t,k)\cdot {x\over {(\log x)^{3/2}}} \] for some positive constant \(C_ 1=C_ 1(t,k)\). The second theorem gives a conditional asymptotic formula for \(V(x;k,\ell;1)\) and a conditional lower bound for \(V(x;k,\ell;1)\) with an explicitly given constant \(C_ 2(t,k)\). ``Conditional'' means that the asymptotic formula contains the term \[ E(x,K)= \sum_{k<K} \max_{(\ell,k)=1} \biggl| \pi(x;k,\ell)- {{li x} \over {\varphi(k)}}\biggr|, \] where \(K= x^{1-\varepsilon}\), \(0<\varepsilon<1/6\). The truth of Halberstam's conjecture would imply a sufficiently good estimate for \(E(x, x^{1-\varepsilon})\). The proofs depend on the half-dimensional sieve and Selberg's upper bound sieve method.
insolubility of special diophantine equations
Wirsing's theorem on sums of nonnegative multiplicative functions
scientific article; zbMATH DE number 556361