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Journal of Discrete Mathematics
Volume 2013 (2013), Article ID 491627, 11 pages
http://dx.doi.org/10.1155/2013/491627
Research Article

Efficient Prime Counting and the Chebyshev Primes

1Institut FEMTO-ST, CNRS, 32 Avenue de l’Observatoire, F-25044 Besançon, France
2Telecom ParisTech, 46 rue Barrault, 75634 Paris Cedex 13, France
3Mathematical Department, King Abdulaziz University, Jeddah, Saudi Arabia

Received 17 October 2012; Revised 28 January 2013; Accepted 29 January 2013

Academic Editor: Pantelimon Stǎnicǎ

Copyright © 2013 Michel Planat and Patrick Solé. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The function where is the logarithm integral and the number of primes up to is well known to be positive up to the (very large) Skewes' number. Likewise, according to Robin's work, the functions and , where and are Chebyshev summatory functions, are positive if and only if Riemann hypothesis (RH) holds. One introduces the jump function at primes and one investigates , , and . In particular, , and for . Besides, for any odd , an infinite set of the so-called Chebyshev primes. In the context of RH, we introduce the so-called Riemann primes as champions of the function (or of the function ). Finally, we find a good prime counting function , that is found to be much better than the standard Riemann prime counting function.

1. Introduction

We recall first some classical definitions and notation in prime number theory [1, 2].(i)The Von Mangoldt function if is a power of a prime and zero otherwise.(ii)The first and the second Chebyshev functions are, respectively, (where : the set of prime numbers) and where ranges over the integers.(i)The logarithmic integral is .(ii)The Möbius function is equal to , respectively, if is, respectively, non-square-free, square-free with an even number of divisors, square-free with an odd number of divisors.(iii)The number of primes up to is denoted .Indeed, and are the logarithm of the product of all primes up to and the logarithm of the least common multiple of all positive integers up to , respectively.

It has been known for a long time that and are asymptotic to (see [2, page 341]). There also exists an explicit formula, due to Von Mangoldt, relating to the nontrivial zeros of the Riemann zeta function [1, 3]. One defines the normalized Chebyshev function to be when is not a prime power, and when it is. The explicit Von Mangoldt formula reads

The function is known to be positive up to the (very large) Skewes’ number [4]. In this paper we are first interested in the jumps (they occur at primes ) in the function . Following Robin’s work on the relation between and RH (Theorem 1), this allows us to derive a new statement (Theorem 7) about the jumps of and Littlewood’s oscillation theorem.

Then, we study the refined function and we observe that the sign of the jumps of is controlled by an infinite sequence of primes that we call the Chebyshev primes    (see Definition 8). The primes (and the generalized primes ) are also obtained by using an accurate calculation of the jumps of , as in Conjecture 12 (and of the jumps of the function , as in Conjecture 14). One conjectures that the function has infinitely many zeros. There exists a potential link between the nontrivial zeros of and the position of the ’s that is made quite explicit in Section 3.1 (Conjecture 16), and in Section 3.2 in our definition of the Riemann primes. In this context, we contribute to the Sloane’s encyclopedia with integer sequences. (The relevant sequences are to 196675 (related to the Chebyshev primes), A197185 to A197188 (related to the Riemann primes of the -type), and A197297 to A197300 (related to the Riemann primes of the -type.)

Finally, we introduce a new prime counting function , better than the standard Riemann’s one, even with three terms in the expansion.

2. Selected Results about the Functions

Let be the th prime number and let be the jump in the logarithmic integral at . For any one numerically observes that . This statement is not useful for the rest of the paper. But it is enough to observe that   and that the sequence is strictly decreasing.

The next three subsections deal with the jumps in the functions   and .

2.1. The Jumps in the Function li

Theorem 1 (Robin). The statement is equivalent to RH [5, 6].

Corollary 2 (related to Robin [5]). The statement is equivalent to RH.

Proof. If RH is true then, using the fact that and that is a strictly growing function when , this follows from Theorem 1 in Robin [5]. If RH is false, Lemma 2 in Robin ensures the violation of the inequality.

Using the fact that , define the jump of index as

Proposition 3. If , then .

Proof. The integral definition of the jump yields The result now follows after observing that by [7, Theorem 18], we have for , and by using the note added in proof of [8] that establishes that for .

By seeing this result it would be natural to make the following conjecture.

Conjecture 4. For all we have .

However, building on Littlewood’s oscillation theorem for we can prove that oscillates about with a small amplitude. Let us recall the Littlewood’s oscillation theorem ([9, Theorem 6.3, page 200], [10, Theorem 34]) where . The omega notations mean that there are infinitely many numbers and constants and , satisfying We now prepare the proof of the invalidity of Conjecture 4 by writing two lemmas.

Lemma 5. For , we have the bounds

Proof. This is straightforward from the integral definition of the jump.

Lemma 6. For large, we have

Proof. We know that by [9, Theorem 6.3, page 200], we have for and large The result follows by considering the primes closest to .

We can now state and prove the main result of this section.

Theorem 7. For large we have

Proof. By Lemma 6 we know there is a constant such that for infinitely many ’s we have By combining with the first inequality of Lemma 5 the minus part of the statement follows after some standard asymptotics. To prove the plus part write , and proceed as before.

2.2. The Jumps in the Function li and the Chebyshev Primes

Definition 8. Let be an odd prime number and the function . The primes such that are called here Chebyshev primes . (Our terminology should not be confused with that used in [11] where the Chebyshev primes are primes of the form , with and an odd prime. We used the Russian spelling Chebyshev to distinguish both meanings.) In increasing order, they are [12, Sequence A196667]{109, 113, 139, 181, 197, 199, 241, 271, 281, 283, 293, 313, 317, 443, 449, 461, 463, …}.

Comment 1. The number of Chebyshev primes less than , is the sequence [12, Sequence A196671]. This sequence suggests the density for the Chebyshev primes. The largest gaps between the Chebyshev primes are {4, 26, 42, 126, 146, 162, 176, 470, 542, 1370, 1516, 4412, 8196, 14928, 27542, 30974, 51588, 62906,…}, [12, Sequence A196672], and the Chebyshev primes that begin a record gap to the next Chebyshev prime are {109, 113, 139, 317, 887, 1327, 1913, 3089, 8297, 11177, 29761, 45707, 113383, 164893, 291377, 401417, 638371, 1045841, …} [12, Sequence A196673].

The results for the jumps of the function are quite analogous to the results for the jumps of the function and stated below without proof.

Let us define the th jump at a prime as

Theorem 9. For large, we have

Corollary 10. There are infinitely many Chebyshev primes .

One observes that the sequence oscillates around and the largest deviations from seem to be unbounded at large . This behaviour is illustrated in Figure 1. Based on this numerical results, we are led to the following conjecture.

491627.fig.001
Figure 1: A plot of the function up to the th prime.

Conjecture 11. The function possesses infinitely many zeros.

Comment 2. The first eleven zeros of occur at the indices {510, 10271, 11259, 11987, 14730, 18772, 18884, 21845, 24083, 33723, 46789} [12, Sequence A1966674], where the corresponding Chebyshev primes are [12, Sequence A1966675] {164051, 231299, 255919, 274177, 343517, 447827, 450451, 528167, 587519, 847607, 1209469}.

Conjecture 12. The jump at primes of the function may be written as , with . In particular, the sign of is that of .

Comment 3. The jump of index (at the prime number ) is There exists a real depending on the index , with such that Using the known locations of the Chebyshev primes of low index, it is straightforward to check that the real reads

This numerical calculations support our Conjecture 12 that the Chebyshev primes may be derived from instead of .

2.3. The Generalized Chebyshev Primes

Definition 13. Let be an odd prime number and the function , . The primes such that are called here generalized Chebyshev primes (or Chebyshev primes of index ).

A short list of Chebyshev primes of index 2 is as follows:{17, 29, 41, 53, 61, 71, 83, 89, 101, 103, 113, 127, 137, 149, 151, 157, 193, 211, 239, 241, …} [12, Sequence A196668].

A short list of Chebyshev primes of index is as follows:{11, 19, 29, 61, 71, 97, 101, 107, 109, 113, 127, 131, 149, 151, 173, 181, 191, 193, 197, 199, …} [12, Sequence A196669].

A short list of Chebyshev primes of index is as follows:{5, 7, 17, 19, 31, 37, 41, 43, 53, 59, 67, 73, 79, 83, 101, 103, 107, …} [12, Sequence A196670].

Conjecture 14. The jump at power of primes of the function may be written as , with . In particular the sign of is that of .

Comment 4. Our comment is similar to the comment given in the context of Conjecture 12 but refers to the generalized Chebyshev primes . To summarize, the jump of the function is accurately defined by a generalized Mangoldt function , that is, , if and otherwise, with as defined in the present proposition. The sign of the function determines the position of the generalized Chebyshev primes.

3. The Chebyshev and Riemann Primes

The next subsection relates the definition of the Chebyshev primes to the explicit Von Mangoldt formula. The following one puts in perspective the link of the Chebyshev primes to RH through the introduction of the so-called Riemann primes.

3.1. The Chebyshev Primes and the Von Mangoldt Explicit Formula

From Corollary 10, one observes that the oscillations of the function around are intimely related to the existence of Chebyshev primes.

Proposition 15. If is a Chebyshev prime (of index 1), then . In the other direction, if , then is a Chebyshev prime (of index 1).

Proof. Proposition 15 follows from the inequalities (analogous to that of Lemma 5)

In what regards the position of the (generalized) Chebyshev primes, our numerical experiments lead to the following conjecture.

Conjecture 16. Let be the normalized Chebyshev function. A prime is a Chebyshev prime of index iff it satisfies the inequality .

Comment 5. It is straigthforward to recover the known sequence of Chebyshev primes (already obtained from Definition 8 or the Conjecture 12) from the new Conjecture 16. Thus, Chebyshev primes (of index 1) are those primes satisfying . Similarly, generalized Chebyshev primes of index (obtained from Definition 13, or the Conjecture 14) also follow from the Conjecture 16.

3.2. Riemann Hypothesis and the Riemann Primes

Under RH, one has the inequality [1] and alternative upper bounds exist in various ranges of values of [8]. In the following, we specialize on bounds for at power of primes .

Definition 17. The champions (left to right maxima) of the function are called here Riemann primes of the type and index .

Comment 6. One numerically gets the Riemann primes of the type and index 1 [12, Sequence A197185]{2, 59, 73, 97, 109, 113, 199, 283, 463, 467, 661, 1103, 1109, 1123, 1129, 1321, 1327, …},the Riemann primes of the type and index [12, Sequence A197186]{2, 17, 31, 41, 53, 101, 109, 127, 139, 179, 397, 419, 547, 787, 997, 1031, …},the Riemann primes of the type and index [12, Sequence A197187]{2, 3, 5, 7, 11, 13, 17, 29, 59, 67, 97, 103, …},and the Riemann primes of the type and index [12, Sequence A197188]{2, 5, 7, 11, 13, 17, 31, …}.
Clearly, the subset of the Riemann primes of the type such that belongs to the set of Chebyshev primes of the corresponding index . Since the Riemann primes of the type maximize , it is useful to plot the ratio . Figure 2 illustrates this dependence for the Riemann primes of index to . One finds that the absolute ratio decreases with the index ; this corresponds to the points of lowest amplitude in Figure 2.

491627.fig.002
Figure 2: The function at the Riemann primes of the type and indices 1 to 4. Points of index 1 are joined.

Under RH, one has the inequality [8, Theorem 10] In the following, we specialize on bounds for at power of primes .

Definition 18. The champions (left to right maxima) of the function are called here Riemann primes of the type and index .

Comment 7. One numerically gets the Riemann primes of the type and index [12, Sequence A197297]{2, 5, 7, 11, 17, 29, 37, 41, 53, 59, 97, 127, 137, 149, 191, 223, 307, 331, 337, 347, 419, …},the Riemann primes of the type and index [12, Sequence A197298]{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 43, 47, 59, 73, 97, 107, 109, 139, 179, 233, 263, …},the Riemann primes of the type and index [12, Sequence A197299]{2, 3, 5, 7, 13, 17, 23, 31, 37, 41, 43, 47, 53, 59, 67, 73, 83, 89, 101, 103, …},and the Riemann primes of the type and index [12, Sequence A197300]{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, …}.
The Riemann primes of the type maximize . In Figure 3, we plot the ratio at the Riemann primes of index to . Again one finds that the absolute ratio decreases with the index ; this corresponds to the points of lowest amplitude in Figure 3.

491627.fig.003
Figure 3: The function at the Riemann primes of the type and indices 1 to 4. Points of index 1 are joined.

In the future, it will be useful to approach the proof of RH thanks to the Riemann primes.

4. An Efficient Prime Counting Function

In this section, one finds that the Riemann prime counting function [13] may be much improved by replacing it by . One denotes , , the offset in the new prime counting function. Indeed, .

By definition, the negative jumps in the function may only occur at . For , they occur at primes (the Chebyshev primes; see Definition 8). For , negative jumps are numerically found to occur at all with an amplitude decreasing to zero. We are led to the following conjecture.

Conjecture 19. Let , . Negative jumps of the function occur at all primes and .

More generally, the jumps of at power of primes are described by the following conjecture.

Conjecture 20. Let be as in Conjecture 19. Positive jumps of the function occur at all powers of primes , , and . Moreover, the jumps are such that and .

A sketch of the function (for ) is given in Figure 4. One easily detects the large positive jumps at (), the intermediate positive jumps at (), and the (very small) negative jumps at primes . This plot can be compared to that of the function displayed in [13].

491627.fig.004
Figure 4: A plot of the function .

Comment 8. The arithmetical structure of we have just described leads to when . Table 1 represents the maximum value that is reached and the position of the extremum, for several small values of and . Thus, the function is a good prime counting function with only a few terms in the summation. This is about a fivefold improvement of the accuracy obtained with the standard Riemann prime counting function (in the range ) and an even better improvement when , already with three terms in the expansion. Another illustration of the efficiency of the calculation based on is given in Table 2, that displays values of at multiples of . See also Figure 4.
It is known that converges for any and may also be written as the Gram series [13] . A similar formula is not established here.

tab1
Table 1: Upper part of the table: maximum error in the new prime counting function for (left-hand part) in comparison to the maximum error using the Riemann prime counting function (right-hand part). Lower part of the table: as above in the range .
tab2
Table 2: Gauss’s and Riemann’s approximation and the approximation . Compare to table III, page 35 in [1].

5. Conclusion

This work sheds light on the structure and the distribution of the generalized Chebyshev primes arising from the jumps of the function . It is inspired by Robin’s work [5] relating the sign of the functions and to RH [5]. Our most puzzling observation is that the nontrivial zeros of the Riemann zeta function are mirrored in the (generalized) Chebyshev primes, whose existence at infinity crucially depends on the Littlewood’s oscillation theorem. In addition, a new accurate prime counting function, based on has been proposed. Future work should concentrate on an effective analytic map between the the zeros and the sequence , in the spirit of our Conjecture 16, and of our approach of RH through the Riemann primes.

Appendices

A. The Chebyshev Primes (of Index 1) Not Exceeding

109, 113, 139, 181, 197, 199, 241, 271, 281, 283, 293, 313, 317, 443, 449, 461, 463, 467, 479, 491, 503, 509, 523, 619, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 761, 769, 773, 829, 859, 863, 883, 887, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1153, 1231, 1237, 1301, 1303, 1307, 1319, 1321, 1327, 1489, 1493, 1499, 1511, 1571, 1579, 1583, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1783, 1787, 1789, 1801, 1811, 1877, 1879, 1889, 1907, 1913, 2089, 2113, 2141, 2143, 2153, 2161, 2297, 2311, 2351, 2357, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2557, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903, 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3083, 3089, 3559, 3583, 3593, 3617, 3623, 3631, 3637, 3643, 3659, 3673, 3677, 3691, 3697, 3701, 3709, 3719, 3727, 3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3823, 3833, 3851, 3853, 3863, 3877, 3881, 3889, 3911, 3917, 3919, 3923, 3929, 3931, 3943, 3947, 3967, 4007, 4013, 4019, 4021, 4027, 4049, 4051, 4057, 4073, 4079, 4091, 4093, 4099, 4111, 4127, 4129, 4133, 4139, 4153, 4157, 4159, 4177, 4219, 4231, 4241, 4243, 4253, 4259, 4261, 4271, 4273, 4283, 4289, 4297, 4327, 4339, 4349, 4357, 4363, 4373, 4519, 4523, 4549, 4567, 4651, 4657, 4663, 4673, 4679, 4691, 4733, 4801, 4817, 5009, 5011, 5021, 5023, 5039, 5051, 5059, 5081, 5087, 5099, 5101, 5107, 5113, 5119, 5237, 5507, 5521, 5527, 5531, 5573, 5581, 5591, 5659, 5669, 5693, 5701, 5711, 5717, 5743, 5749, 5827, 5843, 5849, 5851, 5857, 5861, 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939, 5953, 6271, 6277, 6287, 6299, 6301, 6311, 6317, 6323, 6329, 6337, 6343, 6353, 6359, 6361, 6367, 6373, 6379, 6389, 6397, 6421, 6427, 6449, 6451, 6469, 6473, 6481, 6491, 6571, 6577, 6581, 6599, 6607, 6619, 6703, 6709, 6719, 6733, 6737, 6793, 6803, 6829, 6833, 6841, 6857, 6863, 6869, 6871, 6883, 6899, 6907, 6911, 6917, 6949, 6959, 6961, 6967, 6971, 6977, 6983, 6991, 6997, 7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079, 7103, 7109, 7121, 7127, 7129, 7151, 7159, 7213, 7219, 7229, 7237, 7243, 7247, 7253, 7333, 7351, 7583, 7589, 7591, 7603, 7607, 7621, 7643, 7649, 7673, 7681, 7687, 7691, 7699, 7703, 7717, 7723, 7727, 7741, 7753, 7757, 7759, 7789, 7793, 7879, 7883, 7927, 7933, 7937, 7949, 7951, 7963, 8297, 8839, 8849, 8863, 8867, 8887, 8893, 9013, 9049, 9059, 9067, 9241, 9343, 9349, 9419, 9421, 9431, 9433, 9437, 9439, 9461, 9463, 9467, 9473, 9479, 9491, 9497, 9511, 9521, 9533, 9539, 9547, 9551, 9587, 9601, 9613, 9619, 9623, 9629, 9631, 9643, 9649, 9661, 9677, 9679, 9689, 9697, 9719, 9721, 9733, 9739, 9743, 9749, 9767, 9769, 9781, 9787, 9791, 9803, 9811, 9817, 9829, 9833, 9839, 9851, 9857, 9859, 9871, 9883, 9887, 9901, 9907, 9923, 9929, 9931, 9941, 9949, 9967, 9973, 10007, 10009, 10039, 10069, 10079, 10091, 10093, 10099, 10103, 10111, 10133, 10139, 10141, 10151, 10159, 10163, 10169, 10177, 10181, 10193, 10211, 10223, 10243, 10247, 10253, 10259, 10267, 10271, 10273, 10289, 10301, 10303, 10313, 10321, 10331, 10333, 10337, 10343, 10357, 10369, 10391, 10399, 10427, 10429, 10433, 10453, 10457, 10459, 10463, 10477, 10487, 10499, 10501, 10513, 10529, 10531, 10559, 10567, 10601, 10607, 10613, 10627, 10631, 10639, 10651, 10657, 10663, 10667, 10687, 10691, 10709, 10711, 10723, 10729, 10733, 10739, 10753, 10771, 10781, 10789, 10799, 10859, 10861, 10867, 10883, 10889, 10891, 10903, 10909, 10939, 10949, 10957, 10979, 10987, 10993, 11003, 11119, 11173, 11177, 12547, 12553, 12577, 12583, 12589, 12601, 12611, 12613, 12619, 12637, 12641, 12647, 12653, 12659, 12671, 12689, 12697, 12703, 12713, 12721, 12739, 12743, 12757, 12763, 12781, 12791, 12799, 12809, 12821, 12823, 12829, 12841, 12853, 12911, 12917, 12919, 12923, 12941, 12953, 12959, 12967, 12973, 12979, 12983, 13001, 13003, 13007, 13009, 13033, 13037, 13043, 13049, 13063, 13093, 13099, 13103, 13109, 13121, 13127, 13147, 13151, 13159, 13163, 13171, 13177, 13183, 13187, 13217, 13219, 13229, 13241, 13249, 13259, 13267, 13291, 13297, 13309, 13313, 13327, 13331, 13337, 13339, 13367, 13381, 13399, 13411, 13417, 13421, 13441, 13451, 13457, 13463, 13469, 13477, 13487, 13499, 13513, 13523, 13537, 13697, 13709, 13711, 13721, 13723, 13729, 13751, 13757, 13759, 13763, 13781, 13789, 13799, 13807, 13829, 13831, 13841, 13859, 13877, 13879, 13883, 13901, 13903, 13907, 13913, 13921, 13931, 13933, 13963, 13967, 13999, 14009, 14011, 14029, 14033, 14051, 14057, 14071, 14081, 14083, 14087, 14107, 14783, 14831, 14851, 14869, 14879, 14887, 14891, 14897, 14947, 14951, 14957, 14969, 14983, 15289, 15299, 15307, 15313, 15319, 15329, 15331, 15349, 15359, 15361, 15373, 15377, 15383, 15391, 15401, 15413, 15427, 15439, 15443, 15451, 15461, 15467, 15473, 15493, 15497, 15511, 15527, 15541, 15551, 15559, 15569, 15581, 15583, 15601, 15607, 15619, 15629, 15641, 15643, 15647, 15649, 15661, 15667, 15671, 15679, 15683, 15727, 15731, 15733, 15737, 15739, 15749, 15761, 15767, 15773, 15787, 15791, 15797, 15803, 15809, 15817, 15823, 15859, 15877, 15881, 15887, 15889, 15901, 15907, 15913, 15919, 15923, 15937, 15959, 15971, 15973, 15991, 16001, 16007, 16033, 16057, 16061, 16063, 16067, 16069, 16073, 16087, 16091, 16097, 16103, 16111, 16127, 16139, 16141, 16183, 16187, 16189, 16193, 16217, 16223, 16229, 16231, 16249, 16253, 16267, 16273, 16301, 16319, 16333, 16339, 16349, 16361, 16363, 16369, 16381, 16411, 16417, 16421, 16427, 16433, 16447, 16451, 16453, 16477, 16481, 16487, 16493, 16519, 16529, 16547, 16553, 16561, 16567, 16573, 16607, 16633, 16651, 16657, 16661, 16673, 16691, 16693, 16699, 16703, 16729, 16741, 16747, 16759, 16763, 17033, 17041, 17047, 17053, 17123, 17207, 17209, 17393, 17401, 17419, 17431, 17449, 17471, 17477, 17483, 17489, 17491, 17497, 17509, 17519, 17539, 17551, 17569, 17573, 17579, 17581, 17597, 17599, 17609, 17623, 17627, 17657, 17659, 17669, 17681, 17683, 17713, 17749, 17761, 17791, 17929, 17939, 17959, 17977, 17981, 17987, 17989, 18013, 18047, 18049, 18059, 18061, 18077, 18089, 18097, 18121, 18127, 18131, 18133, 18143, 18149, 18169, 18181, 18191, 18199, 18211, 18217, 18223, 18229, 18233, 18251, 18253, 18257, 18269, 18287, 18289, 18301, 18307, 18311, 18313, 18329, 18341, 18353, 18367, 18371, 18379, 18397, 18401, 18413, 18427, 18433, 18439, 18443, 18451, 18457, 18461, 18481, 18493, 18503, 18517, 18521, 18523, 18539, 18541, 18553, 18583, 18587, 18593, 18617, 18637, 18661, 18671, 18679, 18691, 18701, 18713, 18719, 18731, 18743, 18749, 18757, 18773, 18787, 18793, 18797, 18803, 20149, 20177, 20183, 20407, 20411, 20443, 21601, 21611, 21613, 21617, 21649, 21661, 21821, 21841, 21851, 21859, 21863, 21871, 21881, 21893, 21911, 22039, 22051, 22067, 22073, 22079, 22091, 22093, 22109, 22111, 22123, 22129, 22133, 22147, 22153, 22157, 22159, 22171, 22189, 22193, 22229, 22247, 22259, 22271, 22273, 22277, 22279, 22283, 22291, 22303, 22307, 22343, 22349, 22367, 22369, 22381, 22391, 22397, 22409, 22433, 22441, 22447, 22453, 22469, 22481, 22483, 22501, 22511, 22543, 22549, 22567, 22571, 22573, 22643, 22651, 22697, 22699, 22709, 22717, 22721, 22727, 22739, 22741, 22751, 22769, 22777, 22783, 22787, 22807, 22811, 22817, 22853, 22859, 22861, 22871, 22877, 22901, 22907, 22921, 22937, 22943, 22961, 22963, 22973, 22993, 23003, 23011, 23017, 23021, 23027, 23029, 23039, 23041, 23053, 23057, 23059, 23063, 23071, 23081, 23087, 23099, 23117, 23131, 23143, 23159, 23167, 23173, 23189, 23197, 23201, 23203, 23209, 23227, 23251, 23269, 23279, 23291, 23293, 23297, 23311, 23321, 23327, 23333, 23339, 23357, 23369, 23371, 23399, 23417, 23431, 23447, 23459, 23473, 23563, 23567, 23581, 23593, 23599, 23603, 23609, 23623, 23627, 23629, 23633, 23663, 23669, 23671, 23677, 23687, 23689, 23719, 23741, 23743, 23747, 23753, 23761, 23767, 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73141, 73181, 73189, 74611, 74623, 74779, 74869, 74873, 74887, 74891, 74897, 74903, 74923, 74929, 74933, 74941, 74959, 75037, 75041, 75403, 75407, 75431, 75437, 75577, 75583, 75619, 75629, 75641, 75653, 75659, 75679, 75683, 75689, 75703, 75707, 75709, 75721, 75731, 75743, 75767, 75773, 75781, 75787, 75793, 75797, 75821, 75833, 75853, 75869, 75883, 75991, 75997, 76001, 76003, 76031, 76039, 76103, 76159, 76163, 76261, 77563, 77569, 77573, 77587, 77591, 77611, 77617, 77621, 77641, 77647, 77659, 77681, 77687, 77689, 77699, 77711, 77713, 77719, 77723, 77731, 77743, 77747, 77761, 77773, 77783, 77797, 77801, 77813, 77839, 77849, 77863, 77867, 77893, 77899, 77929, 77933, 77951, 77969, 77977, 77983, 77999, 78007, 78017, 78031, 78041, 78049, 78059, 78079, 78101, 78139, 78167, 78173, 78179, 78191, 78193, 78203, 78229, 78233, 78241, 78259, 78277, 78283, 78301, 78307, 78311, 78317, 78341, 78347, 78367, 79907, 80239, 80251, 80263, 80273, 80279, 80287, 80317, 80329, 80341, 80347, 80363, 80369, 80387, 80687, 80701, 80713, 80749, 80779, 80783, 80789, 80803, 80809, 80819, 80831, 80833, 80849, 80863, 80911, 80917, 80923, 80929, 80933, 80953, 80963, 80989, 81001, 81013, 81017, 81019, 81023, 81031, 81041, 81043, 81047, 81049, 81071, 81077, 81083, 81097, 81101, 81119, 81131, 81157, 81163, 81173, 81181, 81197, 81199, 81203, 81223, 81233, 81239, 81281, 81283, 81293, 81299, 81307, 81331, 81343, 81349, 81353, 81359, 81371, 81373, 81401, 81409, 81421, 81439, 81457, 81463, 81509, 81517, 81527, 81533, 81547, 81551, 81553, 81559, 81563, 81569, 81611, 81619, 81629, 81637, 81647, 81649, 81667, 81671, 81677, 81689, 81701, 81703, 81707, 81727, 81737, 81749, 81761, 81769, 81773, 81799, 81817, 81839, 81847, 81853, 81869, 81883, 81899, 81901, 81919, 81929, 81931, 81937, 81943, 81953, 81967, 81971, 81973, 82003, 82007, 82009, 82013, 82021, 82031, 82037, 82039, 82051, 82067, 82073, 82129, 82139, 82141, 82153, 82163, 82171, 82183, 82189, 82193, 82207, 82217, 82219, 82223, 82231, 82237, 82241, 82261, 82267, 82279, 82301, 82307, 82339, 82349, 82351, 82361, 82373, 82387, 82393, 82421, 82457, 82463, 82469, 82471, 82483, 82487, 82493, 82499, 82507, 82529, 82531, 82549, 82559, 82561, 82567, 82571, 82591, 82601, 82609, 82613, 82619, 82633, 82651, 82657, 82699, 82721, 82723, 82727, 82729, 82757, 82759, 82763, 82781, 82787, 82793, 82799, 82811, 82813, 82837, 82847, 82883, 82889, 82891, 82903, 82913, 82939, 82963, 82981, 82997, 83003, 83009, 83023, 83047, 83059, 83063, 83071, 83077, 83089, 83093, 83101, 83117, 83137, 83177, 83203, 83207, 83219, 83221, 83227, 83231, 83233, 83243, 83257, 83267, 83269, 83273, 83299, 83311, 83339, 83341, 83357, 83383, 83389, 83399, 83401, 83407, 83417, 83423, 83431, 83437, 83443, 83449, 83459, 83471, 83477, 83497, 83537, 83557, 83561, 83563, 83579, 83591, 83597, 83609, 83617, 83621, 83639, 83641, 83653, 83663, 83689, 83701, 83717, 83719, 83737, 83761, 83773, 83777, 83791, 83813, 83833, 83843, 83857, 83869, 83873, 83891, 83903, 83911, 83921, 83933, 83939, 83969, 83983, 83987, 84011, 84017, 84061, 84067, 84089, 84143, 84181, 84191, 84199, 84211, 84221, 84223, 84229, 84239, 84247, 84263, 84313, 84317, 84319, 84347, 84349, 84391, 84401, 84407, 84421, 84431, 84437, 84443, 84449, 84457, 84463, 84467, 84481, 84499, 84503, 84509, 84521, 84523, 84533, 84551, 84559, 84589, 84629, 84631, 84649, 84653, 84659, 84673, 84691, 84697, 84701, 84713, 84719, 84731, 84737, 84751, 84761, 84787, 84793, 84809, 84811, 84827, 84857, 84859, 84869, 84871, 84919, 85093, 85103, 85109, 85121, 85133, 85147, 85159, 85201, 85213, 85223, 85229, 85237, 85243, 85247, 85259, 85303, 85313, 85333, 85369, 85381, 85451, 85453, 85469, 85487, 85531, 85621, 85627, 85639, 85643, 85661, 85667, 85669, 85691, 85703, 85711, 85717, 85733, 85751, 85837, 85843, 85847, 85853, 85889, 85909, 85933, 86297, 86353, 86357, 86369, 86371, 86381, 86389, 86399, 86413, 86423, 86441, 86453, 86461, 86467, 86477, 86491, 86501, 86509, 86531, 86533, 86539, 86561, 86573, 86579, 86587, 86599, 86627, 86629, 87641, 87643, 87649, 87671, 87679, 87683, 87691, 87697, 87701, 87719, 87721, 87739, 87743, 87751, 87767, 87793, 87797, 87803, 87811, 87833, 87853, 87869, 87877, 87881, 87887, 87911, 87917, 87931, 87943, 87959, 87961, 87973, 87977, 87991, 88001, 88003, 88007, 88019, 88037, 88069, 88079, 88093, 88117, 88129, 90023, 90031, 90067, 90071, 90073, 90089, 90107, 90121, 90127, 90191, 90197, 90199, 90203, 90217, 90227, 90239, 90247, 90263, 90271, 90281, 90289, 90313, 90373, 90379, 90397, 90401, 90403, 90407, 90437, 90439, 90473, 90481, 90499, 90511, 90523, 90527, 90529, 90533, 90547, 90647, 90659, 90679, 90703, 90709, 91459, 91463, 92419, 92431, 92467, 92489, 92507, 92671, 92681, 92683, 92693, 92699, 92707, 92717, 92723, 92737, 92753, 92761, 92767, 92779, 92789, 92791, 92801, 92809, 92821, 92831, 92849, 92857, 92861, 92863, 92867, 92893, 92899, 92921, 92927, 92941, 92951, 92957, 92959, 92987, 92993, 93001, 93047, 93053, 93059, 93077, 93083, 93089, 93097, 93103, 93113, 93131, 93133, 93139, 93151, 93169, 93179, 93187, 93199, 93229, 93239, 93241, 93251, 93253, 93257, 93263, 93281, 93283, 93287, 93307, 93319, 93323, 93329, 93337, 93371, 93377, 93383, 93407, 93419, 93427, 93463, 93479, 93481, 93487, 93491, 93493, 93497, 93503, 93523, 93529, 93553, 93557, 93559, 93563, 93581, 93601, 93607, 93629, 93637, 93683, 93701, 93703, 93719, 93739, 93761, 93763, 93787, 93809, 93811, 93827, 93851, 93893, 93901, 93911, 93913, 93923, 93937, 93941, 93949, 93967, 93971, 93979, 93983, 93997, 94007, 94009, 94033, 94049, 94057, 94063, 94079, 94099, 94109, 94111, 94117, 94121, 94151, 94153, 94169, 94201, 94207, 94219, 94229, 94253, 94261, 94273, 94291, 94307, 94309, 94321, 94327, 94331, 94343, 94349, 94351, 94379, 94397, 94399, 94421, 94427, 94433, 94439, 94441, 94447, 94463, 94477, 94483, 94513, 94529, 94531, 94541, 94543, 94547, 94559, 94561, 94573, 94583, 94597, 94603, 94613, 94621, 94649, 94651, 94687, 94693, 94709, 94723, 94727, 94747, 94771, 94777, 94781, 94789, 94793, 94811, 94819, 94823, 94837, 94841, 94847, 94849, 94873, 94889, 94903, 94907, 94933, 94949, 94951, 94961, 94993, 94999, 95003, 95009, 95021, 95027, 95063, 95071, 95083, 95087, 95089, 95093, 95101, 95107, 95111, 95131, 95143, 95153, 95177, 95189, 95191, 95203, 95213, 95219, 95231, 95233, 95239, 95257, 95261, 95267, 95273, 95279, 95287, 95311, 95317, 95327, 95339, 95369, 95383, 95393, 95401, 95413, 95419, 95429, 95441, 95443, 95461, 95467, 95471, 95479, 95483, 95507, 95527, 95531, 95539, 95549, 95561, 95569, 95581, 95597, 95603, 95617, 95621, 95629, 95633, 95651, 95701, 95707, 95713, 95717, 95723, 95731, 95737, 95747, 95773, 95783, 95789, 95791, 95801, 95803, 95813, 95819, 95857, 95869, 95873, 95881, 95891, 95911, 95917, 95923, 95929, 95947, 95957, 95959, 95971, 95987, 95989, 96001, 96013, 96017, 96043, 96053, 96059, 96079, 96097, 96137, 96149, 96157, 96167, 96179, 96181, 96199, 96211, 96221, 96223, 96233, 96259, 96263, 96269, 96281, 96289, 96293, 96323, 96329, 96331, 96337, 96353, 96377, 96401, 96419, 96431, 96443, 96451, 96457, 96461, 96469, 96479, 96487, 96493, 96497, 96517, 96527, 96553, 96557, 96581, 96587, 96589, 96601, 96643, 96661, 96667, 96671, 96697, 96703, 96731, 96737, 96739, 96749, 96757, 96763, 96769, 96779, 96787, 96797, 96799, 96821, 96823, 96827, 96847, 96851, 96857, 96893, 96907, 96911, 96931, 96953, 96959, 96973, 96979, 96989, 96997, 97001, 97003, 97007, 97021, 97039, 97073, 97081, 97103, 97117, 97127, 97151, 97157, 97159, 97169, 97171, 97177, 97187, 97213, 97231, 97241, 97259, 97283, 97301, 97303, 97327, 97367, 97369, 97373, 97379, 97381, 97387, 97397, 97423, 97429, 97441, 97453, 97459, 97463, 97499, 97501, 97511, 97523, 97547, 97549, 97553, 97561, 97571, 97577, 97579, 97583, 97607, 97609, 97613, 97649, 97651, 97673, 97687, 97711, 97729, 97771, 97777, 97787, 97789, 97813, 97829, 97841, 97843, 97847, 97849, 97859, 97861, 97871, 97879, 97883, 97919, 97927, 97931, 97943, 97961, 97967, 97973, 97987, 98009, 98011, 98017, 98041, 98047, 98057, 98081, 98101, 98123, 98129, 98143, 98179, 98227, 98327, 98389, 98411, 98419, 98429, 98443, 98453, 98459, 98467, 98473, 98479, 98491, 98507, 98519, 98533, 98543, 98561, 98563, 98573, 98597, 98627, 98639, 98641, 98663, 98669, 98717, 98729, 98731, 98737, 98779, 98809, 98893, 98897, 98899, 98909, 98911, 98927, 98929, 98939, 98947, 98953, 98963, 98981, 98993, 98999, 99013, 99017, 99023, 99041, 99053, 99079, 99083, 99089, 99103, 99109, 99119, 99131, 99133, 99137, 99139, 99149, 99173, 99181, 99191, 99223, 99233, 99241, 99251, 99257, 99259, 99277, 99289, 99317, 99347, 99349, 99367, 99371, 99377, 99391, 99397, 99401, 99409, 99431, 99439, 99469, 99487, 99497, 99523, 99527, 99529, 99551, 99559, 99563, 99571, 99577, 99581, 99607, 99611, 99623, 99643, 99661, 99667, 99679, 99689, 99707, 99709, 99713, 99719, 99721, 99733, 99761, 99767, 99787, 99793, 99809, 99817, 99823, 99829, 99833, 99839, 99859, 99871, 99877, 99881, 99901, 99907, 99923, 99929, 99961, 99971, 99989, 99991.

B. The Riemann Primes of the Type and Index 1, in the Range

2, 59, 73, 97, 109, 113, 199, 283, 463, 467, 661, 1103, 1109, 1123, 1129, 1321, 1327, 1423, 2657, 2803, 2861, 3299, 5381, 5881, 6373, 6379, 9859, 9931, 9949, 10337, 10343, 11777, 19181, 19207, 19373, 24107, 24109, 24113, 24121, 24137, 42751, 42793, 42797, 42859, 42863, 58231, 58237, 58243, 59243, 59447, 59453, 59471, 59473, 59747, 59753,142231, 142237, 151909, 152851, 152857, 152959, 152993, 153001, 155851, 155861, 155863, 155893, 175573, 175601, 175621, 230357, 230369, 230387, 230389, 230393, 298559, 298579, 298993, 299281, 299311, 299843, 299857, 299933, 300073, 300089, 300109, 300137, 302551, 302831, 355073, 355093, 355099, 355109, 355111, 463157, 463181, 617479, 617731, 617767, 617777, 617801, 617809, 617819, 909907, 909911, 909917, 910213, 910219, 910229, 993763, 993779, 993821, 1062251, 1062293, 1062311, 1062343, 1062469, 1062497, 1062511, 1062547, 1062599, 1062643, 1062671, 1062779, 1062869, 1090681, 1090697, 1194041, 1194047, 1194059, 1195237, 1195247.

C. The Riemann Primes of the Type and Index 1, in the Range

[2, 5, 7, 11, 17, 29, 37, 41, 53, 59, 97, 127, 137, 149, 191, 223, 307, 331, 337, 347, 419, 541, 557, 809, 967, 1009, 1213, 1277, 1399, 1409, 1423, 1973, 2203, 2237, 2591, 2609, 2617, 2633, 2647, 2657, 3163, 3299, 4861, 4871, 4889, 4903, 4931, 5381, 7411, 7433, 7451, 8513, 8597, 11579, 11617, 11657, 11677, 11777, 14387, 18973, 19001, 19031, 19051, 19069, 19121, 19139, 19181, 19207, 19373, 27733, 30089, 30631, 31957, 32051, 46439, 47041, 47087, 47111, 47251, 47269, 55579, 55603, 64849, 64997, 69109, 69143, 69191, 69337, 69371, 69623, 69653, 69677, 69691, 69737, 69761, 69809, 69821, 69991, 88589, 88643, 88771, 88789, 114547, 114571, 115547, 115727, 119489, 119503, 119533, 119549, 166147, 166541, 166561, 168433, 168449, 168599, 168673, 168713, 168851, 168977, 168991, 169307, 169627, 175391, 175573, 175601, 175621, 237673, 237851, 237959, 264731, 288137, 288179, 288647, 293599, 293893, 293941, 293957, 293983, 295663, 295693, 295751, 295819, 298153, 298559, 298579, 298993, 299261, 299281, 299311, 299843, 299857, 299933, 300073, 300089, 300109, 300137, 302551, 302831, 406969, 407023, 407047, 407083, 407119, 407137, 407177, 461917, 461957, 461971, 462013, 462041, 462067, 462239, 462263, 462307, 462361, 462401, 463093, 463157, 463181, 642673, 642701, 642737, 642769, 643369, 643403, 643421, 643847, 678157, 745931, 747199, 747259, 747277, 747319, 747361, 747377, 747391, 747811, 747827, 748169, 748183, 748199, 748441, 750599, 750613, 750641, 757241, 757993, 982559, 983063, 983113, 984241, 984299, 987793, 987911, 987971, 989059, 989119, 989171, 989231, 989623, 989743, 993679, 993763, 993779, 993821, 1061561, 1062169, 1062197, 1062251, 1062293, 1062311, 1062343, 1062469, 1062497, 1062511, 1062547, 1062599, 1062643, 1062671, 1062779, 1062869, 1090373, 1090681, 1090697].

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