Zeroes of Dirichlet functions and irregularities in the distribution of primes
Authors:
Carter Bays and Richard H. Hudson
Journal:
Math. Comp. 69 (2000), 861866
MSC (1991):
Primary 11A15, 11M26, 11Y11, 11Y35
Published electronically:
March 10, 1999
MathSciNet review:
1651741
Fulltext PDF Free Access
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Abstract: Seven widely spaced regions of integers with have been discovered using conventional prime sieves. Assuming the generalized Riemann hypothesis, we modify a result of Davenport in a way suggested by the recent work of Rubinstein and Sarnak to prove a theorem which makes it possible to compute the entire distribution of including the sign change (axis crossing) regions, in time linear in using zeroes of the nonprincipal character modulo 4, generously provided to us by Robert Rumely. The accuracy with which the zeroes duplicate the distribution (Figure 1) is very satisfying. The program discovers all known axis crossing regions and finds probable regions up to . Our result is applicable to a wide variety of problems in comparative prime number theory. For example, our theorem makes it possible in a few minutes of computer time to compute and plot a characteristic sample of the difference with fine resolution out to and beyond the region in the vicinity of discovered by te Riele. This region will be analyzed elsewhere in conjunction with a proof that there is an earlier sign change in the vicinity of .
 [1]
Carter
Bays and Richard
H. Hudson, On the fluctuations of Littlewood for
primes of the form 4𝑛̸=1, Math.
Comp. 32 (1978), no. 141, 281–286. MR 0476615
(57 #16174), http://dx.doi.org/10.1090/S00255718197804766158
 [2]
Carter
Bays and Richard
H. Hudson, Numerical and graphical description of all axis crossing
regions for the moduli 4 and 8 which occur before 10¹²,
Internat. J. Math. Math. Sci. 2 (1979), no. 1,
111–119. MR
529694 (80h:10003), http://dx.doi.org/10.1155/S0161171279000119
 [3]
, A new bound for the smallest with to appear in Math. Comp.
 [4]
Carter
Bays and Richard
H. Hudson, The segmented sieve of Eratosthenes and primes in
arithmetic progressions to 10¹², Nordisk Tidskr.
Informationsbehandling (BIT) 17 (1977), no. 2,
121–127. MR 0447090
(56 #5405)
 [5]
Harold
Davenport, Multiplicative number theory, 2nd ed., Graduate
Texts in Mathematics, vol. 74, SpringerVerlag, New York, 1980.
Revised by Hugh L. Montgomery. MR 606931
(82m:10001)
 [6]
Richard
H. Hudson and Carter
Bays, The mean behavior of primes in arithmetic progressions,
J. Reine Angew. Math. 296 (1977), 80–99. MR 0460261
(57 #255)
 [7]
Richard
H. Hudson, Averaging effects on irregularities in
the distribution of primes in arithmetic progressions, Math. Comp. 44 (1985), no. 170, 561–571. MR 777286
(86h:11074), http://dx.doi.org/10.1090/S00255718198507772867
 [8]
Jerzy
Kaczorowski, Results on the distribution of primes, J. Reine
Angew. Math. 446 (1994), 89–113. MR 1256149
(95f:11070), http://dx.doi.org/10.1515/crll.1994.446.89
 [9]
Jerzy
Kaczorowski, On the distribution of primes (mod 4), Analysis
15 (1995), no. 2, 159–171. MR 1344249
(96h:11095)
 [10]
John
Leech, Note on the distribution of prime numbers, J. London
Math. Soc. 32 (1957), 56–58. MR 0083001
(18,642d)
 [11]
Karl
K. Norton, Upper bounds for 𝑘𝑡ℎ power coset
representatives modulo 𝑛, Acta Arith. 15
(1968/1969), 161–179. MR 0240065
(39 #1419)
 [12]
Herman
J. J. te Riele, On the sign of the difference
𝜋(𝑥)𝑙𝑖(𝑥), Math. Comp. 48 (1987), no. 177, 323–328. MR 866118
(88a:11135), http://dx.doi.org/10.1090/S00255718198708661186
 [13]
Michael
Rubinstein and Peter
Sarnak, Chebyshev’s bias, Experiment. Math.
3 (1994), no. 3, 173–197. MR 1329368
(96d:11099)
 [14]
Daniel
Shanks, Quadratic residues and the
distribution of primes, Math. Tables Aids
Comput. 13 (1959),
272–284. MR 0108470
(21 #7186), http://dx.doi.org/10.1090/S00255718195901084708
 [1]
 Carter Bays and Richard H. Hudson, On the fluctuations of Littlewood for primes of the form 4n+1, Math. Comp. vol. 32 (1978), pp. 199. MR 57:16174
 [2]
 , Numerical and graphical description of all axis crossing regions for the moduli 4 and 8 which occur before Int. J. Math. & Math. Sci. , vol. 2(1979), pp. 111119. MR 80h:10003
 [3]
 , A new bound for the smallest with to appear in Math. Comp.
 [4]
 , The segmented sieve of Erastosthenes and primes in arithmetic progressions to BIT, vol. 17, (1977), pp. 199. MR 56:5405
 [5]
 H. Davenport, Multiplicative Number Theory (2nd ed.), Graduate Texts in Mathematics, vol. 74, Springer, Berlin, 1980. MR 82m:10001
 [6]
 Richard H. Hudson, The mean behavior of primes in arithmetic progressions, J. Reine Angew. Math., vol. 296 (1977),pp. 8099. MR 57:255
 [7]
 , Averaging effects on irregularities in the distribution of primes in arithmetic progressions, Math.Comp., vol. 44 (1985), pp. 199. MR 86h:11074
 [8]
 J. Kaczorowski, Results on the distribution of primes, J. Reine Angew. Math. 446 (1994), 89113. MR 95f:11070
 [9]
 , On the distribution of primes mod , Analysis 15 (1995), 159171. MR 96h:11095
 [10]
 John Leech, Note on the distribution of prime numbers, J. London Math. Soc., vol. 32 (1957), pp. 199. MR 18:642d
 [11]
 Karl K. Norton, Upper bounds for kth power coset representatives modulo n, Acta Arith., vol. 15 (1968/69) pp. 161179. MR 39:1419
 [12]
 Herman te Riele, On the sign change of the difference Math. Comp., vol 48 (1986), pp. 199. MR 88a:11135
 [13]
 Michael Rubinstein and Peter Sarnak, Chebyshev's Bias, Experimental Mathematics, vol. 3 (1994), pp. 199. MR 96d:11099
 [14]
 Daniel Shanks, Quadratic residues and the distribution of primes, Math. Comp., vol 13 (1959), pp. 199. MR 21:7186
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Additional Information
Carter Bays
Affiliation:
Department of Computer Science, University of South Carolina, Columbia, South Carolina 29208
Email:
bays@cs.sc.edu
Richard H. Hudson
Affiliation:
Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
Email:
hudson@math.sc.edu
DOI:
http://dx.doi.org/10.1090/S0025571899011059
PII:
S 00255718(99)011059
Received by editor(s):
March 17, 1997
Received by editor(s) in revised form:
April 1, 1998, and July 6, 1998
Published electronically:
March 10, 1999
Article copyright:
© Copyright 2000 American Mathematical Society
