Second Chebyshev function ($\psi (x)$)

The Second Chebyshev function is the partial sum of the Von Mangoldt function up to a real cutoff $x\ge 0$:For example, $\psi (3)=\log 6$, $\psi (4)=\log 12$, and $\psi (4.5)=\log 12$. At $4=2^2$ the jump has size $\log 2$; between $4$ and $5$ no new prime power greater than one enters the sum. In contrast, the Chebyshev theta function stays at $\vartheta (4)=\log 6$.

There is also an exact connection with the least common multiple. For $N=\lfloor x\rfloor \ge 1$, each prime number $p\le N$ occurs to the highest exponent $k$ for which $p^k\le N$, both in that least common multiple and in the accumulated Von Mangoldt function weights. ThusThis explains the $\log 2520$ value at $x=10$: the least common multiple is $2^3\cdot 3^2\cdot 5\cdot 7$. The Chebyshev theta function instead gives the natural logarithm of the primorial $2\cdot 3\cdot 5\cdot 7=210$. The additional prime powers retain the highest required multiplicities.

Why is this useful for the distribution of prime numbers? The Prime number theorem has the equivalent formswhere $\pi$ is the prime-counting function and $\sim$ denotes asymptotic equivalence. Thus an arithmetic function defined using prime factorization has average weight tending to one over long initial intervals of the positive integers. These are substantive asymptotic equivalences, not consequences of the definitions or the Von Mangoldt divisor identity alone. ^[ref]

The elementary reason the two Chebyshev functions have the same leading behavior is that their difference is small compared with $x$. Every additional prime power has base $p\le \sqrt{x}$, and for each such prime number its total extra weight is at most $\log x$. There are at most $\lfloor \sqrt{x}\rfloor$ such bases, soThis transfers $\psi (x)\sim x$ to $\vartheta (x)\sim x$, or conversely, once either is established. Partial summation then relates the Chebyshev theta function to the prime-counting function. The estimate controls the extra prime powers; proving the leading value itself is the harder Prime number theorem problem.

For the Second Chebyshev function, finite summation gives ${\Psi }_1(x)={\sum }_{n\le x}(x-n)\Lambda (n)$. For $c>1$, Mellin inversion of the kernel $1/(s(s+1))$ givesThe extra denominator makes this smoothing useful: the resulting sum over Nontrivial zeros of the Riemann zeta function is absolutely convergent. This integral differs from the integrated Chebyshev sum, which denotes a sum of the values $\psi (x/n)$.

For $x>1$, moving the Mellin contour past the pole at one, the Nontrivial zeros of the Riemann zeta function, and the remaining poles givesZeros are included with multiplicity. The Riemann–von Mangoldt formula makes the zero sum absolutely convergent. If every nontrivial zero has real part below one, division by $x^2$ and dominated convergence give ${\Psi }_1(x)\sim x^2/2$. Monotonicity of $\psi$ then recovers $\psi (x)\sim x$ by upper and lower difference quotients.

For $\psi (t)={\sum }_{m\le t}\Lambda (m)$, finite rearrangement and the Von Mangoldt divisor identity giveThe Stirling formula, or integral comparison for the logarithm, gives $T(x)=x\log x-x+O(\log (x+1))$ for real $x\ge 1$.

The Chebyshev estimate isIt gives upper and lower bounds of the correct linear order without asserting the Prime number theorem asymptotic $\psi (x)\sim x$.

The central binomial coefficient satisfies $4^m/(2m+1)\le \left(\genfrac{}{}{0ex}{}{2m}{m}\right)\le 4^m$. Prime factors between $m$ and $2m$ give a dyadic upper bound for the Chebyshev theta function. The prime-power exponents in the factorial quotient are sums of zeros and ones, so the central coefficient gives a linear lower bound for the Second Chebyshev function. Higher prime powers contribute only $O(\sqrt{x}\log x)$; hence the prime-counting function has upper and lower bounds of order $x/\log x$.

For $x>1$ away from a jump of $\psi$,where the sum over Nontrivial zeros of the Riemann zeta function is taken symmetrically. A finite-height version has an explicit truncation error controlled by $x/T$ and the distance from $x$ to the nearest prime power.

For $2\le T\le x$, the truncated Perron formula and a contour shift give the displayed form of the Riemann–von Mangoldt explicit formula. The pole at one contributes $x$ and the Nontrivial zeros of the Riemann zeta function contribute the finite sum. One can choose a separated height in $[T,T+1]$ and then restore $T$ using the local zero count for the Riemann zeta function. The error absorbs the half-weight discrepancy at a prime power. Outside this range of $T$, one needs the fuller error with the near-integer terms retained.