Selberg sieve weights

FordefineThe optimizing Selberg weights arefor $d<D$ dividing $P(z)$, and zero otherwise. They satisfy ${\lambda }_1=1$ and diagonalize the main quadratic form to $1/G(D,z)$.

Put $g(n)=\mu (n{)}^2/\phi (n)$. As a Dirichlet convolution, $g=h\ast (n\mapsto 1/n)$, where $h$ is a multiplicative arithmetic function with $h(p)=1/(p(p-1))$, $h(p^2)=-1/(p(p-1))$, and $h(p^j)=0$ for $j\ge 3$. The local sums are one, so ${\sum }_{n}h(n)=1$. The series ${\sum }_n|h(n)|\log (2n)$ converges, since the prime-local errors are $O((\log p)/p^2)$. Consequently $G(z)={\sum }_{d\le z}h(d)H_{\lfloor z/d\rfloor }=\log z+O(1)$ by the harmonic number estimate $H_m=\log m+O(1)$. This identifies the leading constant in the one-dimensional Selberg upper-bound sieve.

Put $L=\sqrt{D}$, $h(t)={\prod }_{p\mid t}(g(p{)}^{-1}-1)$ and $y_t={\sum }_{\begin{array}{c}m\le L\\ t\mid m\end{array}}g(m){\rho }_m$, where $g$ is a multiplicative arithmetic function supported on squarefree integers. For squarefree integers $m,n$,Thus the quadratic form is $\Sigma ={\sum }_{t\le L}h(t)y_t^2$. Möbius inversion gives $g(d){\rho }_d={\sum }_{d\mid t\le L}\mu (t/d)y_t$. On the permitted squarefree integers, the constraint ${\rho }_1=1$ becomes ${\sum }_t\mu (t)y_t=1$. The Cauchy-Schwarz inequality yields $\Sigma \ge 1/J$, where $J={\sum }_{t}1/h(t)$ over those integers; equality holds at $y_t=\mu (t)/(Jh(t))$.

For real coefficients ${\rho }_1=1$ supported on permitted divisors up to $L$, put ${\lambda }_d={\sum }_{[u,v]=d}{\rho }_u{\rho }_v$. Then ${\sum }_{d\mid n}{\lambda }_d=({\sum }_{d\mid n}{\rho }_d{)}^2$ is an upper bound for the indicator function of numbers free of the forbidden prime factors. Its support has $d\le L^2$. If the coefficients vanish on non-squarefree integers and satisfy $|{\rho }_d|\le 1$, then $|{\lambda }_d|\le 3^{\omega (d)}$: each prime factor of $d$ occurs in $u$, in $v$, or in both.

Let $k(d)={\prod }_{p\mid d}g(p)/(1-g(p))$ on permitted squarefree integers. The Selberg sieve weights haveThe numerator is the total weight of the distinct integers $ev\le L$ with $e\mid d$ and $(v,d)=1$: multiplicativity gives ${\sum }_{e\mid d}k(e)={\prod }_{p\mid d}(1-g(p){)}^{-1}$. They are a subset of the terms in $J$.