Note on Legendre-sieve identities at pn+12${p}_{n+1}^{2}$

Abstract We present a finite-sieve framework for studying <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>π</m:mi> <m:mfenced close=")" open="("> <m:mrow> <m:msubsup> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msubsup> </m:mrow> </m:mfenced> </m:math> $\pi \left({p}_{n+1}^{2}\right)$ at the square threshold. Our contributions are threefold: (A) We place the identity π ( x ) = ϕ ( x , a ) + a − 1 with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>a</m:mi> <m:mo>=</m:mo> <m:mi>π</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mrow> <m:mo>⌊</m:mo> <m:mrow> <m:msqrt> <m:mrow> <m:mi>x</m:mi> </m:mrow> </m:msqrt> </m:mrow> <m:mo>⌋</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> $a=\pi (\lfloor \sqrt{x}\rfloor )$ (Legendre–Meissel–Lehmer) on a rigorous inclusion–exclusion footing, specializing to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>x</m:mi> <m:mo>=</m:mo> <m:msubsup> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msubsup> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:math> $x={p}_{n+1}^{2}-1$ and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>x</m:mi> <m:mo>=</m:mo> <m:msubsup> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msubsup> </m:math> $x={p}_{n+1}^{2}$ ; (B) We record two-sided bounds for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>π</m:mi> <m:mfenced close=")" open="("> <m:mrow> <m:msubsup> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msubsup> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mfenced> </m:math> $\pi \left({p}_{n+1}^{2}-1\right)$ and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>π</m:mi> <m:mfenced close=")" open="("> <m:mrow> <m:msubsup> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msubsup> </m:mrow> </m:mfenced> </m:math> $\pi \left({p}_{n+1}^{2}\right)$ obtained by specializing standard explicit inequalities for π ( x ), and we compare these quantities with the Mertens product at sieve level p n ; (C) We partition composites by their least prime factor to obtain <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <m:mi>π</m:mi> <m:mfenced close=")" open="("> <m:mrow> <m:msubsup> <