Zeroes of Dirichlet -functions

and irregularities

in the distribution of primes

Authors:
Carter Bays and Richard H. Hudson

Journal:
Math. Comp. **69** (2000), 861-866

MSC (1991):
Primary 11A15, 11M26, 11Y11, 11Y35

DOI:
https://doi.org/10.1090/S0025-5718-99-01105-9

Published electronically:
March 10, 1999

MathSciNet review:
1651741

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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**, https://doi.org/10.1090/S0025-5718-1978-0476615-8**[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**, https://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****[5]**Harold Davenport,*Multiplicative number theory*, 2nd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York-Berlin, 1980. Revised by Hugh L. Montgomery. MR**606931****[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**, https://doi.org/10.1515/crll.1977.296.80**[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**, https://doi.org/10.1090/S0025-5718-1985-0777286-7**[8]**Jerzy Kaczorowski,*Results on the distribution of primes*, J. Reine Angew. Math.**446**(1994), 89–113. MR**1256149**, https://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**, https://doi.org/10.1524/anly.1995.15.2.159**[10]**John Leech,*Note on the distribution of prime numbers,*J. London Math. Soc., vol. 32 (1957), pp. 1-99. MR**18:642d****[11]**Karl K. Norton,*Upper bounds for 𝑘𝑡ℎ power coset representatives modulo 𝑛*, Acta Arith.**15**(1968/1969), 161–179. MR**0240065****[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**, https://doi.org/10.1090/S0025-5718-1987-0866118-6**[13]**Michael Rubinstein and Peter Sarnak,*Chebyshev’s bias*, Experiment. Math.**3**(1994), no. 3, 173–197. MR**1329368****[14]**Daniel Shanks,*Quadratic residues and the distribution of primes*, Math. Tables Aids Comput.**13**(1959), 272–284. MR**0108470**, https://doi.org/10.1090/S0025-5718-1959-0108470-8

Retrieve articles in *Mathematics of Computation of the American Mathematical Society*
with MSC (1991):
11A15,
11M26,
11Y11,
11Y35

Retrieve articles in all journals with MSC (1991): 11A15, 11M26, 11Y11, 11Y35

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:
https://doi.org/10.1090/S0025-5718-99-01105-9

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