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Myshkin
Myshkin
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Let $\Lambda$ be the von Mangoldt function. I think the following is known to hold under GRH: Given any $q \geq 1$, $(a,q)=1$, and $X \geq 1$, we have $$ \sum_{\substack{1 \leq n \leq X \\ n \equiv a(\text{mod }q) }} \Lambda(n) = \frac{X}{\phi(q)} + O(X^{1/2} \log (qX)). $$$$ \sum_{\substack{1 \leq n \leq X \\ n \equiv a\,(\text{mod }q) }} \Lambda(n) = \frac{X}{\phi(q)} + O(X^{1/2} \log (qX)). $$ I was wondering if someone could give me a reference for this. I keep finding a slight variant of this without the von Mangoldt, but I just wanted this version so that I can directly reference in a paper. Thank you very much!

Let $\Lambda$ be the von Mangoldt function. I think the following is known to hold under GRH: Given any $q \geq 1$, $(a,q)=1$, and $X \geq 1$, we have $$ \sum_{\substack{1 \leq n \leq X \\ n \equiv a(\text{mod }q) }} \Lambda(n) = \frac{X}{\phi(q)} + O(X^{1/2} \log (qX)). $$ I was wondering if someone could give me a reference for this. I keep finding a slight variant of this without the von Mangoldt, but I just wanted this version so that I can directly reference in a paper. Thank you very much!

Let $\Lambda$ be the von Mangoldt function. I think the following is known to hold under GRH: Given any $q \geq 1$, $(a,q)=1$, and $X \geq 1$, we have $$ \sum_{\substack{1 \leq n \leq X \\ n \equiv a\,(\text{mod }q) }} \Lambda(n) = \frac{X}{\phi(q)} + O(X^{1/2} \log (qX)). $$ I was wondering if someone could give me a reference for this. I keep finding a slight variant of this without the von Mangoldt, but I just wanted this version so that I can directly reference in a paper. Thank you very much!

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Johnny T.
Johnny T.
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Let $\Lambda$ be the von Mangoldt function. I think the following is known to hold under GRH: Given any $q \geq 1$, $(a,q)=1$, and $X \geq 1$, we have $$ \sum_{1 \leq n \leq X} \Lambda(n) = \frac{X}{\phi(q)} + O(X^{1/2} \log (qX)). $$$$ \sum_{\substack{1 \leq n \leq X \\ n \equiv a(\text{mod }q) }} \Lambda(n) = \frac{X}{\phi(q)} + O(X^{1/2} \log (qX)). $$ I was wondering if someone could give me a reference for this. I keep finding a slight variant of this without the von Mangoldt, but I just wanted this version so that I can directly reference in a paper. Thank you very much!

Let $\Lambda$ be the von Mangoldt function. I think the following is known to hold under GRH: Given any $q \geq 1$, $(a,q)=1$, and $X \geq 1$, we have $$ \sum_{1 \leq n \leq X} \Lambda(n) = \frac{X}{\phi(q)} + O(X^{1/2} \log (qX)). $$ I was wondering if someone could give me a reference for this. I keep finding a slight variant of this without the von Mangoldt, but I just wanted this version so that I can directly reference in a paper. Thank you very much!

Let $\Lambda$ be the von Mangoldt function. I think the following is known to hold under GRH: Given any $q \geq 1$, $(a,q)=1$, and $X \geq 1$, we have $$ \sum_{\substack{1 \leq n \leq X \\ n \equiv a(\text{mod }q) }} \Lambda(n) = \frac{X}{\phi(q)} + O(X^{1/2} \log (qX)). $$ I was wondering if someone could give me a reference for this. I keep finding a slight variant of this without the von Mangoldt, but I just wanted this version so that I can directly reference in a paper. Thank you very much!

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Johnny T.
Johnny T.
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Reference for Siegel-Walfisz Theorem under GRH

Let $\Lambda$ be the von Mangoldt function. I think the following is known to hold under GRH: Given any $q \geq 1$, $(a,q)=1$, and $X \geq 1$, we have $$ \sum_{1 \leq n \leq X} \Lambda(n) = \frac{X}{\phi(q)} + O(X^{1/2} \log (qX)). $$ I was wondering if someone could give me a reference for this. I keep finding a slight variant of this without the von Mangoldt, but I just wanted this version so that I can directly reference in a paper. Thank you very much!