<p>For <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_640_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="88" /> </InlineMediaObject> <EquationSource Format="TEX">$i\in \{1,2,3\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>∈</mo> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>, let <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_640_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$E_i(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>E</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denote the error term in each of the three theorems of Mertens on the asymptotic distribution of prime numbers. We show that for <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_640_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">$i\in \{1,2\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>∈</mo> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> the Riemann hypothesis is equivalent to the condition <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_640_Article_IEq4.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="118" /> </InlineMediaObject> <EquationSource Format="TEX">$\int _2^X E_i(x)\,\textrm{d}x>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mo>∫</mo> <mn>2</mn> <mi>X</mi> </msubsup> <msub> <mi>E</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="0.166667em" /> <mtext>d</mtext> <mi>x</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_640_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$X>2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>></mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, and we examine assumptions under which the equivalence also holds for <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_640_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">$i=3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>=</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. In addition, we extend our results to analogues of Mertens’ theorems concerning prime sums twisted by quadratic Dirichlet characters or restricted to arithmetic progressions.</p>

On the mean values of the error terms in Mertens’ theorems

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2025年6月16日

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Tianyu Zhao

摘要

For $i\in \{1,2,3\}$ $i \in {1, 2, 3}$ , let $E_i(x)$ $E_{i} (x)$ denote the error term in each of the three theorems of Mertens on the asymptotic distribution of prime numbers. We show that for $i\in \{1,2\}$ $i \in {1, 2}$ the Riemann hypothesis is equivalent to the condition $\int _2^X E_i(x)\,\textrm{d}x>0$ $\int_{2}^{X} E_{i} (x) d x > 0$ for all $X>2$ $X > 2$ , and we examine assumptions under which the equivalence also holds for $i=3$ $i = 3$ . In addition, we extend our results to analogues of Mertens’ theorems concerning prime sums twisted by quadratic Dirichlet characters or restricted to arithmetic progressions.