Variants on Andrica’s Conjecture with and without the Riemann Hypothesis
oleh: Matt Visser
| Format: | Article |
|---|---|
| Diterbitkan: | MDPI AG 2018-11-01 |
Deskripsi
The gap between what we can explicitly prove regarding the distribution of primes and what we suspect regarding the distribution of primes is enormous. It is (reasonably) well-known that the Riemann hypothesis is not sufficient to prove Andrica’s conjecture: <inline-formula> <math display="inline"> <semantics> <mrow> <mo>∀</mo> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>, is <inline-formula> <math display="inline"> <semantics> <mrow> <msqrt> <msub> <mi>p</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </msqrt> <mo>−</mo> <msqrt> <msub> <mi>p</mi> <mi>n</mi> </msub> </msqrt> <mo>≤</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>? However, can one at least get tolerably close? I shall first show that with a logarithmic modification, provided one assumes the Riemann hypothesis, one has <disp-formula> <math display="block"> <semantics> <mrow> <mrow> <msqrt> <msub> <mi>p</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </msqrt> <mo>/</mo> <mo form="prefix">ln</mo> <msub> <mi>p</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo>−</mo> <mrow> <msqrt> <msub> <mi>p</mi> <mi>n</mi> </msub> </msqrt> <mo>/</mo> <mo form="prefix">ln</mo> <msub> <mi>p</mi> <mi>n</mi> </msub> </mrow> <mo><</mo> <mrow> <mn>11</mn> <mo>/</mo> <mn>25</mn> </mrow> <mo>;</mo> <mspace width="2.em"></mspace> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>≥</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mrow> </mrow> </semantics> </math> </disp-formula> Then, by considering more general <inline-formula> <math display="inline"> <semantics> <msup> <mi>m</mi> <mrow> <mi>t</mi> <mi>h</mi> </mrow> </msup> </semantics> </math> </inline-formula> roots, again assuming the Riemann hypothesis, I show that <disp-formula> <math display="block"> <semantics> <mrow> <mroot> <msub> <mi>p</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mi>m</mi> </mroot> <mo>−</mo> <mroot> <msub> <mi>p</mi> <mi>n</mi> </msub> <mi>m</mi> </mroot> <mo><</mo> <mrow> <mn>44</mn> <mo>/</mo> <mo stretchy="false">(</mo> <mn>25</mn> <mspace width="0.166667em"></mspace> <mi>e</mi> <mspace width="0.166667em"></mspace> <mo stretchy="false">[</mo> <mi>m</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mrow> <mo>;</mo> <mspace width="2.em"></mspace> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>≥</mo> <mn>3</mn> <mo>;</mo> <mspace width="0.277778em"></mspace> <mi>m</mi> <mo>></mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mrow> </mrow> </semantics> </math> </disp-formula> In counterpoint, if we limit ourselves to what we can currently prove unconditionally, then the only explicit Andrica-like results seem to be variants on the relatively weak results below: <disp-formula> <math display="block"> <semantics> <mrow> <msup> <mo form="prefix">ln</mo> <mn>2</mn> </msup> <msub> <mi>p</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msup> <mo form="prefix">ln</mo> <mn>2</mn> </msup> <msub> <mi>p</mi> <mi>n</mi> </msub> <mo><</mo> <mn>9</mn> <mo>;</mo> <mspace width="2.em"></mspace> <msup> <mo form="prefix">ln</mo> <mn>3</mn> </msup> <msub> <mi>p</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msup> <mo form="prefix">ln</mo> <mn>3</mn> </msup> <msub> <mi>p</mi> <mi>n</mi> </msub> <mo><</mo> <mn>52</mn> <mo>;</mo> <mspace width="2.em"></mspace> <msup> <mo form="prefix">ln</mo> <mn>4</mn> </msup> <msub> <mi>p</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msup> <mo form="prefix">ln</mo> <mn>4</mn> </msup> <msub> <mi>p</mi> <mi>n</mi> </msub> <mo><</mo> <mn>991</mn> <mo>;</mo> <mspace width="2.em"></mspace> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>≥</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mrow> </mrow> </semantics> </math> </disp-formula> I shall also update the region on which Andrica’s conjecture is unconditionally verified.