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For every $A>0$ there is $B>0$ such that, for $Q\le x^{1/2}/{\log }^{B}x$,Here $\psi (y;q,a)$ is the Chebyshev function in an arithmetic progression and $\varphi$ is the Euler totient function. Partial summation gives the corresponding result for the prime-counting function and the offset logarithmic integral function. The theorem controls the total error over many moduli; it need not give the same bound for each individual modulus.

Fix a positive integer $q$ and an integer $a$ coprime to $q$. We use Dirichlet characters modulo $q$, extended by zero on integers not coprime to $q$, and write ${\chi }_0$ for the principal Dirichlet character. For a nonprincipal Dirichlet character, the sum over one complete period is zero. Its partial sums $A_{\chi }(u)={\sum }_{n\le u}\chi (n)$ are therefore $O(q)$. Partial summation givesThe integral converges locally uniformly there, so these Dirichlet L-functions are holomorphic at one. On the other hand,has a simple pole at one, with residue $\phi (q)/q$.

The assumed nonvanishing of a nonprincipal Dirichlet L-function at one for real Dirichlet characters also excludes vanishing for nonreal ones. To prove this, form the product over all Dirichlet charactersUse the logarithm supplied by the absolutely convergent Euler product. The Orthogonality of Dirichlet characters yieldsThus $\mathcal{F}(\sigma )\ge 1$. If a nonreal Dirichlet character $\chi$ had $L(1,\chi )=0$, its distinct conjugate $\overline{\chi }$ would also have a zero there, since $L(1,\overline{\chi })=\overline{L(1,\chi )}$. These two zeros would outweigh the single pole of the principal factor; all other factors are bounded near one. It would follow that $\mathcal{F}(\sigma )\to 0$ as $\sigma \downarrow 1$, contradicting the lower bound. Together with the assumption for real Dirichlet characters, this proves $L(1,\chi )\ne 0$ for every nonprincipal Dirichlet character.

The first-degree part of the Euler product logarithm givesThe omitted terms are bounded uniformly, since ${\sum }_p{\sum }_{m\ge 2}(m{p}^{m\sigma }{)}^{-1}\ll {\sum }_p{p}^{-2}<\infty$. For each nonprincipal factor, its logarithm remains bounded: choose a holomorphic logarithm near the now established nonzero value $L(1,\chi )$; the Euler-product branch differs from it by a fixed integral multiple of $2\pi i$ on a sufficiently small real interval. For the principal factor,Applying Orthogonality of Dirichlet characters again givesThe right side diverges as $\sigma \downarrow 1$, so the arithmetic progression contains infinitely many primes. This proves Dirichlet's theorem on arithmetic progressions: every reduced residue class modulo $q$ contains infinitely many primes. The argument also covers $q=1$ and $q=2$, when there need not be a nonreal character.

For the sharper statement about reciprocal primes in a fixed arithmetic progression, use the given prime-counting function asymptotic with $q$ fixed:The second equivalence follows by integration by parts in the logarithmic integral function. Partial summation yieldsFor any $\epsilon >0$, choose a fixed $T$ so large that the relative error in $\pi (t;q,a)$ is at most $\epsilon$ for $t\ge T$. The initial integral is a constant, while the remaining integral lies betweenThe boundary term is $O_q(1/\log x)$. Divide by $\log \log x$, let $x\to \infty$, and then let $\epsilon \downarrow 0$. We obtainOnly a relative asymptotic is deduced from the given information; an additive $O(1)$ error, or uniformity for a growing modulus $q$, would require additional estimates.

Let $1(n)=1$ and $ϵ(n)=1_{n=1}$, the identity for Dirichlet convolution. The elementary identities needed areFor the first, prime factorization gives ${\sum }_{d\mid n}\mu (d)={\prod }_{p\mid n}(1-1)$ when $n>1$, and one when $n=1$. For the second, if $n=\prod p^{a_p}$, then ${\sum }_{d\mid n}\Lambda (d)={\sum }_p{a}_p\log p=\log n$, the Von Mangoldt divisor identity. Convolving the second identity with $\mu$ proves the third.

For $U,V\ge 1$, let the subscripts $\le U$, $>U$, $\le V$, $>V$ denote truncations. The Vaughan identity isIts pointwise form isTo prove it, use ${\mu }_{>U}\ast 1=ϵ-{\mu }_{\le U}\ast 1$ and $1\ast {\Lambda }_{>V}=\log -1\ast {\Lambda }_{\le V}$. This givesand rearrangement proves the formula, as in the Vaughan identity proof.

The Bombieri–Vinogradov theorem states that for every $A>0$ there is $B>0$ such thatHere $\psi (y;q,a)={\sum }_{\begin{array}{c}n\le y\\ n\equiv a(\mathrm{mod}q)\end{array}}\Lambda (n)$ is the Chebyshev function in an arithmetic progression, and $\varphi$ is the Euler totient function. In particular, partial summation givesafter choosing the logarithmic saving in the weighted theorem sufficiently large and absorbing the prime power terms. This is the offset logarithmic integral function.

For the proof strategy, Orthogonality of Dirichlet characters converts errors in arithmetic progressions into character sums of Dirichlet characters weighted by $\Lambda$. Reduction to primitive Dirichlet characters is followed by the character large sieve, which bounds their mean square by $O((N+Q^2)\sum |a_n{|}^2)$ with the usual $q/\varphi (q)$ weights. The Vaughan identity separates the weighted sums into short terms, Type I sums and Type II sums. Direct inner-sum estimates handle the Type I sums; the Cauchy-Schwarz inequality and the large sieve control the Type II sums. The Siegel–Walfisz theorem supplies arbitrarily strong logarithmic savings for small moduli, where an average estimate alone would not suffice. A dyadic decomposition, suitable $U,V$ and a sufficiently large $B$ absorb the divisor and logarithmic losses. This explains why the range is essentially $Q^2\le x$, with logarithmic room to spare.

Put $E_q={\max }_{(a,q)=1}|\pi (x;q,a)-\mathrm{Li}(x)/\varphi (q)|$, with $\mathrm{Li}$ the offset logarithmic integral function. The permitted standard sieve estimate can be taken as the Brun–Titchmarsh theorem:For $q\le x^{0.99}$, $\log (x/q)\ge 0.01\log x$. Also $\mathrm{Li}(x)\ll x/\log x$. Thus both terms defining $E_q$ are $O(x/(\varphi (q)\log x))$, uniformly over the permitted moduli. Apply the Cauchy-Schwarz inequality in the formThis is the weighted arithmetic-progression error bound. The prime omega function and Euler totient function weights can therefore be handled using an unweighted mean error and a separate positive sum.