메뉴 건너뛰기




Volumn 465, Issue 2107, 2009, Pages 2197-2216

The first-digit frequencies of prime numbers and Riemann zeta zeros

Author keywords

Benford's law; Counting function; First significant digit; Pattern; Prime number; Riemann zeta function

Indexed keywords

BENFORD'S LAW; COUNTING FUNCTION; FIRST SIGNIFICANT DIGIT; PATTERN; PRIME NUMBER; RIEMANN ZETA FUNCTION;

EID: 67650851490     PISSN: 13645021     EISSN: 14712946     Source Type: Journal    
DOI: 10.1098/rspa.2009.0126     Document Type: Article
Times cited : (34)

References (35)
  • 1
    • 0000956831 scopus 로고
    • The law of anomalous numbers
    • Benford, F. 1938 The law of anomalous numbers. Proc. Am. Philos. Soc. 78, 551-572.
    • (1938) Proc. Am. Philos. Soc , vol.78 , pp. 551-572
    • Benford, F.1
  • 2
    • 12744263576 scopus 로고    scopus 로고
    • One-dimensional dynamical systems and Benford's law
    • doi:10.1090/S0002-9947-04-03455-5
    • Berger, A., Bunimovich, L. A. & Hill, T. P. 2005 One-dimensional dynamical systems and Benford's law. Trans. Am. Math. Soc. 357, 197-220. (doi:10.1090/S0002-9947-04-03455-5)
    • (2005) Trans. Am. Math. Soc , vol.357 , pp. 197-220
    • Berger, A.1    Bunimovich, L.A.2    Hill, T.P.3
  • 3
    • 0032647538 scopus 로고    scopus 로고
    • The Riemann zeta-zeros and eigenvalue asymptotics
    • doi:10.1137/S0036144598347497
    • Berry, M. V. & Keating, J. P. 1999 The Riemann zeta-zeros and eigenvalue asymptotics. SIAM Rev. 41, 236-266. (doi:10.1137/S0036144598347497)
    • (1999) SIAM Rev , vol.41 , pp. 236-266
    • Berry, M.V.1    Keating, J.P.2
  • 4
    • 34547992651 scopus 로고    scopus 로고
    • Riemann zeta functions and quantum chaos
    • doi:10.1143/PTPS.166.19
    • Bogomolny, E. 2007 Riemann zeta functions and quantum chaos. Prog. Theor. Phys. Suppl. 166, 19-44. (doi:10.1143/PTPS.166.19)
    • (2007) Prog. Theor. Phys. Suppl , vol.166 , pp. 19-44
    • Bogomolny, E.1
  • 5
    • 0034608812 scopus 로고    scopus 로고
    • A pseudo zeta function and the distribution of primes
    • doi:10.1073/pnas.97.14.7697
    • Chernoff, P. R. 2000 A pseudo zeta function and the distribution of primes. Proc. Natl Acad. Sci. USA 97, 7697-7699. (doi:10.1073/pnas.97.14.7697)
    • (2000) Proc. Natl Acad. Sci. USA , vol.97 , pp. 7697-7699
    • Chernoff, P.R.1
  • 6
    • 12444279519 scopus 로고
    • Prime numbers and probability
    • Cramér, H. 1935 Prime numbers and probability. Skand. Mat. Kongr. 8, 107-115.
    • (1935) Skand. Mat. Kongr , vol.8 , pp. 107-115
    • Cramér, H.1
  • 7
    • 0002417821 scopus 로고
    • The distribution of leading digits and uniform distribution mod 1
    • doi:10.1214/aop/ 1176995891
    • Diaconis, P. 1977 The distribution of leading digits and uniform distribution mod 1. Ann. Probab. 5, 72-81. (doi:10.1214/aop/ 1176995891)
    • (1977) Ann. Probab , vol.5 , pp. 72-81
    • Diaconis, P.1
  • 10
    • 0000030933 scopus 로고
    • On the probability that a random integer has initial digit A
    • doi:10.2307/2314636
    • Flehinger, B. J. 1966 On the probability that a random integer has initial digit A. Am. Math. Mon. 73, 1056-1061. (doi:10.2307/2314636)
    • (1966) Am. Math. Mon , vol.73 , pp. 1056-1061
    • Flehinger, B.J.1
  • 11
    • 43449086031 scopus 로고    scopus 로고
    • The primes contain arbitrary long arithmetic progressions
    • Green, B. & Tao, T. 2008 The primes contain arbitrary long arithmetic progressions. Ann. Math. 167, 481-547.
    • (2008) Ann. Math , vol.167 , pp. 481-547
    • Green, B.1    Tao, T.2
  • 13
    • 33746062963 scopus 로고
    • The random sieve
    • Hawkins, D. 1957 The random sieve. Math. Mag. 31, 1-3.
    • (1957) Math. Mag , vol.31 , pp. 1-3
    • Hawkins, D.1
  • 14
    • 84966250122 scopus 로고
    • Base-invariance implies Benford's law
    • doi:10.2307/2160815
    • Hill, T. P. 1995a Base-invariance implies Benford's law. Proc. Am. Math. Soc. 123, 887-895. (doi:10.2307/2160815)
    • (1995) Proc. Am. Math. Soc , vol.123 , pp. 887-895
    • Hill, T.P.1
  • 15
    • 84972541029 scopus 로고
    • A statistical derivation of the significant-digit law
    • Hill, T. P. 1995b A statistical derivation of the significant-digit law. Stat. Sci. 10, 354-363.
    • (1995) Stat. Sci , vol.10 , pp. 354-363
    • Hill, T.P.1
  • 16
    • 0001718673 scopus 로고    scopus 로고
    • The first-digit phenomenon
    • Hill, T. P. 1996 The first-digit phenomenon. Am. Sci. 86, 358-363.
    • (1996) Am. Sci , vol.86 , pp. 358-363
    • Hill, T.P.1
  • 17
    • 0004245874 scopus 로고
    • Zurich, Switzerland: AB Academic Publishers pp
    • Hlawka, E. 1984 The theory of uniform distribution. Zurich, Switzerland: AB Academic Publishers pp. 122-123.
    • (1984) The theory of uniform distribution , pp. 122-123
    • Hlawka, E.1
  • 21
    • 0034556676 scopus 로고    scopus 로고
    • Survival distributions satisfying Benford's law
    • doi:10.2307/2685773
    • Leemis, L. M., Schmeiser, W. & Evans, D. L. 2000 Survival distributions satisfying Benford's law. Am. Stat. 54, 236-241. (doi:10.2307/2685773)
    • (2000) Am. Stat , vol.54 , pp. 236-241
    • Leemis, L.M.1    Schmeiser, W.2    Evans, D.L.3
  • 22
    • 0000539930 scopus 로고    scopus 로고
    • Stochastic multiplicative processes with reset events
    • doi:10.1103/PhysRevE.59.4945
    • Manrubia, S. C. & Zanette, D. H. 1999 Stochastic multiplicative processes with reset events. Phys. Rev. E 59, 4945-4948. (doi:10.1103/PhysRevE.59.4945)
    • (1999) Phys. Rev. E , vol.59 , pp. 4945-4948
    • Manrubia, S.C.1    Zanette, D.H.2
  • 24
    • 33645139599 scopus 로고    scopus 로고
    • Benford's law, values of L-functions and the 3x+1 problem
    • doi:10.4064/aa120-3-4
    • Miller, S. J. & Kontorovich, A. 2005 Benford's law, values of L-functions and the 3x+1 problem. Acta Arithmetica 120, 269-297. (doi:10.4064/aa120-3-4)
    • (2005) Acta Arithmetica , vol.120 , pp. 269-297
    • Miller, S.J.1    Kontorovich, A.2
  • 26
    • 0003038640 scopus 로고
    • Note on the frequency of use of the different digits in natural numbers
    • doi:10.2307/2369148
    • Newcomb, S. 1881 Note on the frequency of use of the different digits in natural numbers. Am. J. Math. 4, 39-40. (doi:10.2307/2369148)
    • (1881) Am. J. Math , vol.4 , pp. 39-40
    • Newcomb, S.1
  • 28
    • 34548737045 scopus 로고    scopus 로고
    • Benford's law applied to hydrological data-results and relevance to other geophysical data
    • doi:10.1007/s11004-007-9109-5
    • Nigrini, M. J. & Miller, S. J. 2007 Benford's law applied to hydrological data-results and relevance to other geophysical data. Math. Geol. 39, 469-490. (doi:10.1007/s11004-007-9109-5)
    • (2007) Math. Geol , vol.39 , pp. 469-490
    • Nigrini, M.J.1    Miller, S.J.2
  • 29
    • 0034825396 scopus 로고    scopus 로고
    • Explaining the uneven distribution of numbers in nature: The laws of Benford and Zipf
    • doi:10.1016/S0378-4371 (00) 00633-6
    • Pietronero, L., Tossati, E., Tossati, V. & Vespignani, A. 2001 Explaining the uneven distribution of numbers in nature: the laws of Benford and Zipf. Physica A 293, 297-304. (doi:10.1016/S0378-4371 (00) 00633-6)
    • (2001) Physica A , vol.293 , pp. 297-304
    • Pietronero, L.1    Tossati, E.2    Tossati, V.3    Vespignani, A.4
  • 30
    • 0000109243 scopus 로고
    • On the distribution of first significant digits
    • doi:10.1214/aoms/1177704862
    • Pinkham, R. S. 1961 On the distribution of first significant digits. Ann. Math. Stat. 32, 1223-1230. (doi:10.1214/aoms/1177704862)
    • (1961) Ann. Math. Stat , vol.32 , pp. 1223-1230
    • Pinkham, R.S.1
  • 31
    • 0001295471 scopus 로고
    • The first digit problem
    • doi:10.2307/2319349
    • Raimi, R. A. 1976 The first digit problem. Am. Math. Mon. 83, 521-538. (doi:10.2307/2319349)
    • (1976) Am. Math. Mon , vol.83 , pp. 521-538
    • Raimi, R.A.1
  • 33
    • 3343014432 scopus 로고
    • A visual display of some properties of the distribution of primes
    • doi:10.2307/2312588
    • Stein, M. L., Ulam, S. M. & Wells, M. B. 1964 A visual display of some properties of the distribution of primes. Am. Math. Mon. 71, 516-520. (doi:10.2307/2312588)
    • (1964) Am. Math. Mon , vol.71 , pp. 516-520
    • Stein, M.L.1    Ulam, S.M.2    Wells, M.B.3
  • 35
    • 67650823179 scopus 로고
    • Initial digits for the sequence of primes
    • doi:10. 2307/2316536
    • Withney, R. E. 1972 Initial digits for the sequence of primes. Am. Math. Mon. 79, 150-152. (doi:10. 2307/2316536)
    • (1972) Am. Math. Mon , vol.79 , pp. 150-152
    • Withney, R.E.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.