On the Arithmetic Average of the First <i>n</i> Primes
The arithmetic average of the first <i>n</i> primes, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mover accent="true"><mi>p</mi><mo>¯</mo>...
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2025-07-01
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| author | Matt Visser |
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| author_sort | Matt Visser |
| collection | DOAJ |
| description | The arithmetic average of the first <i>n</i> primes, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mi>n</mi></msub><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mi>n</mi></mfrac></mstyle><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>p</mi><mi>i</mi></msub></mrow></semantics></math></inline-formula>, exhibits very many interesting and subtle properties. Since the transformation from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>p</mi><mi>n</mi></msub><mo>→</mo><msub><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mi>n</mi></msub></mrow></semantics></math></inline-formula> is extremely easy to invert, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>p</mi><mi>n</mi></msub><mo>=</mo><mi>n</mi><msub><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mi>n</mi></msub><mo>−</mo><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><msub><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></semantics></math></inline-formula>, it is clear that these two sequences <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>p</mi><mi>n</mi></msub><mo>⟷</mo><msub><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mi>n</mi></msub></mrow></semantics></math></inline-formula> must ultimately carry exactly the same information. But the averaged sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mi>n</mi></msub></semantics></math></inline-formula>, while very closely correlated with the primes, (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mi>n</mi></msub><mo>∼</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>p</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula>), is much “smoother” and much better behaved. Using extensions of various standard results, I shall demonstrate that the prime-averaged sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mi>n</mi></msub></semantics></math></inline-formula> satisfies prime-averaged analogues of the Cramer, Andrica, Legendre, Oppermann, Brocard, Fourges, Firoozbakht, Nicholson, and Farhadian conjectures. (So these prime-averaged analogues are not conjectures; they are theorems). The crucial key to enabling this pleasant behaviour is the “smoothing” process inherent in averaging. While the asymptotic behaviour of the two sequences is very closely correlated, the local fluctuations are quite different. |
| format | Article |
| id | doaj-art-290bacc34e7d4a37b080742971115b85 |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-290bacc34e7d4a37b080742971115b852025-08-20T03:58:30ZengMDPI AGMathematics2227-73902025-07-011314227910.3390/math13142279On the Arithmetic Average of the First <i>n</i> PrimesMatt Visser0School of Mathematics and Statistics, Victoria University of Wellington, P.O. Box 600, Wellington 6140, New ZealandThe arithmetic average of the first <i>n</i> primes, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mi>n</mi></msub><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mi>n</mi></mfrac></mstyle><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>p</mi><mi>i</mi></msub></mrow></semantics></math></inline-formula>, exhibits very many interesting and subtle properties. Since the transformation from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>p</mi><mi>n</mi></msub><mo>→</mo><msub><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mi>n</mi></msub></mrow></semantics></math></inline-formula> is extremely easy to invert, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>p</mi><mi>n</mi></msub><mo>=</mo><mi>n</mi><msub><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mi>n</mi></msub><mo>−</mo><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><msub><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></semantics></math></inline-formula>, it is clear that these two sequences <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>p</mi><mi>n</mi></msub><mo>⟷</mo><msub><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mi>n</mi></msub></mrow></semantics></math></inline-formula> must ultimately carry exactly the same information. But the averaged sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mi>n</mi></msub></semantics></math></inline-formula>, while very closely correlated with the primes, (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mi>n</mi></msub><mo>∼</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>p</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula>), is much “smoother” and much better behaved. Using extensions of various standard results, I shall demonstrate that the prime-averaged sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mi>n</mi></msub></semantics></math></inline-formula> satisfies prime-averaged analogues of the Cramer, Andrica, Legendre, Oppermann, Brocard, Fourges, Firoozbakht, Nicholson, and Farhadian conjectures. (So these prime-averaged analogues are not conjectures; they are theorems). The crucial key to enabling this pleasant behaviour is the “smoothing” process inherent in averaging. While the asymptotic behaviour of the two sequences is very closely correlated, the local fluctuations are quite different.https://www.mdpi.com/2227-7390/13/14/2279the n-th prime pnaverage of the first n primesprime-averaged conjecturesCramerAndricaLegendre |
| spellingShingle | Matt Visser On the Arithmetic Average of the First <i>n</i> Primes Mathematics the n-th prime pn average of the first n primes prime-averaged conjectures Cramer Andrica Legendre |
| title | On the Arithmetic Average of the First <i>n</i> Primes |
| title_full | On the Arithmetic Average of the First <i>n</i> Primes |
| title_fullStr | On the Arithmetic Average of the First <i>n</i> Primes |
| title_full_unstemmed | On the Arithmetic Average of the First <i>n</i> Primes |
| title_short | On the Arithmetic Average of the First <i>n</i> Primes |
| title_sort | on the arithmetic average of the first i n i primes |
| topic | the n-th prime pn average of the first n primes prime-averaged conjectures Cramer Andrica Legendre |
| url | https://www.mdpi.com/2227-7390/13/14/2279 |
| work_keys_str_mv | AT mattvisser onthearithmeticaverageofthefirstiniprimes |