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Introduction c-Ramanujan primes Results Distribution Conclusion
Generalized Ramanujan Primes
Nadine Amersi, Olivia Beckwith, Ryan Ronan
Advisor: Steven J. Miller
http://web.williams.edu/Mathematics/sjmiller/
Young Mathematicians ConferenceOhio State, August 2011
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Introduction c-Ramanujan primes Results Distribution Conclusion
Prime Numbers
Any integer can be written as a unique product ofprime numbers (Fundamental Theorem of Arithmetic).
Exact spacing of primes unknown but roughly growinglogarithmically (Prime Number Theorem).
We expect linearly increasing intervals to contain anincreasing amount of primes.
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Introduction c-Ramanujan primes Results Distribution Conclusion
Prime Numbers
Any integer can be written as a unique product ofprime numbers (Fundamental Theorem of Arithmetic).
Exact spacing of primes unknown but roughly growinglogarithmically (Prime Number Theorem).
We expect linearly increasing intervals to contain anincreasing amount of primes.
3
Introduction c-Ramanujan primes Results Distribution Conclusion
Prime Numbers
Any integer can be written as a unique product ofprime numbers (Fundamental Theorem of Arithmetic).
Exact spacing of primes unknown but roughly growinglogarithmically (Prime Number Theorem).
We expect linearly increasing intervals to contain anincreasing amount of primes.
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Introduction c-Ramanujan primes Results Distribution Conclusion
Historical Introduction
Bertrand’s Postulate (1845)For all integers x ≥ 2, there exists at least one prime in(x/2, x ].
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Introduction c-Ramanujan primes Results Distribution Conclusion
Ramanujan Primes
DefinitionThe n-th Ramanujan prime Rn: smallest integer such thatfor any x ≥ Rn, at least n primes in any (x/2, x ].
TheoremRamanujan: For each integer n, Rn exists.Sondow: Rn ∼ p2n.Sondow: As n→∞, 50% of primes are Ramanujan.
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Introduction c-Ramanujan primes Results Distribution Conclusion
Ramanujan Primes
DefinitionThe n-th Ramanujan prime Rn: smallest integer such thatfor any x ≥ Rn, at least n primes in any (x/2, x ].
TheoremRamanujan: For each integer n, Rn exists.
Sondow: Rn ∼ p2n.Sondow: As n→∞, 50% of primes are Ramanujan.
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Introduction c-Ramanujan primes Results Distribution Conclusion
Ramanujan Primes
DefinitionThe n-th Ramanujan prime Rn: smallest integer such thatfor any x ≥ Rn, at least n primes in any (x/2, x ].
TheoremRamanujan: For each integer n, Rn exists.Sondow: Rn ∼ p2n.
Sondow: As n→∞, 50% of primes are Ramanujan.
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Introduction c-Ramanujan primes Results Distribution Conclusion
Ramanujan Primes
DefinitionThe n-th Ramanujan prime Rn: smallest integer such thatfor any x ≥ Rn, at least n primes in any (x/2, x ].
TheoremRamanujan: For each integer n, Rn exists.Sondow: Rn ∼ p2n.Sondow: As n→∞, 50% of primes are Ramanujan.
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Introduction c-Ramanujan primes Results Distribution Conclusion
c-Ramanujan Primes
DefinitionThe n-th c-Ramanujan prime Rc,n: smallest integer suchthat for any x ≥ Rc,n have at least n primes in (cx , x ].
For each c and integer n, does Rc,n exist? Yes!
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Introduction c-Ramanujan primes Results Distribution Conclusion
c-Ramanujan Primes
DefinitionThe n-th c-Ramanujan prime Rc,n: smallest integer suchthat for any x ≥ Rc,n have at least n primes in (cx , x ].
For each c and integer n, does Rc,n exist?
Yes!
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Introduction c-Ramanujan primes Results Distribution Conclusion
c-Ramanujan Primes
DefinitionThe n-th c-Ramanujan prime Rc,n: smallest integer suchthat for any x ≥ Rc,n have at least n primes in (cx , x ].
For each c and integer n, does Rc,n exist? Yes!
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Introduction c-Ramanujan primes Results Distribution Conclusion
c-Ramanujan Primes
DefinitionThe n-th c-Ramanujan prime Rc,n: smallest integer suchthat for any x ≥ Rc,n have at least n primes in (cx , x ].
Does Rc,n exhibit asymptotic behavior?
For c → 0, the interval is (0, x ] and Rc,n = pn.For c → 1, the interval would be (x , x ] so Rc,n doesnot exist.
Rc,n ∼ p n1−c
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Introduction c-Ramanujan primes Results Distribution Conclusion
c-Ramanujan Primes
DefinitionThe n-th c-Ramanujan prime Rc,n: smallest integer suchthat for any x ≥ Rc,n have at least n primes in (cx , x ].
Does Rc,n exhibit asymptotic behavior?
For c → 0, the interval is (0, x ] and Rc,n = pn.
For c → 1, the interval would be (x , x ] so Rc,n doesnot exist.
Rc,n ∼ p n1−c
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Introduction c-Ramanujan primes Results Distribution Conclusion
c-Ramanujan Primes
DefinitionThe n-th c-Ramanujan prime Rc,n: smallest integer suchthat for any x ≥ Rc,n have at least n primes in (cx , x ].
Does Rc,n exhibit asymptotic behavior?
For c → 0, the interval is (0, x ] and Rc,n = pn.For c → 1, the interval would be (x , x ] so Rc,n doesnot exist.
Rc,n ∼ p n1−c
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Introduction c-Ramanujan primes Results Distribution Conclusion
c-Ramanujan Primes
DefinitionThe n-th c-Ramanujan prime Rc,n: smallest integer suchthat for any x ≥ Rc,n have at least n primes in (cx , x ].
Does Rc,n exhibit asymptotic behavior?
For c → 0, the interval is (0, x ] and Rc,n = pn.For c → 1, the interval would be (x , x ] so Rc,n doesnot exist.
Rc,n ∼ p n1−c
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Introduction c-Ramanujan primes Results Distribution Conclusion
c-Ramanujan Primes
DefinitionThe n-th c-Ramanujan prime Rc,n: smallest integer suchthat for any x ≥ Rc,n have at least n primes in (cx , x ].
What percent of primes are c-Ramanujan?
Given Rc,n ∼ p n1−c
, it follows Rc,N(1−c) ∼ pN .
1-c
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Introduction c-Ramanujan primes Results Distribution Conclusion
c-Ramanujan Primes
DefinitionThe n-th c-Ramanujan prime Rc,n: smallest integer suchthat for any x ≥ Rc,n have at least n primes in (cx , x ].
What percent of primes are c-Ramanujan?
Given Rc,n ∼ p n1−c
, it follows Rc,N(1−c) ∼ pN .
1-c
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Introduction c-Ramanujan primes Results Distribution Conclusion
c-Ramanujan Primes
DefinitionThe n-th c-Ramanujan prime Rc,n: smallest integer suchthat for any x ≥ Rc,n have at least n primes in (cx , x ].
What percent of primes are c-Ramanujan?
Given Rc,n ∼ p n1−c
, it follows Rc,N(1−c) ∼ pN .
1-c
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Introduction c-Ramanujan primes Results Distribution Conclusion
Preliminaries
Let π(x) be the prime-counting function that gives thenumber of primes less than or equal to x .
Rosser’s Theorem states:
pm = m log m + O(m log log m).
The Prime Number Theorem states:
limx→∞
π(x)x/ log(x)
= 1.
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Introduction c-Ramanujan primes Results Distribution Conclusion
Preliminaries
Let π(x) be the prime-counting function that gives thenumber of primes less than or equal to x .
Rosser’s Theorem states:
pm = m log m + O(m log log m).
The Prime Number Theorem states:
limx→∞
π(x)x/ log(x)
= 1.
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Introduction c-Ramanujan primes Results Distribution Conclusion
Preliminaries
Let π(x) be the prime-counting function that gives thenumber of primes less than or equal to x .
Rosser’s Theorem states:
pm = m log m + O(m log log m).
The Prime Number Theorem states:
limx→∞
π(x)x/ log(x)
= 1.
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Introduction c-Ramanujan primes Results Distribution Conclusion
The Prime Number Theorem
The logarithmic integral function Li(x) is defined by
Li(x) =∫ x
2
1log t
dt .
The Prime Number Theorem gives us
π(x) = Li(x) + O(
xlog2 x
),
i.e., there is a C > 0 such that for all x sufficiently large
−Cx
log x≤ π(x)− Li(x) ≤ C
xlog x
.
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Introduction c-Ramanujan primes Results Distribution Conclusion
The Prime Number Theorem
The logarithmic integral function Li(x) is defined by
Li(x) =∫ x
2
1log t
dt .
The Prime Number Theorem gives us
π(x) = Li(x) + O(
xlog2 x
),
i.e., there is a C > 0 such that for all x sufficiently large
−Cx
log x≤ π(x)− Li(x) ≤ C
xlog x
.
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Introduction c-Ramanujan primes Results Distribution Conclusion
Existence of Rc,n
Theorem (Amersi, Beckwith, Ronan, 2011)For all n ∈ Z and all c ∈ (0,1), the n-th c-Ramanujanprime Rc,n exists.
Proof:The number of primes in (cx , x ] is π(x)− π(cx).Using the Prime Number Theorem and Mean ValueTheorem:
π(x)− π(cx) = Li(x)− Li(cx) + O(x log−2 x)= Li′(yc)(x − cx) + O(x log−2 x)
with yc(x) ∈ [cx , x ].
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Introduction c-Ramanujan primes Results Distribution Conclusion
Existence of Rc,n
Theorem (Amersi, Beckwith, Ronan, 2011)For all n ∈ Z and all c ∈ (0,1), the n-th c-Ramanujanprime Rc,n exists.
Proof:
The number of primes in (cx , x ] is π(x)− π(cx).Using the Prime Number Theorem and Mean ValueTheorem:
π(x)− π(cx) = Li(x)− Li(cx) + O(x log−2 x)= Li′(yc)(x − cx) + O(x log−2 x)
with yc(x) ∈ [cx , x ].
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Introduction c-Ramanujan primes Results Distribution Conclusion
Existence of Rc,n
Theorem (Amersi, Beckwith, Ronan, 2011)For all n ∈ Z and all c ∈ (0,1), the n-th c-Ramanujanprime Rc,n exists.
Proof:The number of primes in (cx , x ] is π(x)− π(cx).
Using the Prime Number Theorem and Mean ValueTheorem:
π(x)− π(cx) = Li(x)− Li(cx) + O(x log−2 x)= Li′(yc)(x − cx) + O(x log−2 x)
with yc(x) ∈ [cx , x ].
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Introduction c-Ramanujan primes Results Distribution Conclusion
Existence of Rc,n
Theorem (Amersi, Beckwith, Ronan, 2011)For all n ∈ Z and all c ∈ (0,1), the n-th c-Ramanujanprime Rc,n exists.
Proof:The number of primes in (cx , x ] is π(x)− π(cx).Using the Prime Number Theorem and Mean ValueTheorem:
π(x)− π(cx) = Li(x)− Li(cx) + O(x log−2 x)
= Li′(yc)(x − cx) + O(x log−2 x)with yc(x) ∈ [cx , x ].
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Introduction c-Ramanujan primes Results Distribution Conclusion
Existence of Rc,n
Theorem (Amersi, Beckwith, Ronan, 2011)For all n ∈ Z and all c ∈ (0,1), the n-th c-Ramanujanprime Rc,n exists.
Proof:The number of primes in (cx , x ] is π(x)− π(cx).Using the Prime Number Theorem and Mean ValueTheorem:
π(x)− π(cx) = Li(x)− Li(cx) + O(x log−2 x)= Li′(yc)(x − cx) + O(x log−2 x)
with yc(x) ∈ [cx , x ].
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Introduction c-Ramanujan primes Results Distribution Conclusion
Existence of Rc,n
π(x)− π(cx) = (1−c)xlog yc
+ O(x log−2 x).
Since log yc = log x − bc, for bc ∈ [0,− log c],
π(x)− π(cx) =(1− c)x
log x − bc+ O
(x
log2 x
).
For sufficiently large x , π(x)− π(cx) is strictlyincreasing and π(x)− π(cx) ≥ n, for all integers n.
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Introduction c-Ramanujan primes Results Distribution Conclusion
Existence of Rc,n
π(x)− π(cx) = (1−c)xlog yc
+ O(x log−2 x).
Since log yc = log x − bc, for bc ∈ [0,− log c],
π(x)− π(cx) =(1− c)x
log x − bc+ O
(x
log2 x
).
For sufficiently large x , π(x)− π(cx) is strictlyincreasing and π(x)− π(cx) ≥ n, for all integers n.
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Introduction c-Ramanujan primes Results Distribution Conclusion
Existence of Rc,n
π(x)− π(cx) = (1−c)xlog yc
+ O(x log−2 x).
Since log yc = log x − bc, for bc ∈ [0,− log c],
π(x)− π(cx) =(1− c)x
log x − bc+ O
(x
log2 x
).
For sufficiently large x , π(x)− π(cx) is strictlyincreasing and π(x)− π(cx) ≥ n, for all integers n.
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Introduction c-Ramanujan primes Results Distribution Conclusion
Bounds on Rc,n
Rc,n ≥ pn
≥ n log n.
Want Rc,n ≤ αcn log(αcn) for large n.
Conjecture αc = 21−c .
To prove, must showπ(x)− π(cx) ≥ n ∀x ≥ αcn log(αcn).
We then obtain bounds on log Rc,n :
(1− βc log log n
log n
)log n ≤ log Rc,n ≤
(1 +
βc log log nlog n
)log n.
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Introduction c-Ramanujan primes Results Distribution Conclusion
Bounds on Rc,n
Rc,n ≥ pn ≥ n log n.
Want Rc,n ≤ αcn log(αcn) for large n.
Conjecture αc = 21−c .
To prove, must showπ(x)− π(cx) ≥ n ∀x ≥ αcn log(αcn).
We then obtain bounds on log Rc,n :
(1− βc log log n
log n
)log n ≤ log Rc,n ≤
(1 +
βc log log nlog n
)log n.
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Introduction c-Ramanujan primes Results Distribution Conclusion
Bounds on Rc,n
Rc,n ≥ pn ≥ n log n.
Want Rc,n ≤ αcn log(αcn) for large n.
Conjecture αc = 21−c .
To prove, must showπ(x)− π(cx) ≥ n ∀x ≥ αcn log(αcn).
We then obtain bounds on log Rc,n :
(1− βc log log n
log n
)log n ≤ log Rc,n ≤
(1 +
βc log log nlog n
)log n.
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Introduction c-Ramanujan primes Results Distribution Conclusion
Bounds on Rc,n
Rc,n ≥ pn ≥ n log n.
Want Rc,n ≤ αcn log(αcn) for large n.
Conjecture αc = 21−c .
To prove, must showπ(x)− π(cx) ≥ n ∀x ≥ αcn log(αcn).
We then obtain bounds on log Rc,n :
(1− βc log log n
log n
)log n ≤ log Rc,n ≤
(1 +
βc log log nlog n
)log n.
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Introduction c-Ramanujan primes Results Distribution Conclusion
Bounds on Rc,n
Rc,n ≥ pn ≥ n log n.
Want Rc,n ≤ αcn log(αcn) for large n.
Conjecture αc = 21−c .
To prove, must showπ(x)− π(cx) ≥ n ∀x ≥ αcn log(αcn).
We then obtain bounds on log Rc,n :
(1− βc log log n
log n
)log n ≤ log Rc,n ≤
(1 +
βc log log nlog n
)log n.
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Introduction c-Ramanujan primes Results Distribution Conclusion
Bounds on Rc,n
Rc,n ≥ pn ≥ n log n.
Want Rc,n ≤ αcn log(αcn) for large n.
Conjecture αc = 21−c .
To prove, must showπ(x)− π(cx) ≥ n ∀x ≥ αcn log(αcn).
We then obtain bounds on log Rc,n :
(1− βc log log n
log n
)log n ≤ log Rc,n ≤
(1 +
βc log log nlog n
)log n.
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Introduction c-Ramanujan primes Results Distribution Conclusion
Bounds on Rc,n
Rc,n ≥ pn ≥ n log n.
Want Rc,n ≤ αcn log(αcn) for large n.
Conjecture αc = 21−c .
To prove, must showπ(x)− π(cx) ≥ n ∀x ≥ αcn log(αcn).
We then obtain bounds on log Rc,n :
(1− βc log log n
log n
)log n ≤ log Rc,n ≤
(1 +
βc log log nlog n
)log n.
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Introduction c-Ramanujan primes Results Distribution Conclusion
Asymptotic Behavior
Theorem (Amersi, Beckwith, Ronan, 2011)For any fixed c ∈ (0,1), the nth c-Ramanujan prime isasymptotic to the n
1−c th prime as n→∞
By the triangle inequality∣∣∣Rc,n − p n1−c
∣∣∣ ≤∣∣∣∣Rc,n −
n1− c
log Rc,n
∣∣∣∣+ ∣∣∣∣ n1− c
log Rc,n −n
1− clog n
∣∣∣∣+
∣∣∣∣ n1− c
log n − n1− c
logn
1− c
∣∣∣∣+
∣∣∣∣ n1− c
log n − p n1−c
∣∣∣∣
≤ γcn log log n
Since n log log npn
→ 0 as n→∞⇒ Rc,n ∼ p n1−c
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Introduction c-Ramanujan primes Results Distribution Conclusion
Asymptotic Behavior
Theorem (Amersi, Beckwith, Ronan, 2011)For any fixed c ∈ (0,1), the nth c-Ramanujan prime isasymptotic to the n
1−c th prime as n→∞
By the triangle inequality∣∣∣Rc,n − p n1−c
∣∣∣ ≤∣∣∣∣Rc,n −
n1− c
log Rc,n
∣∣∣∣+ ∣∣∣∣ n1− c
log Rc,n −n
1− clog n
∣∣∣∣+
∣∣∣∣ n1− c
log n − n1− c
logn
1− c
∣∣∣∣+
∣∣∣∣ n1− c
log n − p n1−c
∣∣∣∣≤ γcn log log n
Since n log log npn
→ 0 as n→∞⇒ Rc,n ∼ p n1−c
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Introduction c-Ramanujan primes Results Distribution Conclusion
Asymptotic Behavior
Theorem (Amersi, Beckwith, Ronan, 2011)For any fixed c ∈ (0,1), the nth c-Ramanujan prime isasymptotic to the n
1−c th prime as n→∞
By the triangle inequality∣∣∣Rc,n − p n1−c
∣∣∣ ≤∣∣∣∣Rc,n −
n1− c
log Rc,n
∣∣∣∣+ ∣∣∣∣ n1− c
log Rc,n −n
1− clog n
∣∣∣∣+
∣∣∣∣ n1− c
log n − n1− c
logn
1− c
∣∣∣∣+
∣∣∣∣ n1− c
log n − p n1−c
∣∣∣∣≤ γcn log log n
Since n log log npn
→ 0 as n→∞
⇒ Rc,n ∼ p n1−c
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Introduction c-Ramanujan primes Results Distribution Conclusion
Asymptotic Behavior
Theorem (Amersi, Beckwith, Ronan, 2011)For any fixed c ∈ (0,1), the nth c-Ramanujan prime isasymptotic to the n
1−c th prime as n→∞
By the triangle inequality∣∣∣Rc,n − p n1−c
∣∣∣ ≤∣∣∣∣Rc,n −
n1− c
log Rc,n
∣∣∣∣+ ∣∣∣∣ n1− c
log Rc,n −n
1− clog n
∣∣∣∣+
∣∣∣∣ n1− c
log n − n1− c
logn
1− c
∣∣∣∣+
∣∣∣∣ n1− c
log n − p n1−c
∣∣∣∣≤ γcn log log n
Since n log log npn
→ 0 as n→∞⇒ Rc,n ∼ p n1−c
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Introduction c-Ramanujan primes Results Distribution Conclusion
Frequency of c-Ramanujan Primes
Theorem (Amersi, Beckwith, Ronan, 2011)In the limit, the probability of a generic prime being ac-Ramanujan prime is 1− c.
Define N = b n1−c c.
|aN
|pN
|
bN
Worst cases:Rc,n = aN and every prime in (aN ,pN ] is c-Ramanujan,Rc,n = bN and every prime in [pN ,bN) is c-Ramanujan.
Goal: π(bN)−π(aN)π(pN)
→ 0 as N →∞.
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Introduction c-Ramanujan primes Results Distribution Conclusion
Frequency of c-Ramanujan Primes
Theorem (Amersi, Beckwith, Ronan, 2011)In the limit, the probability of a generic prime being ac-Ramanujan prime is 1− c.
Define N = b n1−c c.|
aN|
pN|
bN
Worst cases:Rc,n = aN and every prime in (aN ,pN ] is c-Ramanujan,Rc,n = bN and every prime in [pN ,bN) is c-Ramanujan.
Goal: π(bN)−π(aN)π(pN)
→ 0 as N →∞.
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Introduction c-Ramanujan primes Results Distribution Conclusion
Frequency of c-Ramanujan Primes
Theorem (Amersi, Beckwith, Ronan, 2011)In the limit, the probability of a generic prime being ac-Ramanujan prime is 1− c.
Define N = b n1−c c.|
aN|
pN|
bN
Worst cases:
Rc,n = aN and every prime in (aN ,pN ] is c-Ramanujan,Rc,n = bN and every prime in [pN ,bN) is c-Ramanujan.
Goal: π(bN)−π(aN)π(pN)
→ 0 as N →∞.
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Introduction c-Ramanujan primes Results Distribution Conclusion
Frequency of c-Ramanujan Primes
Theorem (Amersi, Beckwith, Ronan, 2011)In the limit, the probability of a generic prime being ac-Ramanujan prime is 1− c.
Define N = b n1−c c.|
aN|
pN|
bN
Worst cases:Rc,n = aN and every prime in (aN ,pN ] is c-Ramanujan,
Rc,n = bN and every prime in [pN ,bN) is c-Ramanujan.
Goal: π(bN)−π(aN)π(pN)
→ 0 as N →∞.
47
Introduction c-Ramanujan primes Results Distribution Conclusion
Frequency of c-Ramanujan Primes
Theorem (Amersi, Beckwith, Ronan, 2011)In the limit, the probability of a generic prime being ac-Ramanujan prime is 1− c.
Define N = b n1−c c.|
aN|
pN|
bN
Worst cases:Rc,n = aN and every prime in (aN ,pN ] is c-Ramanujan,Rc,n = bN and every prime in [pN ,bN) is c-Ramanujan.
Goal: π(bN)−π(aN)π(pN)
→ 0 as N →∞.
48
Introduction c-Ramanujan primes Results Distribution Conclusion
Frequency of c-Ramanujan Primes
Theorem (Amersi, Beckwith, Ronan, 2011)In the limit, the probability of a generic prime being ac-Ramanujan prime is 1− c.
Define N = b n1−c c.|
aN|
pN|
bN
Worst cases:Rc,n = aN and every prime in (aN ,pN ] is c-Ramanujan,Rc,n = bN and every prime in [pN ,bN) is c-Ramanujan.
Goal: π(bN)−π(aN)π(pN)
→ 0 as N →∞.
49
Introduction c-Ramanujan primes Results Distribution Conclusion
Frequency of c-Ramanujan Primes
1 Rosser’s Theorem:
pm = m log m + O(m log log m) x
2 Riemann Hypothesis:
π(x) = Li(x) + O(√
x log x) x
3 Prime Number Theorem X
⇒ π(bN)− π(aN)
π(pN)≤ ξ
log log Nlog N
→ 0 as N →∞
50
Introduction c-Ramanujan primes Results Distribution Conclusion
Frequency of c-Ramanujan Primes
1 Rosser’s Theorem:
pm = m log m + O(m log log m)
x
2 Riemann Hypothesis:
π(x) = Li(x) + O(√
x log x) x
3 Prime Number Theorem X
⇒ π(bN)− π(aN)
π(pN)≤ ξ
log log Nlog N
→ 0 as N →∞
51
Introduction c-Ramanujan primes Results Distribution Conclusion
Frequency of c-Ramanujan Primes
1 Rosser’s Theorem:
pm = m log m + O(m log log m) x
2 Riemann Hypothesis:
π(x) = Li(x) + O(√
x log x) x
3 Prime Number Theorem X
⇒ π(bN)− π(aN)
π(pN)≤ ξ
log log Nlog N
→ 0 as N →∞
52
Introduction c-Ramanujan primes Results Distribution Conclusion
Frequency of c-Ramanujan Primes
1 Rosser’s Theorem:
pm = m log m + O(m log log m) x
2 Riemann Hypothesis:
π(x) = Li(x) + O(√
x log x) x
3 Prime Number Theorem X
⇒ π(bN)− π(aN)
π(pN)≤ ξ
log log Nlog N
→ 0 as N →∞
53
Introduction c-Ramanujan primes Results Distribution Conclusion
Frequency of c-Ramanujan Primes
1 Rosser’s Theorem:
pm = m log m + O(m log log m) x
2 Riemann Hypothesis:
π(x) = Li(x) + O(√
x log x)
x
3 Prime Number Theorem X
⇒ π(bN)− π(aN)
π(pN)≤ ξ
log log Nlog N
→ 0 as N →∞
54
Introduction c-Ramanujan primes Results Distribution Conclusion
Frequency of c-Ramanujan Primes
1 Rosser’s Theorem:
pm = m log m + O(m log log m) x
2 Riemann Hypothesis:
π(x) = Li(x) + O(√
x log x) x
3 Prime Number Theorem X
⇒ π(bN)− π(aN)
π(pN)≤ ξ
log log Nlog N
→ 0 as N →∞
55
Introduction c-Ramanujan primes Results Distribution Conclusion
Frequency of c-Ramanujan Primes
1 Rosser’s Theorem:
pm = m log m + O(m log log m) x
2 Riemann Hypothesis:
π(x) = Li(x) + O(√
x log x) x
3 Prime Number Theorem
X
⇒ π(bN)− π(aN)
π(pN)≤ ξ
log log Nlog N
→ 0 as N →∞
56
Introduction c-Ramanujan primes Results Distribution Conclusion
Frequency of c-Ramanujan Primes
1 Rosser’s Theorem:
pm = m log m + O(m log log m) x
2 Riemann Hypothesis:
π(x) = Li(x) + O(√
x log x) x
3 Prime Number Theorem X
⇒ π(bN)− π(aN)
π(pN)≤ ξ
log log Nlog N
→ 0 as N →∞
57
Introduction c-Ramanujan primes Results Distribution Conclusion
Frequency of c-Ramanujan Primes
1 Rosser’s Theorem:
pm = m log m + O(m log log m) x
2 Riemann Hypothesis:
π(x) = Li(x) + O(√
x log x) x
3 Prime Number Theorem X
⇒ π(bN)− π(aN)
π(pN)≤ ξ
log log Nlog N
→ 0 as N →∞
58
Introduction c-Ramanujan primes Results Distribution Conclusion
Prime Numbers
2 3 5 7 11 13 1719 23 29 31 37 41 4347 53 59 61 67 71 7379 83 89 97 101 103 107109 113 127 131 137 139 149151 157 163 167 173 179 181191 193 197 199 211 223 227
59
Introduction c-Ramanujan primes Results Distribution Conclusion
Ramanujan Primes
2 3 5 7 11 13 1719 23 29 31 37 41 4347 53 59 61 67 71 7379 83 89 97 101 103 107109 113 127 131 137 139 149151 157 163 167 173 179 181191 193 197 199 211 223 227
60
Introduction c-Ramanujan primes Results Distribution Conclusion
Coin Flipping Model (Variation on Cramer Model)
We defineγ = 0.5772 . . ., the Euler-Mascheroni constant,
P, the probability of Heads,N, the number of trials,LN , the longest run of Heads.
E[LN ] ≈ log Nlog(1/P)
−(
12− log(1− P) + γ
log(1/P)
)Var[LN ] ≈ π2
6 log2 (1/P)+
112
61
Introduction c-Ramanujan primes Results Distribution Conclusion
Coin Flipping Model (Variation on Cramer Model)
We defineγ = 0.5772 . . ., the Euler-Mascheroni constant,P, the probability of Heads,
N, the number of trials,LN , the longest run of Heads.
E[LN ] ≈ log Nlog(1/P)
−(
12− log(1− P) + γ
log(1/P)
)Var[LN ] ≈ π2
6 log2 (1/P)+
112
62
Introduction c-Ramanujan primes Results Distribution Conclusion
Coin Flipping Model (Variation on Cramer Model)
We defineγ = 0.5772 . . ., the Euler-Mascheroni constant,P, the probability of Heads,N, the number of trials,
LN , the longest run of Heads.
E[LN ] ≈ log Nlog(1/P)
−(
12− log(1− P) + γ
log(1/P)
)Var[LN ] ≈ π2
6 log2 (1/P)+
112
63
Introduction c-Ramanujan primes Results Distribution Conclusion
Coin Flipping Model (Variation on Cramer Model)
We defineγ = 0.5772 . . ., the Euler-Mascheroni constant,P, the probability of Heads,N, the number of trials,LN , the longest run of Heads.
E[LN ] ≈ log Nlog(1/P)
−(
12− log(1− P) + γ
log(1/P)
)Var[LN ] ≈ π2
6 log2 (1/P)+
112
64
Introduction c-Ramanujan primes Results Distribution Conclusion
Coin Flipping Model (Variation on Cramer Model)
We defineγ = 0.5772 . . ., the Euler-Mascheroni constant,P, the probability of Heads,N, the number of trials,LN , the longest run of Heads.
E[LN ] ≈ log Nlog(1/P)
−(
12− log(1− P) + γ
log(1/P)
)
Var[LN ] ≈ π2
6 log2 (1/P)+
112
65
Introduction c-Ramanujan primes Results Distribution Conclusion
Coin Flipping Model (Variation on Cramer Model)
We defineγ = 0.5772 . . ., the Euler-Mascheroni constant,P, the probability of Heads,N, the number of trials,LN , the longest run of Heads.
E[LN ] ≈ log Nlog(1/P)
−(
12− log(1− P) + γ
log(1/P)
)Var[LN ] ≈ π2
6 log2 (1/P)+
112
66
Introduction c-Ramanujan primes Results Distribution Conclusion
What is P?
Define Pc as the frequency of c-Ramanujan primesamongst the primes,
As N →∞,Pc = 1− c,For finite intervals [a,b], Pc is a function of a and b,Choose a = 105,b = 106.
67
Introduction c-Ramanujan primes Results Distribution Conclusion
What is P?
Define Pc as the frequency of c-Ramanujan primesamongst the primes,As N →∞,Pc = 1− c,
For finite intervals [a,b], Pc is a function of a and b,Choose a = 105,b = 106.
68
Introduction c-Ramanujan primes Results Distribution Conclusion
What is P?
Define Pc as the frequency of c-Ramanujan primesamongst the primes,As N →∞,Pc = 1− c,For finite intervals [a,b], Pc is a function of a and b,
Choose a = 105,b = 106.
69
Introduction c-Ramanujan primes Results Distribution Conclusion
What is P?
Define Pc as the frequency of c-Ramanujan primesamongst the primes,As N →∞,Pc = 1− c,For finite intervals [a,b], Pc is a function of a and b,Choose a = 105,b = 106.
70
Introduction c-Ramanujan primes Results Distribution Conclusion
Distribution of generalized Ramanujan primes
Length of the longest run in [105,106] ofc-Ramanujan primes Non-c-Ramanujan primes
c Expected Actual Expected Actual
0.50 14 20 16 36
71
Introduction c-Ramanujan primes Results Distribution Conclusion
Distribution of generalized c-Ramanujan primes
Length of the longest run in [105,106] ofc-Ramanujan primes Non-c-Ramanujan primes
c Expected Actual Expected Actual0.10 70 58 5 30.20 38 36 7 70.30 25 25 10 120.40 18 21 13 160.50 14 20 16 360.60 11 17 22 420.70 9 14 30 780.80 7 9 46 1540.90 5 11 91 345
72
Introduction c-Ramanujan primes Results Distribution Conclusion
Open Problems
1 Laishram and Sondow: p2n < Rn < p3n for n > 1.Can we find good choices of ac and bc such thatpacn ≤ Rc,n ≤ pbcn for all n?
2 For a given prime p, for what values of c is p ac-Ramanujan prime?
3 Is there any explanation for the unexpecteddistribution of c-Ramanujan primes amongst theprimes?
73
Introduction c-Ramanujan primes Results Distribution Conclusion
Open Problems
1 Laishram and Sondow: p2n < Rn < p3n for n > 1.Can we find good choices of ac and bc such thatpacn ≤ Rc,n ≤ pbcn for all n?
2 For a given prime p, for what values of c is p ac-Ramanujan prime?
3 Is there any explanation for the unexpecteddistribution of c-Ramanujan primes amongst theprimes?
74
Introduction c-Ramanujan primes Results Distribution Conclusion
Open Problems
1 Laishram and Sondow: p2n < Rn < p3n for n > 1.Can we find good choices of ac and bc such thatpacn ≤ Rc,n ≤ pbcn for all n?
2 For a given prime p, for what values of c is p ac-Ramanujan prime?
3 Is there any explanation for the unexpecteddistribution of c-Ramanujan primes amongst theprimes?
75
Introduction c-Ramanujan primes Results Distribution Conclusion
Acknowledgements
This work was supported by the NSF, Williams College,University College London.
We would like to thank Jonathan Sondow and ourcolleagues from the 2011 REU at Williams College.
76